Universität Duisburg-Essen, Essen


Helmut Harbrecht (University of Basel )

Analytical and numerical methods in shape optimization

Shape optimization is indispensable for designing and constructing industrial components.Many problems that arise in application, particularly in structural mechanics and in theoptimal control of distributed parameter systems, can be formulated as the minimizationof functionals which are dened over a class of admissible domains.The application of gradient based minimization algorithms involves the shape functionals’derivative with respect to the domain under consideration. Such derivatives can analyticallybe computed by means of shape calculus and enable the paradigm rst optimize thendiscretize. Especially, by identifying the sought domain with a parametrization of itsboundary, the solution of the shape optimization problem will be equivalent to solving anonlinear pseudo-differential equation for the unknown parametrization.The present talk aims at surveying on analytical and numerical methods for shape opti-mization. In particular, besides several applications of shape optimization, the followingitems will be addressed:

Ralf Kornhuber (Freie Universität Berlin)

Particles in Membranes

The interplay of curvature and particles diffusing in biological membranes is responsiblefor organizing and shaping the membrane and gives rise to a variety of cellular functions.Hybrid models combining a continuum representation of the membrane with discrete,highly coarse grained descriptions of particles have a long history in physics, while mathe-matical analysis is still in its infancy. We present a hierarchy of variational formulationsof existing hybrid models, where the coupling of particles and membrane is formulatedin terms of linear constraints to the minimization of the Canham–Helfrich energy of themembrane. Utilizing concepts from shape calculus, we derive a numerically feasible rep-resentation of the derivative of the minimal Canham–Helfrich energy for given particlepositions with respect to the particle positions. This representation is applied in numericalinvestigations of the clustering behavior of BAR domains and paves the way to Langevindynamics of particles in membranes.This is joint work of Charles M. Elliott (Warwick), Carsten Gräser (FU Berlin), Tobias Kies (FUBerlin), and Ralf Kornhuber (FU Berlin).

Karl Kunisch (Karl-Franzens Universität Graz & RICAM, Linz)

Monotone and primal-dual algorithms for optimization problems involving lp -likefunctionals with p ∈ [0, 1)

Nonsmooth nonconvex optimization problems involving the lp quasi-norm, p ∈ [0, 1), arethe focus of this talk. Two schemes are presented and analyzed, and their performancein practice is discussed: A monotonically convergent scheme and a primal dual active setscheme. The latter heavily relies on a non-standard formulation of the rst order optimalityconditions. Numerical tests include an optimal control problem, models from fracturemechanics and microscopy image reconstruction. - We also remark innite horizon closedloop optimal control problems with lp cost

PLENARIESMarta Lewicka (University of Pittsburgh )Models for thin prestrained structuresVariational methods have been extensively used in the past decades to rigorously derivenonlinear models in the description of thin elastic lms. They allowed to justify or toimprove several classical models, under the hypothesis that rotations minimize the storedenergy density at any point. On the other hand, natural growth or differential swelling orshrinking might lead to models where an elastic body aims at reaching a space-dependentmetric, which may be not realizable. We will describe the effect of such incompatibleprestrain metrics on the singular limits bidimensional models. We will discuss metrics thatvary across the specimen in both the midplate and the thin (transversal) directions. We willalso cover the case of the oscillatory prestrain and exhibit its relation to the non-oscillatorycase via identifying the effective metric, exhibiting the role of the separate entries of theRiemann curvature tensor of the prestrain.6
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PLENARIESPierre Cardaliaguet (Université Paris-Dauphine)Some aspects of Mean Field GamesMean Field Games (MFG) is a new and challenging mathematical topic which analyzes thedynamics of a very large number of interacting rational agents. Introduced by Lasry andLions around 2005, the MFG now appear in many different frameworks: in macroeconomicmodels, in nance, in crowd motions, in vaccination campaign models, etc. We will rstintroduce the basic MFG models and discuss their meaning and their well-posedness. Wewill also discuss why these MFG models are expected to pop up in practice and explainhow they can appear as limit of models with nitely many agents.7
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PLENARIESCarola-Bibiane Schönlieb (University of Cambridge)Deep and shallow learning approaches for regularised inversion in imagingIn this talk we discuss the idea of data-driven regularisers, investigating two parametrisa-tion: total variation type regularisers and deep neural networks. This talk is based on jointworks with J. C. De Los Reyes, L. Calatroni, C. Chung, T. Valkonen, S. Lunz and O. Oektem.8
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PLENARIESTamás Terlaky (Lehigh University, Bethlehem, PA, USA)A novel approach to discrete truss design problemsDiscrete truss sizing problems are challenging to solve due to their combinatorial, non-linear, and non-convex nature. Consequently, truss sizing problems become unsolvableas the size of the truss grows. In this presentation, we focus on modeling and ecientlysolving discrete truss sizing problems, where the cross-sectional areas of the bars takeonly discrete values. We consider various mathematical formulations with the objectiveto minimize the truss weight. The non-convex Euler buckling constraints and Hooke’slaw are also considered. We propose novel Mixed Integer Linear Optimization (MILO)reformulations of the non-convex models. The resulting MILO models, for large realworld trusses, are not solvable with existing MILO solvers. We propose a novel solutionmethodology to solve the MILO models, and present encouraging computational resultswhich demonstrate the power of the novel computational methodology.Based on joint work with: Mohammad Shahabsafa, Ali Mohammad-Nezhad, Luis ZuluagaDept. of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA SichengHe, John T. Hwang, Joaquim R. R. A. Martins Dept. of Aerospace Engineering, University ofMichigan, Ann Arbor, MI, USA9
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PLENARIESBenedikt Wirth (Universität Münster)Optimal transport metrics for applications in data processingOptimal transport and Wasserstein metrics are becoming increasingly popular tools inmany applied elds. In particular, they provide a robust and intuitive measure of dis-crepancy for histograms and mass distributions, which can for instance be exploited indata clustering, data retrieval, image interpolation, deformation analysis, and many moreproblems. Plain Wasserstein metrics can only compare nonnegative measures of equalmass, which often does not reect the requirements of applications. For this reason severalextensions to so-called unbalanced transport with mass changes have been proposed. Iwill present recent work with Bernhard Schmitzer, in which we discuss and analyse thespecial case of unbalanced Wasserstein-1 transport, which is not only computationallyecient, but in which mass change and mass transport also decouple to an extent thatallows a comprehensive understanding of this discrepancy measure.10
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MS 01: KREUZER, SMEARSMS 01: A POSTERIORI ERROR ESTIMATION AND ADAPTIVITYOrganizers: C. Kreuzer, I. SmearsConvergence of adaptive C0IPG methods for the biharmonic problem.Alexander DominicusTU DortmundC0 interior penalty Galerkin (C0IPG) methods for fourth order elliptic problems benets fromsimple standard Lagrange nite element spaces which are also used for second order problems.Although a posteriori estimators are available in the literature, due to the non-conformity of theansatz spaces, a rigorose convergence analysis of adaptive C0IPG methods for elliptic fourth orderproblems appears to be dicult. In this talk we present a basic convergence analysis for adaptiveC0IPG methods for the biharmonic problem which provides convergence without rates. Based onideas in a recent result from Kreuzer and Georgoulis the theory is accomplished by using severalcompactness properties of (broken) Sobolev spaces and a new limit space of the adaptively creatednon-conforming discrete spaces.Rate optimal adaptivity for non-symmetric/indenite problemsMichael FeischlKarlsruhe Institute of Technology (KIT)We develop a framework which allows us to prove the essential general quasi-orthogonality fornon-symmetric and indenite problems as the stationary Stokes problem or certain transmissionproblems. General quasi-orthogonality is a necessary ingredient of rate optimality proofs and isthe major diculty on the way to prove rate optimal convergence of adaptive algorithms for manystrongly non-symmetric or indenite problems. The proof exploits a new connection between thegeneral quasi-orthogonality and LU-factorization of innite matrices.Numerical approximation of planar oblique derivative problems in nondivergence formDietmar GallistlUniversity of TwenteA numerical method for approximating a uniformly elliptic oblique derivative problem in two-dimensional simply-connected domains is proposed. The numerical scheme employs a mixedformulation with piecewise ane functions on curved nite element domains. The direct approxima-tion of the gradient of the solution turns the oblique derivative boundary condition into an obliquedirection condition. A priori and a posteriori error estimates as well as numerical computations onuniform and adaptive meshes are provided.12
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MS 01: KREUZER, SMEARSA posteriori estimates for optimal control problemsFernando GaspozUniversität StuttgartWe consider nite element solutions to quadratic optimization problems, where the state dependson the control via an elliptic partial differential equation. Exploiting that a suitably reduced optimalitysystem satises a Gårding inequality, we derive a priori and a posteriori error estimates for state,dual and control variables.Instance optimal adaptive Crouzeix-Raviart FEMMira SchedensackUniversität MünsterThis talk considers an adaptive algorithm with a modied maximum marking strategy for the Poissonproblem and the Stokes equations in two dimensions approximated with P1 nonconforming nite el-ements, also named after Crouzeix and Raviart. The proof of three properties for the nonconformingapproximation allows to extend the results of the proof of instance optimality for the conformingapproximation of the Poisson problem to this situation. As a consequence, we obtain instanceoptimality also for the nonconforming approximation.A quasi-optimal Crouzeix-Raviart method for linear elasticityPietro ZanottiTU DortmundWe propose a rst-order Crouzeix-Raviart method for the displacement formulation of the linearelasticity system. The method involves a jump penalization to enforce the validity of a discrete Korn’sinequality and is quasi-optimal in the energy norm on shape regular meshes. In the two-dimensionalcase, we derive a robust error estimate for regular solutions of the problem. We also discuss thebehavior of the quasi-optimality constant in the nearly incompressible regime and connect it, if timepermits, to some recent advances in the discretization of the Stokes problem.13
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MS 02: GÖTSCHEL, SIEBENBORNMS 02: ALGORITHMIC CHALLENGES IN PDE-CONSTRAINED OPTIMIZATIONOrganizers: S. Götschel, M. SiebenbornA low-rank in time approach for the differential Riccati equationTobias BreitenKarl-Franzens-Universität GrazOptimal feedback laws for linear quadratic control problems are intimately connected with thedifferential Riccati equation (DRE). Due to the curse of dimensionality, an explicit computation ofthe time-varying matrix valued unknown is infeasible for problems related to partial differentialequations. A common remedy is to combine low-rank methods with an ODE integration scheme.While this allows to eciently compute the solution at each individual time instance, the temporalcomplexity is still tied to the ODE integration scheme. We propose an alternative procedure that isbased on a full space-time discretization of the DRE. For the resulting nonlinear system, we discuss atensor structured Newton iteration. Based on numerical examples, we evaluate the performance ofthe method for several PDE constrained control problems.Parallel-in-time PDE-constrained optimization using PFASSTSebastian GötschelZuse Institute BerlinFor the solution of optimal control problems governed by parabolic PDEs, methods working on thereduced objective functional are often employed to avoid a full spatio-temporal discretization. Theevaluation of the reduced gradient then requires one solve of the state equation, and one backward-in-time solve of the adjoint equation, making the iterative optimization process computationally verydemanding.One approach to decrease the time-to-solution is to utilize the increasing number of CPU coresavailable in current computers. In addition to more common spatial parallelization, time-parallelmethods are receiving increasing interest in recent years. There, iterative multilevel schemes suchas PFASST (Parallel Full Approximation Scheme in Space and Time) are currently state of the art, andachieve a signicant parallel eciency of more than 50In this talk, we investigate approaches to use PFASST for the solution of parabolic optimal controlproblems. Besides enabling time parallelism, the iterative nature of the temporal integrators withinPFASST provides additional exibility for reducing the cost of solving nonlinear equations, re-usingprevious solutions in the optimization loop, and adapting the accuracy of state and adjoint solves tothe optimization progress. We discuss benets and diculties, and present numerical examples.14
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MS 02: GÖTSCHEL, SIEBENBORNReal-time optimization of thermal ablation cancer treatmentsMartin GreplRWTH Aachen UniversityPercutaneous ablation cancer treatments are performed by inserting a probe directly into or closeto the tumor. The probe generates heat and destroys the cancerous tissue. Such treatments arebecoming increasingly popular due to their potential to be applied to nonresectable tumors, as wellas due to the localized nature of the treatment which minimizes the inicted damage to surroundinghealthy tissue and organs.The problem can be formulated as a parametrized optimal control problem governed by a partialdifferential equation, the Pennes bio-heat equation. Our goal is to improve the accuracy and effec-tiveness of ablation treatments by developing reliable and computationally ecient simulationsand optimization routines, which can be used not only preoperative in the planning phase, butalso in real-time during the treatment. To this end, we employ the reduced basis method as asurrogate model for the solution of the optimal control problem and develop rigorous and ecientlycomputable a posteriori error bounds for both the optimal control and the associated optimal costfunctional value. We present numerical results to conrm the validity of our approach.A non-intrusive parallel-in-Time method for simultaneous optimization with unsteady PDEsStefanie GüntherTU KaiserslauternA non-intrusive framework for reducing the overall runtime of conventional gradient-based optimiza-tion algorithms for unsteady PDE-constrained optimization problems will be presented. The newframework applies non-intrusive multigrid iterations to the time-domain of existing unsteady PDEsolvers as well as unsteady adjoint sensitivity solvers. The multigrid iterations enable time-parallelismsuch that workload can be distributed onto multiple processors along the time domain and speedupover conventional time-serial approaches can be achieved through greater concurrency. Additionally,the parallel-in-time multigrid iterations are embedded into a simultaneous optimization framework,namely the One-shot method which incorporates design updates towards optimality after each stateand adjoint update. Thus, optimality and feasibility of the design and the PDE solution are reachedsimultaneously which further reduces the runtime overhead of the optimization algorithm whencompared to a pure simulation. Due to the non-intrusiveness of the proposed framework, transition-ing from a conventional time-serial optimization algorithm to the time-parallel One-shot methodrequires only minimal additional coding. The benet of the new framework will be demonstratedon an advection-dominated model problem which shows signicant speedup over a conventionaltime-serial optimization algorithm.15
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MS 02: GÖTSCHEL, SIEBENBORNPre-dual and splitting algorithms for TVD image reconstruction on surfacesStephan SchmidtJulius-Maximilians-Universität WürzburgMedical imaging and technical 3D scanning is becoming more and more dominant. Thus, there is anincreased interest in image post-processing and edge preserving denoising on surfaces. This edgepreservation is often achieved by total variation (TV) denoising. Summarizing the above, severalalgorithmic diculties arise. On the one hand, the non-smoothness of the TV-norm needs to betaken into account, while on the other hand, the image on the surface needs to be made accessiblewithin computational methods. To this end, we study the TV-denoising with a splitting type algorithmand an alternative approach based on duality. The image will be represented by nite elements onthe surface, with discontinuous Galerkin or Raviart-Thomas elements for the respective primal ordual approaches.Algorithmic aspects of multigrid methods for optimization in shape spacesMartin SiebenbornUniversität HamburgIn many applications, which are modeled by partial differential equations, there is a small numberof spatially distributed materials or parameters distinguished by interfaces. In order to identifythese parameters, it is often more favorable to treat the shape of the interfaces as a variableinstead of the parameter itself. Since the involved materials may form complex contours, highresolutions are required in the underlying nite element discretizations. The challenge is to combinemethods from PDE constraint shape optimization with HPC techniques and prepare algorithms forsupercomputing.In this talk we discuss the interaction of multigrid methods and shape optimization in appropriateshape spaces. The aim is a scalable algorithm for application on supercomputers, which can only beachieved by mesh-independent convergence. The impact of discrete approximations of geometricalquantities, like the mean curvature, on a multigrid shape optimization algorithm with quasi-Newtonupdates is investigated.16
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MS 03: NEITZEL, WACHSMUTHMS 03: ANALYSIS, STABILITY, AND SENSITIVITY OF OPTIMAL CONTROL PROBLEMSOrganizers: I. Neitzel, D. WachsmuthFull stability for a class of control problems of semilinear elliptic partial differential equa-tionsQui Nguyen ThanhJulius-Maximilians-Universität WürzburgWe investigate full Lipschitzian and full Hölderian stability for a class of control problems governedby semilinear elliptic partial differential equations, where all the cost functional, the state equation,and the admissible control set of the control problems undergo perturbations. We establish explicitcharacterizations of both Lipschitzian and Hölderian full stability for the class of control problems.We show that for this class of control problems the two full stability properties are equivalent. Inparticular, the two properties are always equivalent in general when the admissible control set is anarbitrary xed nonempty, closed, and convex set.Optimal control problems in non-convex domains with regularity constraintJohannes PfeffererTechnical University of MunichThis talk is concerned with a tracking type optimal control problem subject to the Poisson equation.The control enters the problem on the right hand side of the partial differential equation. As specialty,the underlying domain is assumed to be non-convex. In this case, it is well known that the solutionto the Poisson equation, and thus the state of the optimal control problem, does not belong toH2(Ω) in general. The lack of regularity is due to the appearance of singular terms in the solutioncaused by the non-convex corners. However, we are interested in optimal states which neverthelessbelong to H2(Ω). Thus, we are imposing a regularity constraint on the state. For instance, this canbe achieved by considering a closed and convex subset of L2(Ω) as control space which only allowsfor H2(Ω)-regular states. In the present talk, we discuss existence and uniqueness of solutions tosuch problems. Moreover, optimality conditions are presented. At the end of the talk, we also stateone possible approach to discretize with nite elements the problems under considerations, andshow related error estimates.The subject of this talk is inspired by a paper recently submitted by Jarle Sogn and Walter Zulehner.17
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MS 04: GUGAT, TRÖLTZSCHMS 04: CONTROL AND OPTIMIZATION FOR PDE-BASED MODELS AND APPLICATIONSOrganizers: M. Gugat, F. TröltzschHow shape calculus prots from regularity theory: probabilistic lifespan optimizationLaura BittnerBergische Universität WuppertalMechanic devices under cyclic loading are exposed to forces like friction, tension and rotationwhich cause stress states in the component’s material. These states inuence the reliability of thecomponent and can be calculated by the PDE of linear elasticity. Since it is impossible to predictexactly, when and where the components surface will break, it is important to combine a stochasticapproach with the common deterministic lifetime calculation.The resulting objective functional J(Ω, Du(Ω)) determines the failure probability depending on theshape Ω and the 1st (or higher) order derivatives of the displacement u(Ω). We intend to minimizethese failure probabilities by shape calculus methods.In contrast to common cases, the shape functionals that arise from this strategy are usually notdened on H1(Ω, n) and can’t be treated by established methods.We solve this problem by application of regularity theory for elliptic PDE: We show the existence anduniqueness of higher order differentiable solutions u(Ω) and shape derivatives u (Ω). Last but notleast, we prove existence of Euler derivatives DJ(Ω) and shape gradients J(Ω) for a big class ofhighly irregular objective functionals.Turnpike theory for boundary control problems with hyperbolic systemsMartin GugatFriedrich-Alexander-Universität Erlangen-NürnbergOften it is possible to steer the state of a dynamic system rapidly to a static state. If the staticstate is the solution of a static optimal control problem, this leads to approximate solutions of thecorresponding dynamic optimal control problem. In this talk we discuss the relation between thestatic optimal controls and the dynamic optimal controls. Results of this type are called turnpikeresults and are well-known since the pioneering work of John von Neumann. In this talk we considerturnpike theory for problems of optimal boundary control of hyperbolic systems. Although forsystems governed by ordinary differential equations turnpike theory is well-established, see forexample the work of Alexander Zaslavski, in the framework of pde-constrained optimal controlproblems turnpike results have been developped only recently, see for example "Optimal Neumanncontrol for the 1D wave equation: Finite horizon, innite horizon, boundary tracking terms and theturnpike property", Gugat M, Trelat E, Zuazua E, Systems & Control Letters 90, 61-70, 2016.18
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MS 04: GUGAT, TRÖLTZSCHNull-controllability of the heat equation in unbounded domainsIvica NakićUniversity of ZagrebWe adapt a strategy for proving null-controllability of parabolic equations by [Tenenbaum-Tucsnak-2011] to a setting of lower semibounded operators with continuous spectrum. This includes inparticular the case of the controlled heat equation in d where the diffusion term −∆ is replaced bya Schrödinger operator −∆ +V . We lay particular emphasis on explicitly estimating the control costin terms of all occurring model parameters. This is then combined with recent scale-free uniquecontinuation principles for spectral subspaces of Schrödinger operators and the behaviour ot thecontrol cost in certain (homogenization) limits is studied. The talk is based on the joint work withMatthias Täufer, Martin Tautenhahn and Ivan Veselić.Optimal kernels of Volterra-type integral operators in nonlinear parabolic equationsMathieu Pascal RosièreTechnische Universität BerlinIn this talk we consider a class of nonlinear parabolic equations where an additional term corre-sponding to a Volterra-type integral operator is added, whose kernel is used as a control function. Weconsider rst and second order optimality conditions for the optimal control of the aforementionedclass of equations with a quadratic objective functional and we investigate stability properties of theassociated optimality system that are related to the convergence of an SQP method.Optimal control of wine fermentation based on partial and ordinary integro-differentialequationsChristina SchenkCarnegie Mellon UniversityAt present many models based on ordinary differential equations already exist to describe theprocess of wine fermentation. However, the dynamics due to yeast cell growth play an importantrole in this fermentation process. That is why we take a closer look at the mass structure of yeast cellsby introducing a nonlinear partial integro-differential equation for the population balance of yeastand ordinary integro-differential equations for the other substrates such as sugar, nitrogen, oxygenand the product, i.e. ethanol. For this application, a high potential for the conservation of energyexists. Therefore, we study energy-optimal control of the cooling process during wine fermentationwhere the dynamic process is represented by the described system of integro-differential equations.The reaction behavior is modeled based on a novel model including a death phase for yeast andthe inuence of oxygen on the process. The derived model is solved numerically using a nitevolume scheme for the discretization of the mass domain and a simultaneous approach for thetemporal discretization of the resulting system of time dependent ordinary differential equations19
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MS 04: GUGAT, TRÖLTZSCHand the control. Ecient optimization techniques are applied to solve the resulting optimal controlproblem.Stationary gas networks with compressor control and random loads: optimization with prob-abilistic constraintsMichael SchusterFAU Erlangen NürnbergStationary Gas Networks with Compressor Control and Random Loads: Optimization with Probabilis-tic ConstraintsWe introduce a stationary model for gas ow based on semilinear isothermal Euler equations ina non-cycled pipeline network. Especially the problem of the feasibility of a random load vector isanalyzed. Feasibility in this context means the existence of a ow vector meeting these loads, whichsatises the physical conservation laws with box constraints for the pressure.An important aspect of the model is the support of compressor stations, which counteract thepressure loss caused by friction in the pipes. The network is assumed to have only one inux node,all other nodes are eux nodes.With these assumptions the set of feasible loads can be characterized analytically. In additionwe show the existence of optimal solutions for some optimization problems with probabilisticconstraints. A numerical example based on real data completes this presentation.The spatial Ramsey model with endogenous productivity growth – an application of nonlocalPDE-constrained optimization in economicsLaura SomorowskyIAAEU & Trier UniversityThe Ramsey model is one of the most popular neoclassical growth models in economics. The primarytime-depending model has been extended by a spatial component in the last few years, meaningthat capital accumulation is modeled as a process not only in time but in space as well. In a newapproach, we consider a Ramsey economy where the value of the capital stock depends not onlyon the respective location but is inuenced by the auence of the surrounding areas as well. Weexpand the common spatial Ramsey model by a nonlocal diffusion term. Moreover, we introducea nonlocal and nonlinear production-productivity operator in order to endogenize the increase ofproductivity in a location, depending on time and the capital stock in this location and the auenceof the surrounding areas. We consider the resulting optimal control problem under partial integro-differential constraints with a bounded and unbounded spatial domain. We analyze the nonlinearpartial integro-differential equation and show the existence of weak solutions. Furthermore, we giveconditions under which we can prove the existence of an optimal control. Finally, we discuss thenumerical results and interpret them in our new economical setting.20
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MS 04: GUGAT, TRÖLTZSCHOptimization of time delays in Pyragas type feedback controlFredi TröltzschTechnische Universität BerlinWe consider a semilinear parabolic delay differential equation with nitely many time delays. Thedelays dene a Pyragas type feedback operator and should be optimized such that the associatedsolution of the partial delay differential equation minimizes the L2-distance to a desired state. Mainemphasis is laid on the differentiability of the mapping that associates the solution of the delayequation to the vector of time delays. We improve a result on local differentiability proved by J. K.Hale and L.A.C. Ladeira (1993) for a short time horizon. We are able to prove this for any nite timehorizon. Based on our differentiability result, rst-order necessary optimality conditions are derivedfor the optimal vector of delays. Moreover, we present numerical examples.21
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MS 05: BREITEN, PFEIFFERMS 05: FEEDBACK CONTROL OF NONLINEAR SYSTEMSOrganizers: T. Breiten, L. PfeifferOptimal control of a 3D heat conductorJulian AndrejUniversität KielWe present an optimal control approach for the linear heat equation in a three dimensional setting,where a model with multiple input and measurement points is considered. Based on a high ordernite element discretization we formulate a transient optimal control problem and compute optimaltrajectories to reach a desired temperature prole in a given time. The gradient for the optimizationproblem is computed using the discrete adjoint method of the forward equation.Using a similar concept we identify parameters of a physical experiment and test the results of thefeed-forward trajectory planning to verify the outcome.Stabilizing ow problems using state-dependent Riccati equationsPeter BennerMax Planck Institute for Dynamics of Complex Technical SystemsStabilizing nonlinear systems using the Linearization Principle and Feedback Control has been asuccessful concept for decades. We will focus here on incompressible ow problems described bythe unsteady Navier-Stokes equations as the considered nonlinear system, and on Riccati-basedfeedback control. As the setpoint may vary during a ow simulation due to perturbations exceedingthe limits of the Linearization Principle, a single Riccati gain may not suce to keep the owprole in a laminar regime. This leads then to so-called state-dependent Riccati equations, wherethe coecients depend on the current state. We introduce a scheme that avoids to solve a newRiccati equation every time the coecients change by using incremental updates that only requirelinear system solves. In closed-form, this scheme leads to a nonlinear feedback controller forimcompressible ow problems.Stabilizing controllers with optimized performance for nonlinear PDEsDante KaliseImperial College LondonWe provide a general framework to ensure stability of a Nonlinear Model Predictive Control (NMPC)scheme by an instantaneous control applied to several linear and nonlinear parabolic partial differ-ential equations (PDEs). In general, stability of an NMPC scheme without terminal constraints mayresult in a long prediction horizon for stabilization. The knowledge of a suboptimal explicit stabilizing22
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MS 05: BREITEN, PFEIFFERfeedback may lead to stability of the NMPC loop with the shortest possible receding horizon. Weillustrate our approach with unstable semilinear parabolic PDEs.Galerkin approximations for nonlinear optimal control problems and feedback controlAxel KroenerHumboldt-Universität zu BerlinIn this talk we formulate a general framework to derive error estimates for the approximation ofthe value function of nonlinear optimal control problems in Hilbert spaces based on a Galerkinapproximation. The framework is applied to optimal control problems of differential delay equations.The numerical approximation of the control in feedback form based on HJB equations is alsoconsidered.Global stabilization of Burgers’ equation by nonlinear Neumann boundary feedback controland its nite element analysisSudeep KunduUniversity of GrazIn this talk, global stabilization results for the Burgers’ equation are established using nonlinearNeumann boundary feedback control law. Then, using C0-conforming nite element method, globalstabilization results for the semidiscrete solution are shown. Moreover, optimal error estimates forthe state variable in L∞(L2), L∞(H1) and L∞(L∞)-norms are obtained. Further, superconvergenceresult is derived for the boundary feedback control law. All the results preserve exponential stabi-lization property. Finally, some numerical experiments are conducted to conrm our theoreticalndings.POD-Based economic model predictive control for heat-convection phenomenaLuca MechelliUniversität KonstanzWe consider an optimal boundary control problem subjected to linear time-dependent convection-diffusion (CD) equation together with bilateral control and pointwise state constraints. Due to thepointwise state constraints, we perform a Lavrentiev regularization and the regularized optimalcontrol problem is solved by a primal-dual active set strategy (PDASS). To speed up the PDASSa reduced-order approach based on proper orthogonal decomposition (POD) is applied and ana-posterori error analysis ensures that the computed (suboptimal) POD solutions are sucientlyaccurate. An Economic Model Predictive Control strategy (EMPC) is considered to treat the long-time horizon and the problem’s parameters’ changes. To improve the model, we combine CD withtime-dependent Navier-Stokes equations in the so-called Boussinesq Approximation (BA). Sincecomputing the solution for the optimal control problem subjected to BA can be costly, we solve23
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MS 05: BREITEN, PFEIFFERBA with an arbitrary control to generate the POD basis for approximating CD, then we apply thePOD-based EMPC algorithm to compute the optimal control for CD. According to the a-posteriorierror estimator, if the POD approximation is not good anymore, we solve again BA with the optimalfeedback control previously computed to generate new POD basis, afterwards we continue to applythe POD-based EMPC algorithm for CD.Optimization based feedback control for time varying systemsSimon PirkelmannUniversität BayreuthIn this talk we present a model predictive control (MPC) scheme for time-varying systems. In MPC afeedback law is synthesized from the successive solution of open-loop optimal control problems.We present conditions under which the resulting solution approximates the solution of a problemon innite horizon. A particular focus is placed on the convergence of the MPC closed-loop trajectorytowards an innite horizon optimal trajectory. The corresponding innite horizon optimality criterionis given by the concept of overtaking optimality.To illustrate the theoretic results we consider a convection-diffusion equation as a model for energyecient heating, cooling and ventilation.Explicit exponential feedback stabilization to trajectories for parabolic equationsSergio RodriguesRICAMAn explicit feedback controller is proposed for stabilization of linear parabolic equations, with a time-dependent reaction-convection operator. The range of the feedback controller is nite-dimensional,and its dimension depends polynomially on a suitable norm of the reaction-convection operator.A sucient condition for stabilizability is given, which involves the asymptotic behavior of theeigenvalues of the (time-independent) diffusion operator, the norm of the reaction-convectionoperator, and the norm of the nonorthogonal projection onto the controller’s range along a suitableinnite dimensional (higher-modes) eigenspace. To construct the explicit feedback, the essentialstep consists in computing the nonorthogonal projection. Numerical simulations are presented,in 1D and 2D, showing the practicability of the controller and its response to measurement errors.Under general conditions on the nonlinearity, the same feedback law is able to locally stabilize asemilinear parabolic system to a given (time-dependent) trajectory.24
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MS 06: LASIECKA, WEBSTERMS 06: HONORING THE WORK OF IGOR CHUESHOVOrganizers: I. Lasiecka, J. WebsterWellposedness and qualitative analysis of ow-structure PDE modelsGeorge AvalosUniversity of Nebraska-LincolnIn this talk, we provide results of wellposedness and long time behavior for solutions of compressibleand incompressible ow-structure partial differential equation (PDE) models which have recentlyappeared in the literature, and which have been studied and/or derived outright by Igor Chueshov.For example, we will consider a compressible ow PDE and its associated state equation for thepressure variable – each evolving within a three dimensional domain O – which are coupled to afourth order plate equation which holds on a at portion Ω of the boundary ∂O. Moreover, since thiscoupled PDE model is the result of a linearization of the compressible Navier-Stokes equations aboutan arbitrary state, the ow system component contains terms which involve a nonzero ambient owprole U. In consequence, this ow-structure PDE system will generally not be dissipative. This talkwill represent joint work with Pelin Guven Geredeli (University of Nebraska-Lincoln).Regularity analysis for the Moore-Gibson-Thompson equationFrancesca BucciUniversita’ degli Studi di FirenzeIn this talk we will report recent results concerning the regularity of solutions to initial/boundaryvalue problems for the Moore-Gibson-Thompson equation (MGT) equation, that is a linearizationof a Partial Differential Equation (PDE) model for ultrasonic wave propagation. The embeddingof the MGT equation in a class of (linear) wave equations with memory, along with the theory ofintegro-differential Volterra equations, provides a perspective and method of proof for the derivationof interior as well as trace regularity estimates of its solutions.(The talk is based on ongoing joint work with Luciano Pandol (Politecnico di Torino, Italy).)On the Moore-Gibson-Thompson equation and its memory relaxationFilippo Dell’OroPolitecnico di MilanoWe discuss the parallel between the third-order Moore-Gibson-Thompson (MGT) equation arisingin acoustics and the equation of linear viscoelasticity with an exponential kernel. We also considerthe MGT equation with memory, a model which accounts for additional nonlocal effects due tomolecular relaxation, and we investigate its asymptotic properties in the critical regime.25
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MS 06: LASIECKA, WEBSTEROn the boundary controllability of hyperbolic systemsMatthias EllerGeorgetown UniversityWalter Littman’s approach to boundary controllability in the context of hyperbolic systems of partialdifferential equations will be reviewed. In the case of constant coecients a fairly general resultis presented, even in the case of characteristics of variable multiplicity. Its proof is based on theknowledge of singularities of the fundamental solutions and on Holmgren’s uniqueness theorem.Extensions of this result to the variable coecient case will be discussed.Pullback and uniform attractors for non-autonomous uid-structure interaction systems.Tamara FastovskaV.N. Karazin Kharkiv National University/ Kharkiv National Automobile and Highway UniversityWe consider a non-autonomous uid-structure interaction system for the uid velocity eld v =v(x, t) = (v1(x, t);v2(x, t);v3(x, t)), the pressure p(x, t), and the transversal displacement of theplateε(t)vt − ν∆v + p = f (x, t) in O×(τ, +∞),divv = 0 in O×(τ, +∞),δ(t)utt + ∆2u + F (u) = p|Ω + g(x, t) in Ω × (τ, ∞)(1.1)for any τ ∈ R. Here ν > 0, O ⊂ R3 is a bounded domain representing a vessel led with viscousincompressible uid with a smooth boundary ∂O = Ω ∪ S, where Ω ∩ S = 0. The at domainΩ represents an elastic plate while S is a rigid wall. Equations (1.1) are supplemented with theboundaryv = 0 on S; v ≡ (v1;v2;v3) = (0; 0;ut ) on Ω,u|∂Ω = ∂u∂n∂Ω= 0.(1.2)and initial conditionsv(x,τ) = vτ(x), u(x,τ) = u0τ (x), ut (x,τ) = u1τ (x).(1.3)We prove that problem (1.1)–(1.3) generates a process on a scale of spaces possessing a pullbackattractor, in particular if limt→∞ε(t) = 0 and limt→∞δ(t) = 0, under appropriate conditions on theparameters of the problem. Moreover, we show that the fractal dimension of the kernel sections ofthe process is nite. In case ε, δ = const we prove the existence of a nite-dimensional uniformattractor.26
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MS 06: LASIECKA, WEBSTEROn the asymptotic behavior of stochastically driven compressible uid owsEduard FeireislCzech Academy of SciencesWe consider the compressible Navier-Stokes system driven by a stochastic forcing. We discuss theasymptotic behavior of such a system for large time, the existence of stationary and/or time periodicsolutionsDecay properties of compressible fuid structure PDE modelsPelin Güven GeredeliHacettepe UniversityIn this talk, we present recently derived results of uniform stability for a coupled partial differentialequation (PDE) system which models a compressible uid-structure interaction of current interestwithin the mathematical literature. The coupled PDE model under discussion will involve a linearizedcompressible, viscous uid ow evolving within a 3-D cavity, and a linear elastic plate–in the absenceof rotational inertia—which evolves on a portion of the uid cavity wall. Since the uid equationcomponent is the result of a careful linearization of the compressible Navier-Stokes equations aboutan arbitrary state, this interactive PDE component will include a nontrivial ambient ow prole, whichtends to complicate the analysis. Moreover, there is an additional coupling PDE which determinesthe associated pressure variable of the uid-structure system. Under a suitable assumption on theambient vector eld, and by obtaining an appropriate estimate for the associated uid-structuregenerator on the imaginary axis, we provide a result of exponential stability for nite energy solutionsof the uid-structure PDE system.Weak-strong uniqueness in uid structure interaction problemSarka NecasovaCzech Academy of SciencesThe relative energy inequality was introduced by Dafermos [D].Deriving the relative energy inequalityfor sucient smooth test functions see [FJN] and proving the weak-strong uniqueness it gives usvery powerful and elegant tool for the purpose of measuring the stability of a solution compared toanother which has a better regularity. The aim of the lecture is show the weak-strong uniquenessin the case of motion of rigid body in a bounded domain lled by incompressible uid with mixedboundary conditions, see [CNM]. References: [D] Dafermos C. M., The second law of thermodynamicsand stability, Arch. Rational Mech. Anal., 70, 167-179, 1979. [FJN] Feireisl E., Jin, B. J., Novotný, A.,Relative Entropies, Suitable Weak Solutions and Weak-Strong Uniqueness for the CompressibleNavier-Stokes System, J. of Math. Fluid Mech., 14 ,717-730, 2012. [CNM] Chemetov, N., Necasova,S., Muha, B., Weak-strong uniqueness for uid-rigid body interaction problem with slip boundarycondition, arxiv27
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MS 06: LASIECKA, WEBSTERWell-posedness and asymptotic properties of solutions to nonlinear PDEs and ODEs in thepresence of state-dependent delayAlexander RezounenkoKharkiv University & AS CRWe discuss a general class of non-linear partial differential equations with general type of boundedtime delays. We focus on the state-dependent type of delays since the type is the most relevant toreal-world applications. Particularly, we are interested in reaction-diffusion equations and systems(in bounded domains) with delays in reaction terms. We investigate the well-posedness in the senseof Hadamard as well as long time asymptotic behaviour of different types of solutions. The Lyapunovstability and the existence of global attractors are discussed. A recent study connected to viral in-hostinfection models with state-dependent delay is also described. The last includes PDE/ODE infectionmodels with /without CTL and antibody immune responses. This talk is dedicated to the memory ofIgor D. Chueshov.Quasistability method for study of uniform attractors of non-autonomous equationsIryna Ryzhkova-GerasymovaV. N. Karazin Kharkiv national universityWe generalize quasistability method for using in study of long-time behaviour of non-autonomousequations with translation compact symbols. As a model problem we consider nonlinear waveequationutt − ∆u + α(t)ut + β(t)g(u) = f (t, x).The time-depending coecient (time symbol) is called translation compact if the completion of{α(t + s), s ∈ } (time symbol space) is compact in an appropriate functional space (e.g. Cb ()).We modify quasistability inequality to account for time symbol, so that it’s essential consequenceswhich hold for autonomous equations take place for non-autonomous equations also. We establishasymptotical compactness and improved smoothness of individual trajectories. In the case whentime symbol space is of nite fractal or Hausdorff dimension, we nd the condition under which auniform attractor has nite fractal dimension.Homogenisation with error estimates of attractors for damped semi-linear anisotropic waveequationsAnton SavostianovDurham UnviersityThe talk is devoted to homogenisation of global and exponential attractors for the damped semi-linear anisotropic wave equation on a bounded 3d domain. First we show how to obtain order-sharp28
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MS 06: LASIECKA, WEBSTERestimates, in a suitable norm weaker than the standard energy norm, between trajectories of the os-cillating system and the homogenised one. These estimates are given in terms of the operator-normdifference between resolvents of the corresponding elliptic operators. Based on this, we derivenorm-resolvent estimates on the Hausdorff distance between the anisotropic (global and exponen-tial) attractors and their homogenised counter-parts. Furthermore, we obtain rst-order correctionfor the homogenised (global and exponential) attractors suggested by asymptotic expansions. Thecorrected homogenised attractors, as expected, are close to the anisotropic attractors already inthe strong energy norm. The corresponding quantitative estimates on the Hausdorff distance withrespect to the energy norm are also obtained. Our results are applied to Dirchlet, Neumann andperiodic boundary conditions.Thresholds for hanger slackening and cable shortening in the Melan equation for suspensionbridgesGianmarco SperonePolitecnico di MilanoThe Melan equation for suspension bridges is derived by assuming small displacements of the deckand inextensible hangers. We determine the thresholds for the validity of the Melan equation whenthe hangers slacken, thereby violating the inextensibility assumption. To this end, we preliminarilystudy the possible shortening of the cables: it turns out that there is a striking difference betweeneven and odd vibrating modes since the former never shorten the cables. These problems arestudied both on beams and plates.Genetic algorithm in uid-structure interactions arising in coupling of elasticity with Navier-Stokes equationKatarzyna SzulcPolish Academy of SciencesWe consider a coupled system of the linearly elastic body immersed in the owing uid which ismodeled by means of incompressible Navier-Stokes equations. For this system we formulate anoptimization problem which amounts to a minimization of a hydro-elastic pressure on the interfacebetween the two environments. The corresponding functional lacks convexity and radial coercivity.The approach taken is based on locating small holes in the elastic domain. These locations areapproximated by the genetic algorithm which uses the probability density in random selection forthe initial population of single holes, pairs or triples, and also to supplement the population inconsecutive generations. The probability density is evaluated based on the values of the topologicalderivative calculated in the elastic subdomain for a given shape functional dened in the exteriorsubdomain lled by uid.29
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MS 06: LASIECKA, WEBSTEROn the periodic Ornstein-Uhlenbeck processPierre VuillermotUniversidade de LisboaThe periodic Ornstein-Uhlenbeck process wandering in Euclidean space is one of the simpleststationary and non-Markovian Gaussian process. In this talk we show that this process may be viewedas a particular Bernstein process which may be associated with a very specic innite hierarchy offorward-backward systems of decoupled linear deterministic parabolic partial differential equations.Crucial to the denition of this hierarchy is the fact that the spectrum of the so-called Hamiltonianof an isotropic system of quantum harmonic oscillators is an explicitly known pure-point spectrum.The particular Bernstein process we are interested in is generated by a weighted average of signedmeasures naturally associated with the spectrum in question, and indeed turns out to be identical inlaw with the periodic Ornstein-Uhlenbeck process.Inertial manifolds for dissipative PDEs: examples and counterexamplesSergey ZelikUniversity of SurreyWe will discuss the problem of existence or non-existence of inertial manifolds (IMs) for semilinearparabolic equations. This will include recent results concerning IMs for Cahn-Hilliard and 1D reaction-diffusion-advection problems as well as the explicit examples where the such manifolds do notexist.30
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MS 07: PIETSCHMANN, WOLFRAMMS 07: INVERSE PROBLEMS AND OPTIMAL CONTROL APPROACHES IN SOCIO-ECONOMIC AP-PLICATIONSOrganizers: J.-F. Pietschmann, M.-T. WolframBayesian parameter estimation for macroscopic pedestrian dynamics modelsSusana GomesImperial College LondonThe fundamental diagram of pedestrian dynamics relates the experimentally observed density ofpedestrians to their average velocity. Although there is a general agreement on its basic shape, itsparametrization depends strongly on the measurement and averaging techniques used as well as theexperimental setup considered. We aim at developing a systematic approach to identify parametersin nonlinear macroscopic crowd motion models using multiple microscopic trajectories. We assumethat each trajectory is a realization of an Ito-McKean process, where the individual velocity dependson the probability density of the process. The probability density satises a nonlinear Fokker-Planckequation itself, leading to a coupling between the microscopic SDE and the macroscopic PDE.Motivated by the fundamental diagram we assume that individuals move with a maximum velocity,which decreases linearly as the probability density approaches the maximum crowd density. Weare interested in identifying the maximum velocity and the maximum crowd density using multipletrajectories. We discuss Bayesian as well as other derivative free optimization methods to estimatethese parameters and present analytic as well as numerical results, which give important insightsinto the dynamics and challenges of this highly nonlinear inverse problem.Optimal inow control of dynamical systems with uncertain demandsSimone GöttlichUniversity of MannheimThe control problem under consideration consits of a deterministic hyperbolic differential equationused to describe the dynamics of a supply systems (e.g. production, energy) coupled to a stochasticdifferential equation to model uncertain demands. We are concerned with optimal control strategiesto nd the optimal system inow such that the stochastic demands are satised. Solution techniquesare based on ecient reformulations of the original stochastic control problem and allow for astraightforward numerical treatment.31
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MS 07: PIETSCHMANN, WOLFRAMProximal methods for Mean Field Games with local couplingsDante KaliseImperial College LondonWe address the numerical approximation of Mean Field Games with local couplings. For power-likeHamiltonians, we consider both unconstrained and constrained stationary systems with densityconstraints in order to model hard congestion effects. For nite difference discretizations of theMean Field Game system, we follow a variational approach. We prove that the aforementionedschemes can be obtained as the optimality system of suitably dened optimization problems. Inorder to prove the existence of solutions of the scheme with a variational argument, the monotonicityof the coupling term is not used, which allow us to recover general existence results. Next, assumingnext that the coupling term is monotone, the variational problem is cast as a convex optimizationproblem for which we study and compare several proximal type methods. These algorithms haveseveral interesting features, such as global convergence and stability with respect to the viscosityparameter, which can eventually be zero. We assess the performance of the methods via numericalexperiments.On the uniqueness of nonlinear diffusion coecientsMatthias SchlottbomUniversity of TwenteWe consider the identication of nonlinear diffusion coecients of the form a(t,u) or a(u) in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established undervery general assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof ofour main result relies on the construction of a series of appropriate Dirichlet data and test functionswith a particular singular behavior at the boundary. This allows us to localize the analysis and toseparate the principal part of the equation from the remaining terms. We therefore do not requirespecic knowledge of lower order terms or initial data which allows to apply our results to a varietyof applications. If time permits, we will discuss some typical examples.Relaxation techniques for PDE-constrained optimizationTristan van LeeuwenUtrecht UniversityPDE-constrained optimization problems arise in many applications, including inverse problemsand optimal control. As optimization over both the control and state parameters is not feasible forlarge-scale problems, one often resorts to a reduced formulation by eliminating the constraints. Theresulting optimization problem is often highly non-linear, which may cause local descent methodsto stall at stationary points away from the global minimizer. Another issue is that it may not evenbe possible to eliminate the constraints as the PDE may be ill-posed (e.g., due to missing boundary32
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MS 07: PIETSCHMANN, WOLFRAMconditions). In this talk, I will discuss ways to relax the constraints and reduce the problem implicitly.The resulting reduced optimization problem can in some case be much less non-linear than theoriginal reduced problem. I will illustrate the approach with a variety of numerical examples.Boltzmann games in heterogeneous consensus dynamicsMattia ZanellaPolitecnico di TorinoWe consider a constrained hierarchical opinion dynamics in the case of leaders’ competition andwith complete information among leaders. Each leaders’ group tries to drive the followers’ opiniontowards a desired state accordingly to a specic strategy. By using the Boltzmann–type controlapproach we analyze the best–reply strategy for each leaders’ population. Derivation of the cor-responding Fokker-Planck model permits to investigate the asymptotic behaviour of the solution.Heterogeneous followers populations are then considered where the effect of knowledge impactsthe leaders’ credibility and modies the outcome of the leaders’ competition.33
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MS 08: AVDONIN, MAKSIMOVMS08: INVERSION, ESTIMATION AND CONTROL OF UNCERTAIN DISTRIBUTED DYNAMICALSYSTEMSOrganizers: S. Avdonin, V. MaksimovControl of time-delay stochastic systems with uncertaintiesBoris AnanyevIMM UB of RASConsider a linear controlled uncertain time-delay system with stochastic disturbances and a mea-sured output. The noises in the state and output systems are independent and their matrices containtime-varying parametric uncertainties. The initial condition is assumed to be a zero-mean Gaussianrandom vector along with some initial function. The procedure of control with nal quadratic costis divided into two parts. For the rst one the control equals zero. The rst part consists of robustlter design which ends at some random stopping-time τ. We employ hereafter a Riccati equationapproach to solve the robust Kalman ltering for time-delay systems. The obtained two-parametricestimator is independent of the delay factor and it reduces to the standard Kalman ltering algorithmin the case of systems without uncertainties and delay. After the rst stage we pass to the secondone, where the system cannot be observed and the separation principle is used in order to obtain aminimax control on [τ,T ] and a value of the cost. At last we solve an optimal stopping-time problemin order to nd optimal τ and thereby minimize the maximal value of the cost. The work is supportedby the Russian Science Foundation, project no. 16-11-10146.State estimation problem for impulsive control system under uncertaintyTatiana FilippovaKrasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sci-encesThe nonlinear dynamical control system with uncertainty in initial states and parameters is studied.It is assumed that the dynamic system has a special structure in which the system nonlinearity is dueto the presence of quadratic forms in the system velocities. The case of combined controls is studiedhere when both classical measurable control functions and also the controls generated by vectormeasures are allowed. Here we present several theoretical schemes and the related estimatingalgorithms allowing to nd the upper bounds for reachable sets of the studied control system. Inour research we use and further develop the basic results and techniques of the ellipsoidal calculusand of the theory of evolution equations for set-valued states of dynamical systems and of relateddifferential inclusions having in their description the uncertainty of set-membership kind. So weenlarge the class of nonlinear control systems for which it is possible to nd the upper estimatesof their reachable sets. Numerical results of system modeling based on the proposed methodsare also included. The research was supported by the Russian Science Foundation (RSF ProjectNo.16-11-10146).34
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MS 08: AVDONIN, MAKSIMOVSome problems of feedback control of the distributed systemsVyacheslav MaksimovKrasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sci-encesIn the recent years, a part of mathematical control theory, namely, the theory of control for dis-tributed systems, has been intensively developed. There exists a number of monographs devotedto control problems for distributed systems . In these works, the emphasis is on the problems ofopen-loop or feedback control in the case when all system’s parameters are precisely specied.But, the investigation of control problems for systems with uncontrollable disturbances (game orrobust control problems) is also natural. Similar problems have been insuciently investigated;in our opinion, this is connected with the fact that the well-known Pontryagin maximum principleis not really suitable for solving such problems. In the early 1970es, N.N. Krasovskii suggested aneffective approach to solving guaranteed control problems. This approach is based on the formalismof positional strategies. The goal of this report is to demonstrate the essence and abilities of thisapproach. Toward this aim, we investigate the problem of tracking an uncontrolled input (a trajec-tory) by means of feedback laws formalized in the form of positional strategy. In the process, weconsider “classical” linear parabolic and hyperbolic equations in the case when the information ontheir solutions forth comes (with error) at discrete times.35
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MS 09: BOTKIN, TUROVAMS 09: MODELLING AND CONTROL IN MECHANICAL AND BIOMEDICAL SYSTEMSOrganizers: N. Botkin, V. TurovaQuick creation of dangerous disturbances for ight systemsNikolai BotkinTechnical University of MunichThis paper addresses a method of constructing repulsive disturbances in linear differential gamesusing a dynamic programming approach. The method constructs, in reverse time, a “repulsive”sequence of polyhedrons such that the enemy player can keep the state vector outside of thesepolyhedrons if the initial position lies outside of the polyhedron corresponding to the initial timeinstant. The polyhedrons are stored as unordered systems of linear inequalities, and the compu-tations involve solving a large number of linear programming problems. For this purpose, a fastalgorithm for low-dimensional linear programs is used. Several examples of computing dangerousdisturbances are presented. This includes a simple linear differential game to demonstrate the mainfeatures of the method. Moreover, a linearized model of the longitudinal aircraft motion containingaircraft’s dynamics, servomechanisms, pilot and disturbance inputs, and a controller is considered.Finally, a nonlinear model of aircraft takeoff in windshear conditions is addressed. Simulationsshowing the eciency of the method are presented.This work was supported by the DFG grant TU427/2-1 and HO4190/8-1. Computer resources forthis project have been provided by the Gauss Centre for Supercomputing/Leibniz SupercomputingCentre under grant: pr74lu.Leadership kernels and trajectory control in the presence of windshearNikolai BotkinTechnical University of MunichThis paper addresses problems of aircraft control in the presence of wind disturbances. It is assumedthat the state variables of the model are constrained to dene an appropriate ight domain (AFD).The dynamics of the aircraft is considered as a differential game where the rst player (pilot) appliescontrol inputs, and the second player (wind) produces the worst wind gusts. The maximal viablesubset, viability kernel, of the AFD is computed. This yields a feedback control that keeps trajectoriesin the viability kernel, and hence in the AFD, for all admissible disturbances. It should be emphasizedthat the notion of viability kernel is typical for control systems. In the case of differential games, theterms leadership kernel is more suitable. The leadership kernel assumes that the second player(wind) knows current controls of the pilot and uses feedback counter strategies, which is reasonablein context of computing guaranteeing controls.36
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MS 09: BOTKIN, TUROVAThe paper presents theoretical and numerical aspects of nding leadership kernels for near-to-realistic aircraft models. Nontrivial examples are demonstrated.This work was supported by the DFG grant TU427/2-1 and HO4190/8-1. Computer resources forthis project have been provided by the Gauss Centre for Supercomputing/Leibniz SupercomputingCentre under grant: pr74lu.Robust trajectory controller and its implementation on a ight simulatorJohannes DiepolderTU Munich - Campus GarchingWe present an approach for the implementation of a robust trajectory control structure based onviability kernel in a realistic ight simulator. This viability kernel is obtained from solving a stateconstrained differential game between the aircraft control (rst player) and a wind disturbance(second player). From the viability kernel solution a state feedback law can be derived which ensuresthat despite the wind disturbance, the aircraft does not leave the save set of states. For the numericalsolution of the state constrained differential game the associated value function is approximatedby a grid function. Due to the curse of dimensionality when using this grid approximation thenumber of states to be used is restricted (so far we were able to obtain solutions for up to sevenstates). Therefore, we compute the viability kernel for a lower dimensional model including thetranslational dynamics and a reference model for the attitude dynamics. The implementation on theight simulator is then based on a nonlinear dynamic inversion control structure with referencemodel following control using the same reference model as for the solution of the differentialgame.Continuum model of blood circulation in brainAndrei KovtaniukTechnical University of MunichNumerical modeling of blood circulation in brain is very important for the simulation of possibleinuring effects such as critical drop of oxygen in brain tissue or abnormal local pressure concentra-tion. Very often, the capillary network of the brain is considered as a homogenized medium. Thus adomain containing a large number of holes that simulate the ends of arterioles and venules, playingthe role of blood inlets and outlets, is considered. The pressure distribution is described by a Poissonequation with the prescribed pressures on the boundaries of holes, which leads to the considerationof the problem in a ne perforated domain. Instead of that, we propose to introduce a linear combi-nation of source terms into the right-hand side of the Poisson equation and to t the correspondingcoecients in such a way that the boundary conditions would be well approximated. We rigorouslyformulate this problem and prove its unique solvability. An algorithm for nding the unknown source37
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MS 09: BOTKIN, TUROVAintensities and the resulting pressure distribution is proposed. Numerical experiments are discussed.This work was supported by the Klaus Tschira Stiftung, Buhl-Strohmaier Stiftung, and Würth Stiftung.Rapid generation of extremal disturbances in linear conict control problemsKirill MartynovTechnische Universität MünchenIn this talk, we present a fast method for generation of feedback disturbances for linear conictcontrol systems. The approach is based on construction of repulsive sets, i.e. domains for whichthere exists a feedback disturbance producing trajectories violating state constraints. In our method,repulsive sets are approximated by simple geometric objects, parallelotops, which ensure lowcomputational efforts. The method consists in the integration of an ODE system describing centersand matrices of the parallelotops. Such computations can run in real time on a common processorso that high-dimensional linear models can be treated. Nonlinear models can be processed byapplying sequential linearization techniques.Several examples focused on aircraft control are presented. First, a simple low-dimensional lineardifferential game demonstrates the method. Second, a nonlinear aircraft model with reference tothe problem of take-off in wind shear conditions is addressed. Third, generation of disturbances fora highly-nonlinear adaptive aircraft controller is investigated. Simulation results demonstrate theeciency of disturbances constructed.This work was supported by the DFG grant TU427/2-1 and HO4190/8-1. Computer resources forthis project have been provided by the Gauss Centre for Supercomputing / Leibniz SupercomputingCentre under grant: pr74luMathematical modeling and viability theory-based feedback control of impaired cerebralautoregulation in premature infantsVarvara TurovaTechische Universität MünchenCerebral autoregulation is extremely important for keeping cerebral blood ow (CBF ) at a constantlevel. Impaired cerebral autoregulation is a risk factor for cerebral hemorrhages in preterm infantsbecause of sudden uctuations of CBF , which can damage unstable cerebral blood vessels. Themost important factors inuencing CBF are arterial carbon dioxide partial pressure (pCO2), meanarterial pressure (M AP), and venous pressure (V P). A mathematical model of impaired cerebralautoregulation accounting for these factors is presented in this paper. An effective heuristic feedbackcontrol for keeping deviations of CBF within tolerance limits despite unpredictable disturbances ofpCO2, M AP, andV P, is proposed. The ecacy of this feedback control, which can be interpretedas a medication affecting CBF , is proven using viability theory. Simulation results demonstrating38
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MS 09: BOTKIN, TUROVAthe quality of the feedback control proposed are shown.This work was supported by the Klaus Tschira Stiftung, Buhl-Strohmaier Stiftung, and Würth Stiftung.39
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MS 10: FESTA, GÖTTLICH, KNAPPMS 10: MODELING AND OPTIMIZATION OF NETWORKED SYSTEMSOrganizers: A. Festa, S. Göttlich, S. KnappA measure theoretic approach for Vehicular Trac Control on NetworksRaul De MaioUniversity of Rome "La Sapienza"In this talk we present a measure theoretic approach for vehicular trac problem on networks. Inparticular, aiming to describe trac ow on road networks with long-range driver interactions,weshow results on existence and uniqueness of solution for nonlinear transport equation dened onan oriented network where the nonlocal velocity elds is assumed. In the same framework, wepresent examples of control problems such as trac lights and self-driving cars and advantagesoffered by this approach.A semi-Lagrangian scheme for HJ equations on networksAdriano FestaINSA RouenWe present a semi-Lagrangian scheme for the approximation of a class of Hamilton-Jacobi-Bellmanequations on networks. The scheme is explicit and stable under large time steps. We discuss aconvergence theorem and an error estimate, which is also veried by numerical tests. Finally, weapply the scheme to simulate problems modeling trac ows.Non-local conservation laws: A Godunov type scheme and network modelsJan FriedrichUniversität MannheimWe present a Godunov type numerical scheme for a class of scalar conservation laws with non-localux arising for example in trac ow modeling. The considered scheme delivers more accuratesolutions than the widely used Lax-Friedrich type scheme. In contrast to existing work, we considera non-local mean velocity instead of a mean density to adapt it to networks. We provide L∞ andbounded variation estimates for the sequence of approximate solutions obtained by the proposedscheme. We also demonstrate the well-posedness of the considered class of scalar conservationlaws. In addition, we propose an approach to consider the class of scalar conservation laws on anetwork and provide some numerical examples.40
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MS 10: FESTA, GÖTTLICH, KNAPPDiscretized feedback control for hyperbolic balance lawsStephan GersterRWTH Aachen UniversityPhysical systems such as water and gas networks are usually operated in a state of equilibrium andfeedback control is employed to damp small perturbations over time. We consider ow problemson networks, described by hyperbolic balance laws, and analyze the stabilization of steady states.Sucient conditions for exponential stability in the continuous and discretized setting are presented.Computational experiments illustrate the theoretical ndings.Dynamic boundary control game with a star of vibrating stringsMartin GugatFriedrich-Alexander-Universität Erlangen-NürnbergConsider a star-shaped network of vibrating strings. Each string is governed by the wave equation.At the central node, the states are coupled by algebraic node conditions in such a way that theenergy is conserved. At each boundary node of the network there is a player that performs Dirichletboundary control action and in this way inuences the system state. We consider the correspondingantagonistic game, where each player minimizes her quadratic objective function that is the sum ofa control cost and a tracking term for the nal state. We show that under suitable assumptions aunique Nash equilibrium exists and give an explicit representation of the equilibrium strategies.M. Gugat, S. Steffensen: Dynamic boundary control games with networks of strings, ESAIM: COCV,DOI: https://doi.org/10.1051/cocv/2017082Production network models with stochastic capacities: semi-Markov and load-dependentapproachesStephan KnappUniversity of MannheimWe focus on production network models based on coupled ordinary and partial differential equationscombined with time-dependent random capacity functions. The partial differential equations arescalar conservation laws and of hyperbolic type coupled with ordinary differential equations at theboundaries.In a rst step, the random capacity function of every processor is an external given stochastic process,a semi-Markov process that allows intermediate capacity states in the range of total breakdownto full capacity. The operating and down times can be arbitrarily distributed, provided they keeppositive.In general, the assumption that capacity drop probabilities are independent of the production andexternal given by a stochastic process is too restrictive. This motivates to introduce an inuence from41
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MS 10: FESTA, GÖTTLICH, KNAPPthe production to the capacity process as well and we obtain a bidirectional relation between the pro-duction and the random capacity process. For this purpose, we embed the stochastic load-dependentproduction network model into the theory of piecewise deterministic Markov processes.We present solution concepts for both stochastic production network models and show the well-posedness. Caused by the complexity of the model, we state suitable simulation methods andperformance measures to evaluate and interpret the results.Crowd dynamics in domains with boundariesElena RossiInria Sophia Antipolis - MéditerranéeIn this talk, we present a new approach to the macroscopic modelling of moving crowds, based on aclass of conservation laws able to describe non local interactions and taking into consideration thepresence of boundaries. The apparent conict between boundaries and non locality is solved throughthe introduction of an ad hoc operator, aware of walls and obstacles, describing the interactionsamong individuals at different positions. Besides the well posedness of this class of non localinitial boundary value problems, in any space dimensions, see [2], we develop an ad hoc numericalalgorithm to compute the solutions to these equations, see [1]. We provide convergence tests andevaluate the behaviour of the model in realistic crowd dynamics situations. References [1] R. M.Colombo and E. Rossi. Modeling crowd movements in domains with boundaries. In preparation. [2]R. M. Colombo and E. Rossi. Non local conservation laws in bounded domains. Preprint 2017.On a degenerate parabolic model for gas transport in pipeline networksLucas Schoebel-KroehnTU DarmstadtAs a model for gas transport in pipeline networks on practically relevant length and time scales, weconsider a degenerate parabolic system of partial differential algebraic equations which describethe conservation of mass and the dissipation of energy due to friction at the pipe walls. Based onvariational arguments, we establish existence of weak solutions and we discuss the discretization bynite elements and an implicit time stepping scheme. Numerical tests are presented for illustrationof the theoretical results.A comparison of different models coupled by gas turbines for power generationAleksey SikstelRWTH Aachen UniversityA gas ow through a turbine generating electrical power is modeled by coupled p-systems and Eulerequations respectively. The p-systems are equipped with isentropic and isothermal pressure laws42
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MS 10: FESTA, GÖTTLICH, KNAPPwhile the Euler equations with an ideal gas law. The validity of the simpler p-system is investigatedquantitatively for typical scenarios. The gas turbine is modeled as coupling conditions. Explicitcoupling conditions for the in and outow, which rely on the solution of a nonlinear equation, arederived and validated.Control strategies for road risk mitigation in kinetic and hydrodynamic trac modellingMattia ZanellaPolitecnico di TorinoWe present a Boltzmann–type kinetic approach to the modellig of road trac dynamics, whichincludes control strategies at the level of microscopic binary interactions aimed at the mitigation ofspeed-dependent road risk factors. Such a description is meant to mimic a system of driver–assistvehicles, which by responding locally to the actions of their drivers can impact on the large–scaletrac dynamics, including those related to the collective road risk and safety. Furthermore, wepresent the derivation of the corresponding hydrodynamic equations for the conserved quantitiesof the kinetic model, where the derived control term results embedded in the denition of aconstrained ux function. Suitable numerical methods are necessary to observe the describedhierarchy of scales.43
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MS 11: HOFMANN, PLATOMS11: NEW TRENDS IN VARIATIONAL AND LAVRENTIEV REGULARIZATION FOR ILL-POSEDPROBLEMSOrganizers: B. Hofmann, R. PlatoAdaptive discretization for the problem of identication of laser beam quality parametersTeresa ReginskaInstitute of Mathematics PASThe problem of identication of laser beam quality parameters can be reduce to nding the waist ofthe axial prole dened by the radii of the beam. The regularized solutions of the Cauchy problemfor the Helmholtz equation can be employed to approximate the axial prole at some points. Sowe look for an approximate minimum of a function describing the axial prole given on a discreteset of points where its values are given with some errors. Presented is a new method for ndingan approximate minimum of a real function f . The initial problem is replaced by that of ndingparameters v ∈ Rd such that F (v, ·) approximates f (·). Here F is appropriately chosen and F (v, ·)are functions whose minima can easily be calculated. A modication of the iterative Tikhonovregularization is applied in which the set of points (where noisy value of f are taken) changes atevery step of iteration; The convergence of the method is proved but the rate of convergence is stillan open problem.On the characterization of unstable ultra-short laser pulse trains with D-SCANDaniel GerthTU ChemnitzSince the reaction time of electronics spans several cycles of a modern high-end pulse laser, suchpulses cannot be measured directly and indirect methods are averaged over an unknown numberof individual pulses. If there are differences from one pulse to another we speak of an unstablepulse train. A problem that is known in the literature but only recently gained more attention isthat unstable pulse trains my lead to incorrect reconstructions and thus misinterpretation of theresults.It is impossible to recover individual pulses in an unstable pulse train. Instead, one is interested inkey quantities such as the average duration of each pulse. In this talk we explain these technicalitiesin more details and show how the model for D-SCAN for stable pulses is used to estimate theaverage lengths of pulses in an unstable pulse train. The key to this is treating an in practice knownquantity as unknown and comparing its reconstruction with the data from the measurement setup.Mathematically, the problem corresponds to the solution of a complex-valued autoconvolutionequation with nontrivial unknown kernel function where additionally only the absolute values of theright-hand side are known and the phase information is missing.44
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MS 11: HOFMANN, PLATOOn ill-posedness concepts, stable solvability and saturationBernd HofmannTechnische Universität ChemnitzWe consider different concepts of well-posedness and ill-posedness and their relations for solvingnonlinear and linear operator equations in Hilbert spaces. First, the concepts of Hadamard andNashed are recalled which are appropriate for linear operator equations. For nonlinear operatorequations, stable respective unstable solvability is considered, and the properties of local well-posedness and ill-posedness are investigated. Those two concepts consider stability in image spaceand solution space, respectively, and both seem to be appropriate concepts for nonlinear operatorswhich are not onto and/or not, locally or globally, injective. Several example situations for nonlinearproblems are considered, including the prominent autoconvolution problems and other quadraticequations in Hilbert spaces. It turns out that for linear operator equations, well-posedness andill-posedness are global properties valid for all possible solutions, respectively. The special role of thenullspace is pointed out in this case. Finally, non-injectivity also causes differences in the saturationbehavior of Tikhonov and Lavrentiev regularization of linear ill-posed equations. This is examined atthe end of this study. This talk presents joint work with Robert Plato (University of Siegen). Researchis partially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant HO 1454/10-1.New convergence rates for variational Lavrentiev regularization of nonlinear monotone ill-posed problemsRobert PlatoUniversität SiegenWe consider nonlinear ill-posed equations Fu = f in Hilbert spaces H, where F : H → H ismonotone on a closed convex subset M⊂H. For given data f δ ∈ H, f δ − f ≤ δ, a standardapproach is Lavrentiev regularization Fvδα +αvδα = f δ , with α > 0 small. In practical applications likeparameter estimation problems, the considered operator is monotone on M ⊊ H only. Since theregularized equation may not have a solution in M then, we replace this equation by the regularizedvariational inequality and consider uδα ∈ M satisfying〈Fuδα + αuδα − f δ,w − uδα 〉 ≥ 0 for each w ∈ M.In this talk we present new estimates of the error uδα − u∗ for suitable choices of α = α(δ), ifthe solution u∗ ∈ M of Fu = f is source-representable. This is joint work with B. Hofmann (TUChemnitz).45
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MS 11: HOFMANN, PLATOIRGNM Ivanov type method with a posteriori choice of regularization parameter under atangential cone condition in Banach spaceMario Luiz Previatti de SouzaAlpen-Adria-Universität KlagenfurtThis talk deals with a combined analysis of regularization and discretization of inverse problems inBanach spaces with partial differential equations (PDEs). The quantities - parameters and states -have to be discretized, e.g., by the nite element method, and the error due to this discretization hasto be estimated and controlled by error estimators and mesh renement. A challenge is to take intoaccount the interplay between mesh size, regularization parameter and data noise level. The PDEssetting is relevant to the adaptive discretization of the regularized problems. I will show convergenceresult and an algorithm of the Iteratively Regularized Gauss Newton Method (IRGNM) in its Ivanovversion with a posteriori choice of regularization parameter under a tangential cone condition inBanach space setting. I will present how to obtain the a posteriori estimates and to achieve theaccuracy by adaptive discretization using goal oriented error estimators. This is illustrated for aninverse source problem for a nonlinear elliptic boundary value problem with L∞ source. Thereare numerous applications with the Banach space setting assigned by the regularity of the soughtcoecients and features like sparsity, e.g., medical imaging.The Arnoldi process for ill-posed problemsLothar ReichelKent State UniversityThe Arnoldi process is the basis for the GMRES method, which is one of the most popular iterativemethods for the solution of large linear systems of algebraic equations that stem from the discretiza-tion of a linear well-posed problem. The Arnoldi process and GMRES also can be applied to thesolution of ill-posed problems. This talk discusses properties of Tikhonov regularization and iterativemethods, that are based on the Arnoldi process, for the solution of linear ill-posed problems.Variational method for multiple parameter identication in elliptic PDEsNhan Tam Quyen TranUniversity of HamburgIn this talk I present the inverse problem of identifying simultaneously the diffusion matrix Q ∈L∞(Ω)d×d , source term f ∈ L2(Ω) and boundary condition g ∈ L2(∂Ω) in the Neumann boundaryvalue problem for an elliptic partial differential equation (PDE)− · (Q Φ) = f in Ω ⊂ d ,Q Φ · ìn = g on ∂Ω46
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MS 11: HOFMANN, PLATOfrom a measurement zδ ∈ L2(Ω) of the solution Φ ∈ H1(Ω), where ìn is the unit outward normalon the boundary ∂Ω of the open bounded connected domain Ω in the Euclidean space d . Avariational method based on energy functions with Tikhonov regularization is here proposed to treatthe identication problem. We discretize the PDE with the nite element method and prove theconvergence as well as analyze error bounds of this approach. To illustrate the theoretical results, anumerical case study is presented which supports our analytical ndings.47
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MS 12: GONG, JIN, LIMS 12: NUMERICAL ANALYSIS OF PDE CONSTRAINED OPTIMAL CONTROLOrganizers: W. Gong, B. Jin, B. LiOptimal control of instationary gas transportHerbert EggerTU DarmstadtWe consider the optimal control of instationary gas transport in a pipeline network. The well-posedness of the governing system of partial differential-algebraic equations is discussed andthe existence of minimizers is established. For the numerical solution, we consider a Galerkinapproximation of the state equation by mixed nite elements, we establish well-posedness of thisdiscretization, and prove the existence of minimizers for the corresponding discretized optimalcontrol problem. We further discuss the ecient numerical minimization via projected Newton-typealgorithms and present computational test to illustrate the performance of the proposed algorithmsand to demonstrate their viability for online control of typical situations arising during intradayoperation of a gas network.Convergence of adaptive nite element method for PDE-constrained optimal control prob-lemsWei GongAcademy of Mathematics and Systems Science, Chinese Academy of SciencesIn this talk we present our recent results on convergene of adaptive nite element method forPDE-constrained optimal control problems. The study of adaptive nite element method for optimalcontrol problems started from the end of last century but the convergence analysis is rather recent.We use variational discretization for the control variable, thus the study of the rst order optimalitysystem reduces to the study of the state and adjoint state equations. Then the well-establishedconvergence analysis of AFEM for single PDE can be adapted to prove the convergence and optimalityof the state and adjoint state of optimal control problems. The control variable in the convergenceanalysis is eliminated based on a duality argument which gives higher order a priori bound for L2norm error than the energy norm error, this introduces an assumption on the smallness of initialmesh. The convergence of the control variable is thus a direct consequence of above results. Toprove the optimality of the control variable we can use L2-norm based AFEM, so we have to assumeH2 regularity of the governing state equation.48
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MS 12: GONG, JIN, LIError estimates of an optimal control problem for fractional diffusionBangti JinUniversity College LondonFractional diffusion has received immense attention in recent years. However, the related optimalcontrol problems are scarcely studied. In this talk, I present a complete numerical analysis for adistributed optimal control problem for fractional diffusion, with box constraint on the control. Thefully discrete scheme is obtained by applying the conforming linear Galerkin nite element methodin space, L1 scheme/backward Euler convolution quadrature in time, and the control variable by avariational type discretization. I shall give sharp convergence rates for the numerical solutions ofthe optimal control problem. Numerical experiments are provided to support the theoretical results.This is a joint work with Buyang Li and Zhi Zhou.Integration based prole likelihood calculation for parameter estimation in PDEsBarbara KaltenbacherAlpen-Adria-Universität KlagenfurtPDE models are widely used in engineering and natural sciences to describe spatio-temporal pro-cesses. The parameters of the considered processes are often unknown and have to be estimatedfrom experimental data. Uncertainty in the estimates due to partial observations and measurementnoise, can be assessed using prole likelihoods, a reliable but computationally intensive approach.In this talk, we present the integration based approach for prole likelihood calculation developedby Chen and Jennrich, 2002, and adapt it to inverse problems with PDE constraints. While existingmethods for prole likelihood calculation in parameter estimation problems with PDE constraintsrely on repeated optimization, the proposed approach exploits a dynamical system evolving alongthe likelihood prole. We derive the dynamical system for the unreduced estimation problem,prove convergence and study the properties of the integration based approach for the PDE case.To evaluate the proposed method, we compare it with state-of-the-art algorithms for an applica-tion in systems biology. We observe a good accuracy of the method as well as a signicant speedup as compared to established methods. Joint work with Romana Boiger, Alpen-Adria-UniversitätKlagenfurt, as well as Jan Hasenauer and Sabrina Hroß, Helmholtz Zentrum München.Improved error estimates for nite element solutions of parabolic dirichlet boundary con-trol problemsBuyang LiHong Kong Polytechnic UniversityThe parabolic Dirichlet boundary control problem and its nite element discretization are consideredin convex polygonal and polyhedral domains. We improve the existing results on the regularity ofthe solutions by establishing and utilizing the maximal Lp -regularity of parabolic equations under49
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MS 12: GONG, JIN, LIinhomogeneous Dirichlet boundary conditions. Based on the proved regularity of the solutions, weprove O(h1−1/q−ε) convergence for the semi-discrete nite element solutions for some q > 2, withq depending on the maximal interior angle at the corners and edges of the domain and ε being apositive number that can be arbitrarily small.Second-order analysis and numerical approximation for bang-bang bilinear control prob-lemsDaniel WachsmuthInstitut für MathematikWe consider bilinear optimal control problems, whose objective functionals do not depend on thecontrols. Hence, bang-bang solutions will appear. We investigate sucient second-order conditionsfor bang-bang controls, which guarantee local quadratic growth of the objective functional in L1.In addition, we prove that for controls that are not bang-bang, no such growth can be expected.Finally, we study the nite-element discretization, and prove error estimates of bang-bang controlsin L1-norms.50
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MS 14: HERZOG, KOSTINAMS 14: OPTIMAL EXPERIMENTAL DESIGNOrganizers: R. Herzog, E. KostinaOptimal sensor placement for detection and localization of seismic sourcesChristian BoehmETH ZürichEarthquakes, landslides and nuclear explosions excite waves that can be measured in form ofseismograms at remote receiver locations. The waveform data contains information about boththe internal structure of the Earth, and the characteristics of the source. Seismic source inversioninfers the location, origin time, source-time function and source mechanism from the measuredseismograms. This can be stated as an inverse problem governed by the elastic wave equation.We present strategies for experimental design to determine the optimal locations of seismic sta-tions that minimize the uncertainty in the inferred source parameters. To this end, we consider aprobability density function characterizing the likelihood of a source location, and apply A-optimalexperimental design using the expected value of the trace of the posterior covariance to select anoptimal subset of seismic stations.Assuming a-priori knowledge of complex 3D Earth structure, and exploiting reciprocity allows usto solve the forward problem for arbitrary source locations without signicant computational cost.Furthermore, we compare sequential optimal experimental design approaches and sparsifyingconstraints to determine a suitable number of seismic stations.Numerical examples illustrate several important applications, such as improving earthquake hazardmanagement and Tsunami early warning systems, or monitoring nuclear explosions.An OED problem for interface identicationRoland HerzogTU ChemnitzInterface identication refers to a class of parameter estimation problems where the unknown isthe location of an interface, e.g., between two values of a material parameter, which may appear asa coecient in a partial differential equation. Compared to classical parameter estimation problems,the unknown is therefore not a point in a nite dimensional vector space, but can be considered anelement of an innite dimensional manifold of shapes. Its tangent space, which contains potentialdirections of parameter variations, can be conceived as a space of velocity elds acting on the shapeboundary.The proper formulation and numerical solution of OED problems for interface identication, whichwe will address in this talk, therefore imposes a number of challenges compared to classical OEDproblems in vector spaces.51
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MS 14: HERZOG, KOSTINAThe use of the population Fisher information matrix for parameter estimation of nonlinearmixed-effects modelsFelix JostUniversität MagdeburgParameter estimation for nonlinear mixed-effects models is widely used in pharmaceutical andbiomedical research and population pharmacokinetic and pharmacodynamic analysis. In clinicalstudies measurements from a population of patients or healthy volunteers are collected and usedto personalize a suited mathematical model via parameter estimation. In comparison to pointestimators (least squares problems), in which one set of parameters is estimated, in mixed-effectsmodelling the population shares the same model but each patient has an set of parameters de-pending on individual and population parameters. In this talk we give an introduction to estimationmethods and the population Fisher information matrix (FIM) for nonlinear mixed-effects models.Additionally, we propose a Gauss-Newton algorithm for parameter estimation of this kind of modelsin which the Hessian of the objective function is approximated by the population FIM. The algorithmis implemented as an prototype in CasADi.Optimum experimental design based on a second-order analysis of parameter estimatesEkaterina KostinaHeidelberg UniversityA successful application of model-based simulation and optimization of dynamic processes requiresan exact calibration of the underlying mathematical models. Here, fundamental tasks are the estima-tion of unknown model coecients by means of real observations and design of optimal experiments.The goal of the design of optimal experiments is the identication of those measurement times andexperimental conditions, which allow a parameter estimate with a maximized statistical accuracy.The design of optimal experiments problem can be formulated as an optimization problem, wherethe objective function is given by a suitable quality criterion based on the sensitivity analysis ofthe parameter estimation problem. In this talk we present a new objective function, called theQ-optimality function, which is based on a second order sensitivity analysis of parameter estimates.The robustness properties of the new objective function in terms of parameter uncertainties is in-vestigated and compared to a worst-case formulation of the design of optimal experiments problem.Numerical experiments show that the design of experiments based on Q-optimality leads to a drasticimprove of the Gauss-Newton convergence rate for the underlying parameter estimation problems.The talk is based on the joint work with M. Nattermann.52
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MS 14: HERZOG, KOSTINAFirst-order methods for regularized optimal experimental design problemsEric LeglerTU ChemnitzThe task of parameter identication of a certain model by carrying out experiments is closelyconnected to the question of nding the most valuable experimental conditions. The identicationof these experiments leads to so-called Optimal Experimental Design problems. Among otherformulations, it is possible to cast these problems in the form min F (Λ(w)) + G(w) with a measurew ∈ C∗(X), where F and G are convex and Λ is a linear operator. Therefore this problem is convexbut nonsmooth since G and in some cases also F are not differentiable. In order to reformulate thisproblem in Hilbert space, we regularize the problem by an additional term and solve a problem witha density w ∈ L2(X) instead. After discretization we compare various rst-order methods.Experimental design for ill-posed problemsMarta SauterUniversität HeidelbergExperimental design for parameter estimation is based on the minimization of an appropriatefunction of the covariance matrix subject to constraints on experimental conditions. The covariancematrix is implicitly dened by a generalized inverse of the Jacobian of the underlying parameterestimation problem under condition that the Jacobian has full rank. In this case the parameters canbe identied using experimental data. Here we consider the situation when the Jacobian does nothave full rank, hence the covariance matrix can be computed, and experimental design methodscannot be applied directly. In order to compute optimal experiments for such ill-posed parameterestimation problems we rst suggest to apply the Tikhonov regularization to estimate the biasedparameters. The regularized parameter estimation problem can be solved with generalized Gauss-Newton methods. A statistical analysis of the solution yields the mean square error which is takenas a new criterion for experimental design. Numerical results for the Diels-Adler reaction show thatthis approach allows to exploit the information from regularized problem for parameter estimateswithout regularization and to take into account existing correlations between parameters. Moreover,the local identiability of parameters can be re-established with the new design criterion withoutusing correlations of these parameters.Sensitivity analysis and approximation of sparse sensor placement problemsDaniel WalterTechnische Universität MünchenWe consider the estimation of an unknown parameter q entering a partial differential equation. Inpractice, q can not be inferred directly but only through nitely many noisy measurements of the53
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MS 14: HERZOG, KOSTINAassociated state y . We propose to determine a suitable measurement setup by solving a sparsesensor placement problemminω∈M+(Ωo )Ψ(I(ω) + I0) + β ω M .(P)Here, the distribution of the measurement sensors in the observation domain Ωo is described by apositive Borel-measure ω. By Ψ we denote a suitable optimal design criterion which assesses thequality of a design measure ω through properties of the associated Fisher-information matrix I(ω).The matrix I0 represents possible a priori knowledge on the unknown parameter and · Mis thecanonical total variation norm.In this talk, we consider the stability and sensitivity of minimizers to P with respect to the problemdata. In an appropriate sense, we show the existence of rst-order derivatives of the optimaldesign measure with respect to perturbations of the optimal design criterion and discuss theirnumerical computation. Furthermore we propose a suitable discretization for P and present severalconvergence results. The results are illustrated by numerical examples.54
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MS 15: SCHMITZER, WIRTHMS 15: OPTIMAL TRANSPORT AND APPLICATIONSOrganizers: B. Schmitzer, B. WirthError estimates for numerical approximations of optimal transportSoeren BartelsAlbert-Ludwigs-Universität Freiburg im BreisgauNumerical schemes for two instances of optimal transportation problems are devised and analyzed.The rst one concerns linear transportation costs and the discretization and iterative solution ofthe corresponding Monge-Kantorovich problem. The approximation error for the optimal cost iscontrolled by a discrete primal-dual gap which leads to a priori and a posteriori error estimates. Asplitting algorithm is used to solve the nondifferentiable primal and dual problems. For transporta-tion problems with superlinear cost function we consider the numerical approximation of transportplans within suitable spaces of measures and devise an active set strategy for its ecient numericalsolution.Gradient ows that converge to global minimizers in the many-particle limitLénaïc ChizatInriaWe consider the problem of minimizing a loss function on a Hilbert space, the latter being parametrizedin a non-linear way through sums of "particles". This non-convex problem is very common in statisticsand signal processing (e.g., neural networks with one hidden layer, low-rank semidenite program-ming, gridless spikes deconvolution). To shed light on "when" and "why" we can expect gradientdescent-based algorithms to succeed on these problems, we suggest to study the gradient ow whenthe number of particles grows to innity. We prove that, in some important cases, when initializedwith sucient variety, this many-particle gradient ow converges to a global minimizer, whenever itconverges.This work is at the crossroads of non-convex optimization, variational problems on measures forstatistics, and Wasserstein gradient ows (a by-product of optimal transport theory). This is jointwork with Francis Bach.On Entropy-transport problems and the Hellinger-Kantorovich distanceMatthias LieroWeierstrass Institute for Applied Analysis and Stochastics (WIAS)In this talk, we will present a general class of variational problems involving entropy-transportminimization with respect to a couple of given nite measures with possibly unequal total mass.55
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MS 15: SCHMITZER, WIRTHThese optimal entropy-transport problems can be regarded as a natural generalization of classicaloptimal transportation problems. With an appropriate choice of the entropy/cost functionals theyprovide a distance between measures that exhibits interesting geometric features. We call thisdistance Hellinger-Kantorovich distance as it can be seen as an interpolation between the Hellingerand the Kantorovich-Wasserstein distance. The link to the entropy-transport minimization problemsrelies on convex duality in a surprising way. Moreover, a dynamic Benamou-Brenier characterizationalso shows the role of these distances in dynamic processes involving creation or annihilation ofmasses. Finally, we will give a characterization of geodesic curves and of convex functionals anddiscuss other geometric properties. This is joint work with Vaios Laschos (TU Berlin), AlexanderMielke (WIAS Berlin and HU Berlin) and Giuseppe Savaré (U Pavia).Statistics for optimal transport: Inference, algorithms and applicationsAxel MunkGeorg-August-Universität GöttingenWe discuss some recent statistical laws for empirical optimal transport distances on discrete spacesand its consequences for statistical inference. These laws are given as dual optimal transportproblems in a multidimensional normal variable. Our proofs are based on a combination of sensitivityanalysis from optimization and discrete empirical process theory. We discuss several strategies,e.g., resampling, to overcome the computational burden to simulate such laws. In particular, weexamine empirically an upper bound for such limiting distributions on discrete spaces based on aspanning tree approximation which can be computed explicitly. This can be used for generating newrandomized computational schemes for the computation of optimal transport itself and we giveprecise error bounds on such schemes. Our results are illustrated in computer experiments and onbiological data from super-resolution cell microscopy.This is joint work with Jörn Schrieber, Max Sommerfeld and Carla Tameling.Adaptive grid methods for branched transportation networksCarolin RoßmanithApplied Mathematics Münster,Westfälische Wilhelms-Universität MünsterTransportation problems and the need of nding an optimal network to transport mass appear inseveral elds of application. Here, we consider a branched transport model which aims at nding anoptimal path between a given initial and nal distribution of mass to the lowest possible costs, wheremass is preferentially transported in bulks. Unfortunately, the non-convexity of the cost functionalcauses the task of performing realistic numerical simulations to be challenging in many cases.The problem can be reformulated as a Mumford-Shah-type optimisation problem and thereforeadmits a convex relaxation via so-called functional lifting, which as a penalty comes along with anincreased dimensionality and a large number of constraints. We provide an ecient adaptive grid56
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MS 15: SCHMITZER, WIRTHdiscretisation method for the variational model which is capable of dealing with the above-mentioneddisadvantages and perform some numerical simulations.Unbalanced optimal transportBernhard SchmitzerWWU MünsterOptimal transport induces a geometrically intuitive metric on the space of probability measures andis a powerful tool for image and data analysis. With the evolution of ecient numerical methods itis becoming increasingly popular. However, in many models the assumption that all measures haveunit mass and that mass is exactly preserved locally are too restrictive, for instance in biochemicalgrowth processes. Hence, in recent years, ‘unbalanced’ transport problems, that allow creation orannihilation of mass during transport, have received increased attention. In this talk we presentseveral formulations for such problems, ecient numerical methods and illustrate applications andadvantages of unbalanced metrics.This is joint work with Lénaïc Chizat, Gabriel Peyré, François-Xavier Vialard and Benedikt Wirth.Compressed motion sensingChristoph SchnörrHeidelberg UniversityWe study the recovery of a sparse time-varying signal from linear measurements of a single staticsensor, that are taken at two different points of time. This set-up can be modelled as observing asingle signal using two different sensors – a real one and a virtual one induced by signal motion.We examine the recovery properties of the resulting combined sensor. Assuming the sensor matrixto be in general position, we impose a weak a condition of sucient change of the signal, besidesthe usual sparsity assumption, under which not only the signal can be uniquely recovered withoverwhelming probability by linear programming, but also the correspondence of signal values(signal motion) can be established between the two points of time. In particular, we show that in ourscenario the performance of an undersampling static sensor is doubled or, equivalently, that thesucient number of measurements of a static sensor can be halved.Applications of optimal transport in spatial statisticsDominic SchuhmacherUniversität GöttingenIn spatial statistics data in d is modelled by suitable random structures, such as point patterns,maps, or unions of convex bodies. Often such structures exhibit spatial dependence which decaysas a function of the distance in space. Amenable data comes from plant ecology (positions of plants57
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MS 15: SCHMITZER, WIRTHform point patterns), disease mapping (risk maps for rare diseases), materials science (meso-scalestructure of porous material), and many other elds.In this talk I give a brief introduction to spatial statistics and present some theory and applicationsof optimal transport in this context.58
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MS 16: AVALOS, GÜVEN GEREDELIMS16: QUALITATIVE ANALYSIS AND CONTROL THEORETIC PROPERTIES OF EVOLUTIONARYPARTIAL DIFFERENTIAL EQUATIONSOrganizers: G. Avalos, P. Güven GeredeliExponential stability of partially damped plate equationsRobert DenkUniversität KonstanzWe consider the transmission problem for a coupled system of undamped and structurally dampedplate equations in two suciently smooth and bounded subdomains. It is shown that, independentlyof the size of the damped part, the damping is strong enough to produce uniform exponential decayof the energy of the coupled system.Viscoelastic wave equations with supercritical nonlinearitiesYanqiu GuoFlorida International UniversityThe talk presents a study of the history value problem of a viscoelastic wave equation which featuresa fading memory term as well as a supercritical source term and a frictional damping term. Well-posedness, asymptotic behavior, as well as singularity formulation are discussed for the model. Thespecial features of the model lie in that the source term has a supercritical growth rate and thememory term accounts to the full past history that goes back to negative innity.Exact locally distributed controllability of string bounding a linear potential uidScott HansenIowa State UniversityA variable coecient string or beam equation is used to model the exible portion of the boundaryof the domain of a two dimensional linear potential uid.Exact controllability is proved when control is applied either at an end point or over an open intervalof positive length. The method of proof is based on application of Ingham’s inequality togetherwith mini-max estimates of eigenvalues. Related results wherein the uid surrounds a string ormembrane are described.Feedback control of the acoustic pressure in HIFU propagation.Irena LasieckaInstitute of Systems Research, Polish Academy of Sciences59
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MS 16: AVALOS, GÜVEN GEREDELIThis talk will discuss boundary feedback control associated with PDE models arising in HIFU modles– which are PDEs of third order in time. This leads to a notion of a non-standard Riccati equationswhich provide suitable gain operators for the feedback control. Singularity of the control actioncompromises the usual regularity of the associated Riccati operators making the analysis challengingparticularly in the case of boundary controls. In this latter case, the loss of regularity is “double” –due to singularity caused by the appearance if time derivatives in control function and also due tothe intrinsic loss associated with unbounded and un-closeable trace operators. In order to constructa viable theory one needs to develop suitable regularity theory within the framework of non-smoothoptimization. It will be shown how the propagation of hidden trace regularity in hyperbolic dynamicsallows to build suitable concepts. This talk is based on joint work with Francesca Bucci from Universitadi Firenze.A generalization of the Aubin-Lions-Simon compactness Lemma for problems on moving do-mainsBoris MuhaFaculty of Science, University of ZagrebThis work addresses an extension of the Aubin-Lions-Simon compactness result to generalizedBochner spaces L2(0,T ; H(t)), where H(t) is a family of Hilbert spaces, parameterized by t. Acompactness result of this type is needed, e.g., in the study of the existence of weak solutions to aclass of nonlinear evolution problems governed by partial differential equations dened on movingdomains. We identify the conditions on the regularity of the domain motion in time, i.e., on thedependence of the functions spaces on time, under which our extension of the Aubin-Lions-Simoncompactness result holds. Concrete examples of the application of the new compactness theoremare presented. They include a classical problem for the incompressible, Navier-Stokes equationsdened on a given non-cylindrical domain, and a class of uid-structure interaction problems for theincompressible, Navier-Stokes equations, coupled to the elastodynamics of a Koiter shell. Both theno-slip coupling condition, and the Navier slip coupling condition, are discussed. The compactnessresult presented in this talk is crucial in obtaining constructive existence proofs to nonlinear, movingboundary problems, using Rothe’s method.Poro-visco-elasticity in biomechanicsMarcella NoormanNC State UniversityModeling of uid ows through porous deformable media is relevant for many applications inbiology, medicine and bioengineering, including tissue perfusion and uid ow inside cartilages andbones. These uid-structure mixtures are described mathematically by nonlinear poro-visco-elasticsystems, driven by mixed boundary conditions. We investigate the well-posedness of solution, as wellas sensitivity analysis with respect to data, as precursor to control problems subject to these coupled60
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MS 16: AVALOS, GÜVEN GEREDELIsystems. Our theoretical results and numerical analysis provide a novel hypothesis concerning thecauses of damage in biological tissues. I will discuss our results and their applications to ocularperfusion and conned compression tests for biological tissues.Pseudo-backstepping and stabilization of higher order PDEsTurker OzsariIzmir Institute of TechnologyWe consider the boundary feedback stabilization problem for higher order PDEs such as KdV,Kawahara, and fourth order Schrödinger equations on a nite interval. A well-known technique forstabilizing PDEs from the boundary is constructing a backstepping controller which is obtained froman integral transformation involving a specially constructed kernel. However, this method fails asthe order of the PDE gets higher, and there are serious mathematical challenges in nding the rightkernel. The purpose of this talk is to introduce "pseudo-backstepping," a technique using only animperfect kernel, which is obtained by relaxing some of the conditions enforced in the standardbackstepping method. We prove that the boundary controllers constructed via these imferfectkernels still exponentially stabilize the system with the cost of a slower rate of decay. Finally, thismethod allows us to answer some open problems in the stabilization theory for higher order PDEs.This research has been supported by TÜBİTAK 1001 Grant #117F449 and IZTECH BAP Grant #2017IYTE14.Decay estimates for a Korteweg-de-Vries-Burgers equation with time delayCristina PignottiUniversity of L’AquilaWe consider a KdV-Burgers equation with indenite damping and time delay in the whole real line.Under appropriate conditions on the damping mechanism and the time delay feedback, globalwell-posedness and exponential decay estimates are established for the linearized equation and thenonlinear model. Joint work with Vilmos Komornik (Université de Strasbourg)Flow-induced instability of a cantilever in axial owJustin WebsterUniversity of Maryland Baltimore CountyFlow-induced instability of a exible object is of great interest in engineering—both the onset (utter),as well as the qualitative properties of post-utter dynamics. In particular, the case of a cantilever inaxial ow (ag-like) has been vigorously researched in the last 15 years. Yet, there have been very fewrigorous analyses of the appropriate uid-structure interaction. This is largely due to the challengingnature of the ow boundary conditions (at the elastic interface and in the wake), which are mixed61
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MS 16: AVALOS, GÜVEN GEREDELIand dynamic in nature; moreover, structural nonlinearity is required to capture the post-utterdynamics, and the appropriate inextensible beam is highly nonlinear and nonlocal.Here, we present three approaches to analyzing these ow-structure dynamics. We rst invoke asimplifying ow assumption (piston theory) and demonstrate that the corresponding nonlinear,non-dissipative beam dynamics are semigroup well-posed with a compact global attractor. Followingthis, we consider the full linear ow-beam system and discuss its well-posedness in the contextof recent ow-structure work addressing the Kutta-Joukowsky condition (near the free end of thebeam). Time-permitting, we will discuss a heuristic way of studying the system which rewrites theeffect of the ow acting on the structure via an integral (memory) term.62
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MS 17: BERTRAND, BIRKMS 17: SELECTED ASPECTS OF MODELLING THERMALLY INDUCED DAMAGE AND FRACTUREOrganizers: F. Bertrand, C. BirkA priori und a posteriori Fehleranalyse der SBFEMFleurianne BertrandUniversität Duisburg-EssenIn this talk, we study the treatment of Linear Elastic Fracture Mechanics with the Scaled Bound-ary Finite Element Method (SBFEM). First, a priori estimates are developped using standard FEMtechniques. Then, an a posteriori error estimator using a stress reconstruction is presented.Optimal control of a damage model with penaltyLivia BetzUniversität Duisburg-EssenThe talk is concerned with a damage model including two damage variables, a local and a non-localone, which are coupled through a penalty term in the free energy functional. After introducing theprecise model, we prove existence and uniqueness for the viscous regularization thereof. We thenintroduce the optimal control problem and establish that the control-to-state operator is Hadamarddirectionally differentiable, but not Gâteaux differentiable. Hence, standard adjoint calculus is notapplicable. However, it turns out that, under additional assumptions, an optimality system of strongstationary type can be derived.Thermal stress analysis using the scaled boundary nite element methodCarolin BirkUniversität Duisburg-EssenSudden temperature changes lead to thermally induced stresses that cause fracture and damagein brittle materials such as ceramics. Here, complex networks of cracks may develop rapidly. Thesimulation of crack propagation processes requires numerical methods that can represent stresssingularities accurately and eciently, such as the scaled boundary nite element method (SBFEM).The SBFEM is a semi-analytical technique which combines a numerical solution in the circumferentialdirections with an anlytical solution in the radial direction of the considered domain. It facilitates thederivation of generalized polygon elements and thus provides great exibility in meshing. In a crackpropagation simulation, polygon meshes based on SBFEM require minimal re-meshing in the vicinityof crack tips. The principles of the scaled boundary nite element method for transient thermalanalysis and thermal stress analysis will be summarized in this contribution and illustrated usingvarious examples. Recent developments with respect to extension of the SBFEM towards nonlinearproblems will also be addressed.63
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MS 17: BERTRAND, BIRKNumerical simulation of mechanical damage processes and associated optimal control prob-lemsMarita HoltmannspötterUniversität Duisburg-EssenIn this talk a numerical solution method for an optimal control problem of a specic viscous two-eldgradient damage model will be presented. The mechanical damage model features two damagevariables which are coupled by a penalty term in the gradient enhanced free energy functional.The minimization of the free energy as well as the consideration of the evolution of the damage intime result in state equations which are nonlinear and nonsmooth in general. Therefore necessaryoptimality conditions are dicult to obtain.Under certain assumptions it is possible to derive an approximate gradient which is used to apply agradient based optimization algorithm. We focus on solving the discretized problem and presentsupporting test results.Weakly symmetric stress reconstruction and a posteriori error estimation for elasticityMarcel MoldenhauerUniversität Duisburg-EssenWe present an a posteriori error estimator for the conforming nite element method of the linearand nonlinear elasticity problem based on a nonsymmetric H(div)-conforming approximation of thestress tensor. We derive an a posteriori error estimator which gives us a completely computableupper bound of the error. First we want to show results for linear elasticity in the compressible andincompressible limit. Afterwards we extend this idea to hyperelastic material models.64
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MS 18: PAGANINI, STURMMS 18: SHAPE OPTIMIZATION: THEORY AND PRACTICEOrganizers: A. Paganini, K. SturmOptimal design of the shape of a column against buckling revisitedMichel DelfourUniversité de MontréalLagrange considered the design of vertical columns that can accommodate the largest verticalload before buckling. This problem was generalized to a family indexed by p>0 (solid p=2, hollowp=1). Cox and Overton (SIAM J. Math. 1992) proved existence of a maximizing prole for prolesbounded below and above by strictly positive constants. For the hollow column their numericalresults indicated the possibility that the cross sectional area of the column might be zero in twopoints. Their a priori hypothesis was challenged by Egorov (J. Math. Phys. 2003). In this paper werevisit the clamped clamped circular column. We rst provide numerical computations that indicatethat for 0Multimaterial topology optimization based on the topological derivativePeter GanglGraz University of TechnologyThe majority of shape and topology optimization algorithms considers the optimization with respectto two different states (e.g. material and void). In many engineering applications, however, one isinterested in nding an optimal material distribution consisting of three or more different materials.We present an extension of the level-set algorithm introduced by Amstutz and Andrae in 2006, whichis based on the topological derivative, to the case of three and more materials. We show numericalresults obtained by applying the algorithm to the optimization of an electric motor where the task isto nd the optimal distribution of ferromagnetic material, air and permanent magnets.Homogenization-based topology optimization for high-resolution microstructuresJeroen GroenTechnical University of DenmarkWe present a projection method to obtain high-resolution manufacturable structures from coarse-scale, homogenization-based topology optimization results[1]. The focus of this work is on com-pliance minimization of linear-elasticity problems, for which the optimal solution is in the spaceof layered materials. Pantz and Trabelsi introduced a method to project the microstructures fromhomogenization-based topology optimization[2]. The microstructures are oriented along the direc-tions of lamination such that a well-connected design is achieved. This approach paves the way for65
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MS 18: PAGANINI, STURMcoarse-scale topology optimization where the projection can be performed on a high-resolutionmesh, without a need for cumbersome and expensive multi-scale formulations. In a recent work wehave simplied the projection procedure, and introduced procedures for controlling the shape ofthe projected design[1]. This allowed for high-resolution ( 1 million elements in 2D), near-optimaland manufacturable designs, obtained within a few minutes on a standard PC. In the current workwe will demonstrate extensions of the method into 3D, and discuss the potential of the method overstandard topology optimization.[1] J.P. Groen, O. Sigmund. Homogenization-based topology optimization for high-resolution manu-facturable micro-structures. IJNME 2017;:1–18 [2] Pantz O, Trabelsi K.A post-treatment of the homoge-nization method for shape optimization. SIAM Journal on Control and Optimization 2008;47(3):1380–1398Topology optimization in Navier–Stokes ow with the phase eld approachChristian KahleTU MünchenWe investigate the problem of nding optimal topologies of uid domains. In a given domain Ωwe search for a topology of a uid domain, such that an objective is minimized that might dependon the velocity eld and the pressure eld inside the domain and that depends on the topology.In addition we use integral constraints on the optimizing topology. The problem class especiallycontains the problem of minimizing the drag of an obstacle in free ow.Using the phase eld approach, we describe the distribution of the uid and non-uid domain bya phase eld variable ϕ ∈ H1(Ω) ∩ L∞(Ω) that encodes the uid domain by ϕ(x) = 1 and thenon-uid domain by ϕ(x) = −1.Additionaly we use a porosity approach to extend the Navier–Stokes equation from the uid domainto the non-uid domain by assuming a material with high density inside and adding a Darcy term inthe Navier–Stokes equation, that are now stated on the whole domain Ω.Due to the regularity of the optimization variable ϕ, we apply the variable metric projection typemethod proposed in [L. Blank and C. Rupprecht, SICON 2017, 55(3)].Applications of the distributed shape derivative in shape optimizationAntoine LaurainUniversity of São PauloThe concept of shape derivative is fundamental in shape optimization, and used as the basis of manynumerical algorithms. In view of Zolésio’s structure theorem, the shape derivative is usually writtenas a boundary integral depending on the normal perturbations of the boundary, if the boundaryis suciently smooth. Alternatively, the shape derivative can be written as a domain integral, inwhich case it is called distributed shape derivative. This representation is actually more convenient66
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MS 18: PAGANINI, STURMthan the boundary expression for handling shapes with low regularity. In this talk we will discusssome interesting theoretical features of the distributed shape derivative, and compare it with theboundary expression. We will also show numerical applications and results, in particular for levelset methods.Second order directional shape derivatives of integrals on submanifoldsAnton SchielaUniversität BayreuthWe compute rst and second order shape sensitivities of integrals on smooth submanifolds usinga variant of shape differentiation. The result is a quadratic form in terms of one perturbationvector eld that yields a second order quadratic model of the perturbed functional. We discuss thestructure of this derivative, derive domain expressions and Hadamard forms in a general geometricframework, and give a geometric interpretation of the arising terms.Weak shape Hessians in applicationsStephan SchmidtJulius-Maximilians-Universität WürzburgMany PDE constrained optimization problems fall into the category of shape optimization, meaningthe geometry of the domain is the unknown to be found. Most natural applications are dragminimization in uid dynamics, but many tomography and image reconstruction problems also fallinto this category.The talk introduces shape optimization as a special sub-class of PDE constraint optimization problems.The main focus here will be on generating Newton-like methods for large scale applications. The keyfor this endeavor is the derivation of the shape Hessian, that is the second directional derivative ofa cost functional with respect to geometry changes in a weak form based on material derivativesinstead of classical local shape derivatives. To avoid human errors, a computer aided derivationsystem is also introduced.The methodologies are tested on problem from uid dynamics and geometric inverse problems.Second-order shape derivatives along normal trajectories, and approximation methodsJean-Léopold ViéÉcole Nationale des Ponts et ChausséesComputing derivatives with respect to a shape is essential for shape optimization. Following thework of Zolesio and co-workers a perturbation of a shape can be dened by the maximal ow ofa regular vector eld. In this context, the structure of the second order shape derivatives is welldescribed [3,4].67
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MS 18: PAGANINI, STURMThe goal of this work is to derive along time trajectories - similarly to this approach - in the case ofthe level-set method [1], when the vector eld is aligned with the outer normal of the shape.The present work focuses on the shape derivatives in this particular framework, with the help of thebicharacteristic method for solving the Hamilton-Jacobi equation [2,5]. At the end, this leads to asecond order derivative which has a different structure from those of the derivatives obtained in theprevious contexts.In a second time we focus on computing the second-order derivative of the compliance, which is wellknown in shape optimization. Computing the second-order shape derivative of this criteria basicallyrequires to invert the stiffness matrix, which make it computationnaly expensive. This is the reasonwhy we then consider different methods for approximating the second-order shape derivative ofthis criteria.Shape optimisation with nearly conformal mappingsFlorian WechsungUniversity of OxfordIn shape optimisation the shapes are often discretised using meshes. When updating the shape,we want to move the initial mesh instead of generating a new mesh, as the process of remeshing iscostly.In two dimensions conformal mappings are good candidates for mesh deformations as they keepangles constant and can easily be characterised by the Cauchy-Riemann equations. We propose amethod that augments the inner-product and norm on a function space in a way that deformationsthat lead to stretched mesh elements are penalized. We can make use of this new inner-productboth in rst order methods (steepest descent, L-BFGS) as well as in second order methods (Newton)to obtain fast optimisation methods that automatically choose mesh updates that retain the meshquality of the initial mesh.We present an analytical result stating that in a certain limit the chosen mesh updates are conformal.Furthermore we show several numerical examples that illustrate the performance of the resultingoptimisation methods for simple toy problems as well as classical problems in aerodynamic shapeoptimisation.68
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MS 19: BRUNE, SCHLOTTBOMMS 19: STATISTICAL AND VARIATIONAL METHODS FOR INVERSE PROBLEMSOrganizers: C. Brune, M. SchlottbomModel parameter learning for quantitative photoacoustic tomography reconstructionYoeri BoinkUniversity of TwentePhotoacoustic tomography (PAT) is a hybrid biomedical imaging technique that combines high opticaltissue contrast with high ultrasound resolution. The goal of quantitative PAT is to retrieve the imageof optical absorption that gives contrast between blood vessels and surrounding tissue.Many PAT systems require calibration measurements in order to enhance accuracy of reconstructions.Moreover, the inverse problem is based on physical models that require input of parameterssuch as sound speed and transducer sensitivity in the acoustic reconstruction. Also, the physicalmodel is sometimes simplied to speed-up the reconstruction process, for instance when diffusionapproximation of the radiative transfer equation is used for the optical reconstruction.In this talk, we will explore a data-driven approach to learn model parameters involved in quantitativephotoacoustic reconstruction. An easy learning procedure involves tting a linear convolution kernelto enhance acoustic reconstruction. More involved procedures account for parameters that have anonlinear effect in the forward model.Bounded variation regularization: convergence analysis in higher dimensionsMiguel del AlamoUniversity of GoettingenIn inverse problems, variational methods based on bounded variation (BV) penalties are well-knownfor yielding edge-preserving reconstructions, which is a desirable feature in many applications.Despite its practical success, the theory behind BV-regularization is poorly understood: hereby wemean in particular convergence guarantees.In this talk we present a variational estimator that combines a BV penalty and a multiscale constraint,and prove that it converges to the truth at the optimal rate. Our theoretical analysis relies on a properstatistical modeling of noise in the inverse problem, as well as on rened interpolation inequalitiesbetween function spaces. We also illustrate the performance of these variational estimators inMonte Carlo simulations.69
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MS 19: BRUNE, SCHLOTTBOMTests for qualitative features in the random coecients modelKonstantin EckleLeiden UniversityThe random coecients model is an extension of the linear regression model that allows forunobserved heterogeneity in the population by modeling the regression coecients as randomvariables. Given data from this model, the statistical challenge is to recover information about thejoint density of the random coecients which is a multivariate and ill-posed problem. Because ofthe curse of dimensionality and the ill-posedness, pointwise nonparametric estimation of the jointdensity is dicult and suffers from slow convergence rates. Larger features, such as an increase ofthe density along some direction or a well-accentuated mode can, however, be much easier detectedfrom data by means of statistical tests. In this article, we follow this strategy and construct testsand condence statements for qualitative features of the joint density, such as increases, decreasesand modes. We propose a multiple testing approach based on aggregating single tests which aredesigned to extract shape information on xed scales and directions. Using recent tools for Gaussianapproximations of multivariate empirical processes, we derive expressions for the critical value. Weapply our method to simulated and real data.On Tikhonov regularization under conditional stabilityHerbert EggerTU DarmstadtWe consider the stable solution of nonlinear ill-posed problems by Tikhonov regularization in Hilbertscales. Order optimal convergence rates are established for a-priori and a-posteriori parameterchoice strategies under a conditional stability assumption for the inverse problem. The role of ahidden source condition is investigated and the relation to previous results for regularization inHilbert scales is elaborated. The applicability of the results is discussed for some model problems,and the theoretical results are illustrated by numerical tests.New aspects of l1-regularizationDaniel GerthTU ChemnitzIn the last decade, sparsity-regularization became an important method in theory and practice ofInverse and Ill-posed problems. Its most prominent example, the l1-regularization, leads to linearconvergence rates if the true solution is sparse. Convergence rates without assuming sparsity ofthe true solution have been shown provided that the standard basis elements of l1 are in the rangeof the adjoint of the forward operator. In the talk, we review modications of this range inclusionthat have been proposed recently and show that, if the forward operator is injective and the imagespace is reexive, one can derive convergence rates whenever l1-regularized solutions exist. While70
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MS 19: BRUNE, SCHLOTTBOMgenerally impossible to give these rates explicitly, we show that using the discrepancy principleone can always achieve these rates. We extend these results also to the case that the image spaceis non-reexive. Furthermore, we show that l1-regularization is even applicable when the truesolution does not belong to l1, but lies in l2ł1. In particular, l1-regularization does not saturate as isthe case for classical Tikhonov regularization. Our theoretical results are supported by numericalexperiments.Imaging subsurface saltbodies – a geometric inverse problemTristan van LeeuwenUtrecht UniversityDetailed images of the subsurface can be obtained from seismic data by solving an inverse problem.As usual, there are not sucient measurements to recover the parameters of interest uniquelyand hence regularization is needed. In simple geological settings one may settle for recoveringthe global trends of the properties by imposing a smoothness constraint. In the presence of largeimpedance contrasts, such a smoothing is no longer suitable and other regularization is required(e.g. Total Variation). Of special interest are subsurface salt bodies which have constant (known)impedance and can thus be characterized by their shape alone. The inverse problem now consistsof recovering this shape. In this talk I will discuss the application of a parametric level-set methodfor this geometric inverse problem in seismic imaging. As an extension we consider joint recovery ofboth the shape of the salt body and the parameters of the surrounding sediments.71
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MS 20: STETTNERMS 20: STOCHASTIC MODELING AND CONTROLOrganizers: Ł. StettnerOptimal control of partially observable Piecewise Deterministic Markov ProcessesNicole BäuerleKarlsruher Institut für Technologie (KIT)In this talk we consider a control problem for a partially observable Piecewise Deterministic MarkovProcess of the following type: After the jump of the process the controller receives a noisy signalabout the state and the aim is to control the process continuously in time in such a way thatthe expected discounted cost of the system is minimized. We solve this optimization problem byreducing it to a discrete-time Markov Decision Process. This includes the derivation of a lter forthe unobservable state. Imposing sucient continuity and compactness assumptions we are ableto prove the existence of optimal policies and show that the value function satises a xed pointequation. A generic application is given to illustrate the results.Non-smooth verication for impulse control problemsChristoph BelakUniversity of TrierStochastic impulse control problems are continuous-time optimization problems in which a stochas-tic system is controlled through nitely many impulses causing a discontinuous displacement ofthe state process. The objective is to choose the impulses optimally so as to maximize or mini-mize a reward or cost functional of the state process. This type of optimization problem arises inmany branches of applied probability and economics such as optimal portfolio management undertransaction costs, optimal forest harvesting, inventory control, and real options analysis.In this talk, I will give an introduction to optimal impulse control and discuss classical solutiontechniques. I will then introduce a new method to solve impulse control problems based on super-harmonic functions and a stochastic analogue of Perron’s method, which allows to construct optimalimpulse controls under a very general set of assumptions. Finally, I will show how the general resultscan be applied to a problem of optimal investment in the presence of constant and proportionaltransaction costs.Linear-Quadratic control for stochastic equations driven by Rosenblatt processesTyrone E. DuncanUniversity of Kansas72
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MS 20: STETTNERFor some modeling problems it is not reasonable to assume that the noise processes are Gaussian.However a limited amount of results are available for stochastic analysis for non-Gaussian processes.The Rosenblatt processes are a natural generalization from Gaussian processes and some appli-cations have been noted. A control problem for a linear system driven by a Rosenblatt process isconsidered to determine optimal controls.Nonlinear factor analysis for identifying features of epileptogenesis after traumatic braininjuryDominique DuncanUniversity of Southern CaliforniaMost people with epilepsy have acquired forms of the disorder, and the development of antiepilep-togenic interventions could potentially prevent or cure epilepsy in many of them. We investigatethe development of post-traumatic epilepsy (PTE) following traumatic brain injury (TBI), becausethis condition offers the best opportunity to know the time of onset of epileptogenesis in patients.Epileptogenesis is common after TBI, and because much is known about the physical history of PTE,it represents a near-ideal human model in which to study the process of developing seizures. Afundamental challenge in discovering biomarkers of epileptogenesis is that this process is likelymultifactorial, crossing multiple modalities. Investigators must have access to numerous high quality,well-curated data points and study subjects for biomarker signals to be detectable above the noiseinherent in complex phenomena, such as epileptogenesis and TBI. We have developed and appliedanalytic tools for denoising, spike detection, dimensionality reduction, and pattern classicationof the EEG using nonlinear factor analysis, particularly, power spectrum analysis, Diffusion Maps,and Unsupervised Diffusion Component Analysis. Based on heterogeneous biomarkers, we de-scribe novel analytic tools designed to study epileptogenesis after TBI with the goal of tracking theprobability of developing epilepsy over time.Asymptotic stability of Boltzmann-type equations on convex setsHenryk GackiUniversity of Silesia in Katowice(Joint work with Prof. Łukasz Stettner, Polish Academy of Sciences)Some problems of the mathematical physics can be written as differential equations for functionswith values in the space of measures.We will discuss a equation drawn from the kinetic theory of gases. This equation was stimulated bythe problem of the stability of solutions of the following version of the Boltzmann equation (see [1])dψdt+ ψ =n1cnPnψ(1.4)73
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MS 20: STETTNERwhere ψ : + → Msig ( +) is an unknown function, c1,..., cn is a nite sequence of real summingsup to one and P1,..., PNare operators acting on the space of probability measures. Pkfor k ≥ 2describes the simultaneous collision of k particles and P1 the inuence of external forces.It will be a generalized version of (1.4) (see [2]) with an innite sum of the right hand side. Thiscorresponds to the situation in which the number of colliding particles is not bounded.[1] H. Gacki, Applications of the Kantorovich–Rubinstein maximum principle in the theory ofMarkov semigroups, Dissertationes Math., 448, 1–59, (2007).[2] H. Gacki and Ł. Stettner, Asymptotic stability of some nonlinear evolutionary equation, toappear.Invariance formulas for stopping times of squared Bessel processJacek JakubowskiUniversity of WarsawWe present the invariance formulas for stopping times of a squared Bessel process R with positiveindex. For a stopping time τ satisfying some relatively mild assumptions and every positive Borelfunction f we have the invariance formula¾f (τ + R(τ)ζ(µ))(ζ(µ))2µ1{τ<∞} = ¾f (ζ(µ))(ζ(µ))2µ,where ζ(µ) = 12γµand γµ is a gamma random variable independent of σ(τ, R(τ)). The applications ofinvariance formulas are presented. Hence new results on the distribution of a killed squared Besselprocess, a conditional inverse local time and the rst hitting time of a graph by a squared Besselprocess are delivered.Optimal uniform approximation of Lévy processes on Banach spaces with nite variationprocessesRafal LochowskiWarsaw School of EconomicsLet Xt , t ≥ 0, be a càdlag Lévy process on a Banach space V (i.e. a process with a.s. càdlag pathsand independent and stationary increments), and let AXbe the family ofV -valued processesYt ,t ≥ 0, adapted to the natural ltration of X. By |·| we denote the norm in V . For T > 0 and twoprocessesY , Z : Ω ×T → B, where T is an index set such that [0,T ]⊂T, we denoteY − Z ∞,[0,T ]:= sup0≤t ≤T|Yt − Zt |74
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MS 20: STETTNERandTV (Y , [0,T ]) := supnsup0≤t0<t1<···<tn ≤Tni=1Yti −Yti−1 .In this talk we will deal with the following optimisation problem. For given non-decreasing functionψ : [0, +∞) → [0, +∞) and T , θ > 0, calculate (or at least estimate up to universal constants)VX (ψ, θ,T ) := ¾ infY ∈AX{ψ(X −Y ∞,[0,T ]) + θ ·TV (Y , [0,T ])} .(1.5)To make the problem non-trivial we assume that ¾ |X1| < +∞. Such problems arise for examplein nancial mathematics when the process X denotes optimal hedging strategy while Y denotesapproximation of this strategy in the presence of (proportional) transaction costs. We will presentformulas for VX (ψ, θ,T ) expressed in terms of simpler functionals of X and apply them to astandard Brownian motion on n or a symmetric α-stable process on .BSDEs on random horizon – applications to quadratic hedgingMariusz NiewęgłowskiWarsaw University of TechnologyWe consider BSDE’s on random interval driven by general martingale of the formYt = η +σt(a(Yu )dNu + g(Yu )d 〈M〉u ) +σtψu dMt + Lσ − Ltwhere N is a bounded counting process, M is a martingale and L is a martingale orthogonal with M.We prove existence and uniqueness of solutions for such BSDE’s. We show that one can constructsolution by solving corresponding recursive system of BSDE on random intervals and piecing themtogether appropriately. This generalizes BSDE’s considered by Carbone et.al and El Karoui and Huang.Then we prove that under some Markovianity assumption solution of the above BSDE are associatedwith system of Cauchy problems. This results are then applied to quadratic hedging problems i.e.risk-minimization of claims described by general dividend process.Monte Carlo algorithm for optimal control of Markov processesJan PalczewskiUniersity of LeedsWe develop two Regression Monte Carlo algorithms (value and performance iteration) to solvegeneral problems of optimal stochastic control of discrete-time Markov processes. We formulateour method within an innovative framework that allow us to prove the speed of convergence of ournumerical schemes. We rely on the Regress Later approach unlike other attempts which employthe Regress Now technique. We exploit error bounds obtained in our proofs, along with numericalexperiments, to investigate differences between the value and performance iteration approaches.75
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MS 20: STETTNERIntroduced in Tsitsiklis and VanRoy [2001] and Longstaff and Schwartz [2001] respectively, their char-acteristics have gone largely unnoticed in the literature; we show however that their differences areparamount in practical solution of stochastic control problems. Finally, we provide some guidelinesfor the tuning of our algorithms.Linear-quadratic control for bilinear evolution equations with Gauss-Volterra processesBozenna Pasik-DuncanUniverity of KansasSome control problems are explicitly solved for bilinear evolution equations where the noise is aGauss-Volterra process. The controls are chosen from the family of linear feedback gains. The optimalgain is different from the control problem for a linear equation with a quadratic cost functional.Some examples are given.Risk-sensitve portfolio optimisation: weighted approachMarcin PiteraJagiellonian UniversityIn this talk we discuss the long-run risk sensitive optimisation in discrete-time. Using the span-contraction framework we show how to deal with the associated Bellman equation (MPE) anddiscuss it’s intricacies. In particular, we show how to extend the classical bounded and uniformlyergodic framework by considering appropriate Lyapunov weight functions, and how to link solutionto Bellman equation to various portfolio optimisation problems.The talk is based on a join work with prof. L. Stettner.Dual characterization of super-replication prices in uncertain volatility models with fric-tionAgnieszka RygielCracow University of EconomicsWe study the problem of super-replication of contingent claims in a discrete time nancial marketmodel with transaction costs and volatility uncertainty, considering a general multi asset market. Thedistribution of stocks prices is not assumed to be (completely) known a priori. Our sole assumptionon each stock price dynamic is that the absolute value of the log-returns is bounded from below andabove. We show that the pricing of European options with convex payoff can be reduced to studypricing of suitable multinomial model. Using classical convex optimization results we obtain a dualrepresentation of the super-replication cost for a basket option. The results generalize papers [1] and[2]. References [1] Bank, P., Dolinsky, Y., Gökay, S.: Super-replication with nonlinear transaction costsand volatility uncertainty. The Annals of Applied Probability 26 (3), 1968-1726, (2016). [2] Dolinsky, Y.,76
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MS 20: STETTNERSoner, H. M.: Duality and convergence for binomial markets with friction. Finance and Stochastics 17(3), 447-475, (2013).Optimal consumption and investment for a large investor: an intensity-based control frame-workFrank SeifriedUniversity of TrierWe introduce a stochastic control framework where in addition to controlling the local coecientsof a jump-diffusion process, it is also possible to control the intensity of switching from one state tothe other. Building upon this framework, we investigate optimal consumption and investment of alarge investor as a benchmark example.Long run control with degenerate observationŁukasz StettnerInstitute of Mathematics PASWe assume that a given discrete time controlled Markov process (xn) is observed using degenerateobservation yn = h(xn), where h not necessarily invertible function. The behaviour of such system isdescribed using so called ltering process, which is a measure valued process πn such that πn(B) =P [xn ∈ B|Y n], whereY n is a sigma eld of available observations till time n. The state process iscontrolled basing on available observation. When the observation space is at most denumerablewe can get an explicit recursive formula for πn and consequently πn forms a controlled Markovprocess. Using suitable limit procedure we can show that it is also true for general observation spacehowever we don’t have explicite formulae. Our problem is to maximize long run average cost per unittime functional. For this purpose we adapt resulta of [1] concerning nondegenerate problem andobtain rst existence of solutions to suitable Belman equation for at most denumerable observationcase. Then using limit procedure is solve the problem in the case of general observation space. Theproblem is closely related to ergodicity of ltering process.References:[1] L. Stettner, Ergodic Control of Partially Observed Markov Processes with Equivalent Probabilities,Appl. Math. 22,1 (1993), 25–38.Bilateral incomplete information stopping problemKrzysztof SzajowskiWrocław University of Science and TechnologyThis paper treats decision problem related to the observation of a Markov process by two decisionmakers. Admissible strategies are stopping moments. The receiving payment decision makers are77
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MS 20: STETTNERdened by stopped and accepted state of the process. The players’ decision to stop has a variableeffect which depends on the type of the decision makers (players). The type β player’s stoppingdecision assign the state of the process with chance β and it gives the state to the opponent withprobability 1 − β. It is a random mechanism which decides the type of the player. The knowledgeabout the type of the players is not public and by this way, the payers have also different information.The details of the description allow formulating the problem as a Bayesian game with sets ofstrategies based on the stopping times. It is an extension of the Dynkin’s game related to theobservation of a Markov process with random assignment mechanism of states to the players. Someexample related to the best choice problem is analyzed.78
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MS 21: KRÖNER, WINKLERMS 21: THEORY AND NUMERICS FOR OPTIMAL CONTROL PROBLEMS WITH PDESOrganizers: A. Kröner, M. WinklerGlobal minima for semilinear optimal control problemsAhmad Ahmad AliUniversity of HamburgWe consider an optimal control problem subject to a semilinear elliptic PDE together with itsvariational discretization. We provide a condition which allows to decide whether a solution of thenecessary rst order conditions is a global minimum. This condition can be explicitly evaluated atthe discrete level. Furthermore, we prove that if the above condition holds uniformly with respect tothe discretization parameter the sequence of discrete solutions converges to a global solution of thecorresponding limit problem. Numerical examples with unique global solutions are presented.An algorithmic approach for time-optimal control problems with bang-bang controlsLucas BonifaciusTechnische Universität MünchenWe consider time-optimal control problems subject to linear parabolic equations. Due to the stateconstraint and the absence of a regularization term in the objective, it is in general dicult tosolve these problems algorithmically. We propose an equivalent reformulation leading to a bileveloptimization problem: In the outer loop we have to determine the root of a certain value function andin the inner loop we need to solve convex optimal control problems subject to control constraints,only. Different methods will be discussed for the solution of both loops. For the classical case thatthe terminal set is given as the L2 ball centred at some desired state, we can employ a Newtonmethod for the outer loop that typically requires just a few iterations to reach machine precision.In numerical examples we see that this approach is capable of solving the time-optimal controlproblem up to high precision in short time. Moreover, our approach is at least competitive with aregularization strategy for the original problem.Optimization with abs-linearization for non-smooth PDE problemsOlga EbelUniversity PaderbornWe present a new algorithmic approach for solving optimal control problems constrained by non-smooth PDEs where the non-smoothness is assumed to be given by combinations of the piecewisenon-differentiable functions abs(), min() and max(). The key idea of the presented optimizationmethod is to locate stationary points by appropriate decomposition of the original problem into79
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MS 21: KRÖNER, WINKLERseveral smooth branch problems, which can be solved by classical means. Suitable optimalityconditions and subsequent exploitation of information given by the respective dual variables lead tothe next branch and thus to successive reduction of the function value. A discussion of numericalresults for selected model problems is also involved.Optimal control of a critical semilinear wave equation in 3dHannes MeinlschmidtRICAMWe consider optimal control problems subject to a defocusing H1 critical semilinear wave equationon a domain in three spatial dimensions. The prototype of such an equation is given byy (t) − y(t) − y5 = u, y(0) = y0, y (0) = y1with homogeneous Dirichlet- or Neumann boundary data and the control u. The critical exponent5 makes global analysis for the equation under consideration very dicult, and global existencefor u = 0 has been shown only recently. In this talk, we explore what we can achieve in an optimalcontrol setting for this equation.Discretization error estimates for normal derivatives on boundary concentrated meshesJohannes PfeffererTechnical University of MunichThis talk is concerned with nite element error estimates for the solution of linear elliptic equations.More precisely, we focus on approximations and related discretization error estimates for thenormal derivative of the solution. In order to illustrate the ideas, we consider the Poisson equationwith homogeneous Dirichlet boundary conditions and use standard linear nite elements for itsdiscretization. The underlying domain is assumed to be polygonal but not necessarily convex.Approximations of the normal derivatives are introduced in a standard way as well as in a variationalsense. On quasi-uniform meshes, one can show that these approximate normal derivatives possessa convergence rate close to one in L2(∂Ω) as long as the singularities due to the corners are mildenough. Using boundary concentrated meshes, we show that the order of convergence can even bedoubled in terms of the mesh parameter.As an application, we use these results for the numerical analysis of Dirichlet boundary controlproblems, where the control variable corresponds to the normal derivative of some adjoint variable.Finally, the predicted convergence rates are conrmed by numerical examples.80
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MS 21: KRÖNER, WINKLERShape optimization for a viscous Eikonal equation with applications in electrophysiologyPhilip TrautmannKFU GrazThis talk is concerned with a shape optimization problem involving a viscous Eikonal equation as thestate equation. This problem is motivated by an inverse problem from cardiac electrophysiology. Theviscous Eikonal equation under consideration is a semilinear elliptic equation. Its solution modelsthe arrival time of a electromagnetic wave in the heart which is initiated at several activation sites.The heart is modeled by a domain and the activation sites are given by small domains inside theheart. The state equation is posed on the heart without the activation sites and thus has Dirichletboundary conditions on the surface of the activation sites. Given measurements on the surface ofthe heart the locations of the activations sites are sought-after. This constitutes a shape optimizationproblem. First the wellposeness of the state and adjoint state equation is discussed. Then shapederivative of the involved cost functional is derived. Based on this derivative a perturbation eld iscalculated which is used to translate the activation sites. The talk is concluded with two numericalexperiments in 2D and on a realistic geometry of a heart in 3D.81
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MS 22: KALTENBACHER, WALDMS 22: TIME-DEPENDENT INVERSE PROBLEMSOrganizers: B. Kaltenbacher, A. WaldOptical ow regularization for dynamic inverse problemsMarta BetckeUniversity College LondonIn dynamic inverse problems the data acquisition speed dictates the temporal resolution of theimaging system. More frequently than not, the underlying dynamics happens at a rate much higherthan the data acquisition resulting in a severely subsampled data per time frame. To compensate forthe missing information, the reconstruction problem is usually formulated in a variational frameworkand regularization is included. The choice of the latter is paramount to the quality of the nal dynamicimage reconstruction. In this talk we are going to discuss a particular choice of spatio-temporalregularization for motion dominated dynamics based on optical ow. We discuss different solversfor the resulting non-convex optimisation problem and present results for applications in X-ray andPhotoacoustic imaging.Modelling and algorithms in dynamic imagingBernadette HahnUniversität WürzburgMotion compensation represents an important time-dependent problem in tomography. Mostmodalities record the data sequentially, including computerized tomography, magnetic resonanceimaging, etc. Therefore, temporal changes of the object lead to undersampled and/or inconsistentmeasurements. Consequently, suitable models and algorithms have to be developed in order toprovide artefact free reconstructions. In particular, they have to incorporate some prior informationabout the dynamic behavior. This talk presents recent advances concerning modelling and algorithmsin dynamic imaging.Spatio-temporal concentration estimation in magnetic particle imaging using a priori mo-tion informationTobias KluthUniversität BremenMagnetic particle imaging (MPI) is a new imaging modality to determine the concentration ofnanoparticles from their nonlinear magnetization behavior. Highly dynamic applied magnetic eldsallow a rapid data acquisition in 3D. The applied magnetic elds are characterized by a eld freepoint (FFP) moving along a predened trajectory which mainly denes the eld of view. Due to82
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MS 22: KALTENBACHER, WALDsafety limitations the size of this eld of view is limited. To overcome this issue, homogeneous offsetelds (focus elds) were introduced to shift the measured region in space to increase the totaleld of view. This can be realized with an increased measurement time in a patch by patch strategyor continuously in time. Despite the fact that data is acquired rapidly, temporal changes of theparticle concentration can already be relevant in single cycles of the FFP. As a result the dynamicsare not negligible when increasing the eld of view by using multiple FFP cycles. In this talk wediscuss the dynamic MPI problem with time-dependent particle concentration and the incorporationof suitable motion constraints in the concentration reconstruction which is illustrated by initialnumerical results.All-at-once versus reduced version of Landweber-Kaczmarz for parameter identication intime dependent problemsTram NguyenAlpen-Adria-Universität KlagenfurtA large number of inverse problems in applications ranging from engineering via economics tosystems biology can be formulated as a state space system with a nite or innite dimensionalparameter that is supposed to be identied from additional continuous or discrete observations. Inthis talk we will compare reduced and all-at-once versions of Landweber-Kaczmarz iteration for areformulation of the problem as a system resulting from splitting the time line into subintervals.Modeling the system function in MPIAnne WaldSaarland UniversityMagnetic particle imaging is a novel fast, dynamic medical imaging technique for, e.g., blood owvisualization. Magnetic nanoparticles are injected into the blood stream and the main objective is toreconstruct their concentration. The particles respond to an applied dynamic magnetic eld andthe response is measured in the shape of induced currents in a set of receive coils. The forwardoperator in MPI is given by an integral equation of the rst kind. The integration kernel is calledthe system function, which describes the potential of magnetic particles to induce a signal in thereceive coils by a change in their magnetization. The particles’ change in their magnetization dependson the applied magnetic eld. A simple approach is the equilibrium model, which states that themean magnetization only depends on the absolute value of the applied eld. However, our aim isto include the dynamic behavior of the magnetization (leading in particular to relaxation effects),which can be done by using the Landau-Lifshitz-Gilbert equation in combination with an uncertaintyapproach in the model to reduce its complexity.83
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MS 22: KALTENBACHER, WALDCoecient inverse problems for Kelvin-Voigt viscoelasticityMasahiro YamamotoThe University of TokioWe consider the Kelvin-Voigt viscoelasticity equation in a bounded smooth domain, which is anon-stationary Lame system with memory term. For suitable subboundary Γ, we discussCoecient inverse problems: Determine spatially varying coecients by extra data of solution onΓ × (0,T ).We prove Lipschitz conditional stability estimates for the inverse problems provided that we canchoose a nite number of suitable initial values and boundary values. The key is a Carleman estimatefor the parabolic system with time-integral term.The work is based on the joint work with Professor Oleg Yu. Imanuvilov (Colorado State University).84
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MS 23: BETZ, CHRISTOFMS 23: VARIATIONAL INEQUALITIES AND NONSMOOTH PDE-CONSTRAINED OPTIMIZATIONOrganizers: L. Betz, C. ChristofBouligand-Landweber iteration for a non-smooth ill-posed problemChristian ClasonUniversität Duisburg-EssenWe consider an inverse source problem for an elliptic partial differential equation with a non-differentiable non-linearity. The corresponding solution mapping is merely directionally but notGâteaux differentiable, which makes standard regularization methods inapplicable. We present anovel iterative regularization method of Landweber type that makes use of specic elements in theBouligand subdifferential in place of the non-existent Fréchet derivative and show its convergencein the noise-free setting and its regularization property if the iteration is stopped according to thediscrepancy principle. Numerical examples illustrate the behavior of the proposed method.Optimal control of a non-smooth evolution equation with viscous regularizationTobias GeigerJulius-Maximilians-Universität WürzburgWe study the optimal control of an evolution equation with non-smooth dissipation. The solutionmapping of this system is non-smooth and hence the analysis is quite challenging. Our aim is tond an optimality system and we present two approaches to get such a system. The rst one isto regularize the dissipation via approximation by a smooth function. The second one is a timediscretization of the state equation. In both cases we get optimality systems for an approximationof the optimal control problem. By passing to the limit we obtain optimality conditions for theoriginal non-smooth problem. Finally, we compare the systems that were derived by the differentapproaches and show some numerical examples.Solving inverse optimal control problems to global optimalityFelix HarderBTU Cottbus-SenftenbergWe consider a class of bilevel optimization problems which can be interpreted as inverse optimalcontrol problems. The lower-level problem is an optimal control problem and it has a convex andparametrized objective function.The upper-level problem is used to identify the parameters of thelower-level problem. We reformulate the inverse optimal control problem using the value functionof the lower-level problem. The feasible set of this reformulation is relaxed by using piecewise aneapproximations of the value function. This allows us to compute global optimal solutions of the85
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MS 23: BETZ, CHRISTOForiginal non-smooth and non-convex inverse optimal control problem. The global convergence ofour algorithm can be shown rigorously. Finally, the theory is illustrated by means of some numericalexamples.A non-smooth trust-region method for optimal control of variational inequalitiesChristian MeyerTU DortmundWe present a new trust-region approach for the optimization of non-smooth objectives. It is inparticular designed to optimize composite functions consisting of a smooth outer objective anda non-smooth control-to-state operator as it appears in the implicit programming approach foroptimal control of variational inequalities (VIs). The construction of the trust-region sub problems isbased on a distinction of cases depending on the trust-region radius and employs the Bouligand-subdifferential once the trust-region radius becomes small. For the example of a particular VI ofthe second kind, we demonstrate how to compute the Bouligand subdifferential and to solve theassociated trust-region subproblems. We show that accumulation points of the sequence of iteratesare Clarke-stationary.Subgradient calculus for the obstacle problemAnne-Therese RaulsTU DarmstadtThe obstacle problem is an important prototype of an elliptic variational inequality which appearsin the mathematical formulation of applications from physics, nance and other elds. The maindiculty to handle when dealing with constraints of obstacle type in optimization problems is thenondifferentiability of the solution operator.In this talk we investigate how specic elements of the Bouligand subdifferential respective to thesolution operator of the obstacle problem can be computed. Using density of Gâteaux differentiabilitypoints for Lipschitz functions, we construct an abstract sequence of differentiability points whosederivatives converge to a subgradient. In order to show this convergence, a precise analysis of therelevant set-valued mappings connected to the Gâteaux derivatives is necessary. The limit and thusthe subgradient itself is determined by the solution operator of a Dirichlet problem on a quasi-opendomain and it is independent of the approximating sequence.Based on these theoretic results, we discuss strategies for obtaining inexact subgradients resultingfrom discretization and approximation schemes.86
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MS 23: BETZ, CHRISTOFOptimal control of static contact in nite strain elasticityMatthias StoeckleinUniversität BayreuthNonlinear elastic problems usually appear in modeling the deformations of nonlinear materials inclassical mechanics. These deformations can be described as minimizers of an respective energyfunctional. Solving such problems is already a highly challenging task owing to the nonlinearity andnonconvexity of elastic energy functionals. Extending this to contact problems and an optimal controlapproach results in a nonsmooth, nonconvex, nonlinear and constrained optimization problem. Toaddress these kinds of problems we have to apply suitable regularization and optimization methodsin function space. The talk will therefore deal with the theoretical foundations of static contactproblems in nonlinear elasticity and corresponding optimal control problems. The focus here will lieon theoretical results we have achieved so far such as existence theory and convergence results forthe regularized problem. Also, a short discussion about suitable algorithms to solve such problemsnumerically will be given.Optimal control of thermoviscoplasticityAilyn StötznerTU ChemnitzElastoplastic deformations play a tremendous role in industrial forming. Many of these processeshappen at non-isothermal conditions. Therefore, the optimization of such problems is of interestnot only mathematically but also for applications.In this talk we will present the analysis of the existence of a global solution of an optimal controlproblem governed by a thermovisco(elasto)plastic model. We will point out the diculties arisingfrom the nonlinear coupling of the heat equation with the mechanical part of the model. Finally, wewill discuss some numerical results.Fixed domain approaches in variational inequalities and free boundary problemsDan TibaRomanian AcademyWe report on recent results concerning the numerical approximation of certain variational inequali-ties and free boundary problems. The main point is to avoid the geometric diculties related to thetracking of the unknown free boundary. From this point of view, one may compare free boundaryproblems with shape optimization problems. The proposed methods have a xed domain characterand are based on the ctitious domain approach. The applications concern elliptic and parabolicvariational inequalities, as well as solid- uid interface problems. The results are relevant both fromthe computational and the theoretical points of view.87
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MS 23: BETZ, CHRISTOFHyperbolic quasi-variational inequality of Maxwell type for high-temperature superconduc-torsIrwin YouseptUniversity of Duisburg EssenBean’s critical-state model describes the irreversible physical phenomena of penetration and exitof magnetic ux in type-II superconductors. The model postulates a nonlinear and nonsmoothconstitutive relation between the current density and the electric eld through the so-called criticalcurrent. By the nature of superconductivity, conrmed by experimental measurements, the criticalcurrent depends not only on the magnetic eld but also on the temperature distribution, which makesthe corresponding mathematical analysis intricate. In this talk, we discuss the well-posedness of anevolutionary Maxwell system governed by the Bean critical-state law with a nonlinear temperatureand magnetic eld dependence in the critical current. As a nal result, a well-posed hyperbolicquasi-variational inequality is presented and shown to be equivalent to the evolutionary Maxwellsystem.88
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CONTRIBUTED TALKSCONTRIBUTED TALKSOptimum brain cooling to reduce TBI damages using inverse heat transfer methodAli Abbas NejadShahrood University of TechnologyThe brain is one of the most important organs in a biological body. With the interruption of cardiopul-monary circulation in many cardiac surgical procedures or accidental events leading to cerebralcirculation arrest, an imbalance between energy production and consumption will occur. Meanwhile,the cooling function of the blood ow on the hot tissue will be stopped, while metabolic heat gener-ation in the tissues still keeps running for a while. Under such adverse situations, the potential forcerebral protection through hypothermia has been intensively investigated in clinics by loweringbrain temperature to restrain the cerebral oxygen demands. In this paper, the conjugate gradientmethod, coupled with an adjoint equation approach, is used to solve the inverse heat conductionproblem using Pennes bioheat equation in the axisymmetric coordinate system and estimate thetime-dependent heat ux using temperature distribution at a point brain to achieve the temperaturereduction about 5 degrees within 30 minutes. Two cases containing one-layer and three-layer, areconsidered. To solve this problem the general coordinate method is used. The obtained results forfew selected examples show the good accuracy of the presented method. Also the solutions havegood stability even if the input data includes noise.Bregman iterated variational regularization for nested primal-dual algorithms: with the ap-plication for an atmospheric tomography problemErdem AltuntacUniversite Libre de BruxellesThe problem of minimization of the least squares functional with Bregman distance associated withthe non-smooth total variation (TV) penalizer, and an indicator function is considered to be solvediteratively by some nested primal-dual algorithm.It is assumed that the exact solution of the linear inverse ill-posed problem satises a variationalsource condition (VSC). The regularization parameter obeying Morozov‘s discrepancy principleprovides tight convergence rates of the regularized solution of the minimization problem againstthe exact solution in terms some concave, positive denite index function. Convergence of theregularized solution of the minimization problem to the exact solution of the inverse problem,and convergence of the iteratively regularized solution to the exact solution are both analyzedseparately.Theoretical development is applied for an atmospheric tomography problem named as GPS-Tomography.GPS-tomography involves the reconstruction of some quantity (e.g. humidity), pointwise within avolume, from geodesic X-ray measurements transmitted by nonuniformly distributed transducers89
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CONTRIBUTED TALKS(satellites). These measurements are sparse and uctuate randomly with receiver availability. Thetask here is the reconstruction of the 3-dimensional spatially varying index of refraction of theatmosphere, from a set of line integrals which are fan-beam projections.Shape gradients for three-dimensional contact problems with Tresca frictionBastien ChaudetUniversité LavalPart of an ongoing research on shape optimization for hyperelastic contact problems, this presenta-tion deals with three dimensional elastic bodies in contact with given friction (Tresca model). Theproposed method is based on shape gradients combined with a level set representation of theshape. Due to the irregularity (non-linearity, non-differentiability) of the boundary conditions, thecomputation of shape gradients requires a specic treatment. Indeed, since the solution of thecontact problem is non-differentiable with respect to the shape, we introduce a regularized Lagrangemultiplier approach. The main advantages of this approach are its variational formulation, whichtakes the form of a non-linear variational equality, and the shape-differentiability of the regularizedsolution obtained.After proving the convergence of the regularized solutions to the original one, we express the shapegradient of a general functional for the regularized problem, and try to establish sucient conditionsfor those regularized shape gradients to converge.Numerical experiments, based on the nite element method for the augmented Lagrangian for-mulation and nite differences for the advection of the level set, will be presented. The methodbenets from an original mesh cutting algorithm allowing sharp representation of the boundary ateach iteration of the optimization process.Magnetic induction tomography in 3D using a shape optimization approachOliver DornThe University of ManchesterThere exists a large variety of problems in industry and technology where the goal is to noninvasivelydetect and identify objects or structural imperfections hidden inside conductive materials. MagneticInduction Tomography has been discussed lately as a promising tool for addressing this task.However, due to the conductive nature of the material, very low frequencies need to be used in orderto achieve sucient penetration depths. This makes the corresponding inverse problem highly ill-posed. Often, classical regularization schemes do not seem appropriate in these applications due totheir inherent oversmoothing property which makes it dicult to identify and correctly characterizethe sought objects. As an alternative approach, we propose here a novel 3D shape optimizationscheme for detecting and characterizing shapes of inhomogeneities buried in conductive box-like domains at various scales. A level set technique in 3D, combined with an ecient forward90
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CONTRIBUTED TALKSsolver for Maxwell’s equations in frequency domain, is applied for practically performing the shapeoptimization driven by a reduction of a given data mist functional. An ecient line search strategyis proposed for each individual step which avoids the need of an excessive number of additionalforward solves for nding appropriate step-sizes.Stochastic optimization of uid ow simulation in porous media by incorporating controlledsource electromagnetics dataOliver DornThe University of ManchesterThe forecasting of production and injection in active reservoirs is an important component of modernproduction technologies adopted by oil companies. The modeling of the underlying multiphaseow in porous media requires the estimation of various distributed system parameters over theentire production cycle. As data for these estimations usually well-logs, water and hydrocarbonpressure and production data, and time-lapse seismic data are employed. Additional modalitiessuch as controlled source electromagnetics have also been discussed recently which can helpimproving the estimation of those system parameters that are correlated with electromagneticproperties of the reservoir, in particular the unknown water saturation. In the talk we propose amodied ensemble Kalman lter approach for incorporating such additional electromagnetic datainto the time-dependent estimation approach. It relates water saturation inside the 3D reservoirto electromagnetic measurements at the surface by solving an additional inverse problem forMaxwell’s equations. Numerical results are presented which demonstrate that the incorporation ofsuch additional electromagnetic data signicantly improves the estimation of reservoir propertiesand the nal match of production data.Cut-sharing in stochastic dual dynamic programmingChristian FüllnerKarlsruhe Institut of TechnologyStochastic Dual Dynamic Programming (SDDP) is a widely used method to solve multi-stage stochasticlinear programming problems, introducing sampling to the Benders decomposition method and,hence, allowing to deal with a large number of scenarios and stages. Yet, in its classical formSDDP relies on interstage independent random vectors so that Benders cuts can be shared amongdifferent scenario subproblems at the same stage. Recently, techniques have been developed toenable cut sharing also for some types of interstage dependency, mainly assuming uncertainty inthe right-hand side of the problem modelled by ane linear or at least convex interstage dependentstochastic processes. We build upon this work and further generalize the cut sharing methodologyto a broader class of uncertainty models. A real-life power system example is examined to illustratethe effectiveness of the proposed techniques.91
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CONTRIBUTED TALKSAn optimal control problem governed by a parabolic obstacle problemDominik HafemeyerTechnische Universität MünchenWe consider an optimal control problem with an parabolic obstacle problem as constraint. Given atime intervall I and a suciently regular domain Ω ⊂ N with N ∈ {2, 3} it has the form{0 ≤ y − Ψ ⊥ ∂t y − ∆y + f (y) − u ≥ 0 in I × Ω,y |I ×∂Ω = 0, y(0) = y0.(VI)Here Ψ is a suciently regular obstacle, u ∈ L∞(I × Ω) and f is a “nice” nonlinearity. The optimalcontrol problem has the formmin(y,u)12y − yQ2L2(I ×Ω)+α2u2L2(I ×Ω)such that (y,u) satisfy (VI).(P)We regularize (VI) by a family of appropriate semilinear parabolic differential equations. We discretizethose equations by a dG(0)cG(1) scheme. We deduce a sharp L∞ discretization error estimate,which is independent of the regularization parameter. We can now combine this discretization errorestimate for (VI) with an regularization error estimate to give a complete a priori error estimatebetween the solution of the variational inequality and the regularized, discretized solution. We thentransfer those results to the optimal control problem (P) to obtain estimates.Maximal discrete sparsity in parabolic optimal control with measuresEvelyn HerbergUniversität HamburgWe consider a parabolic optimal control problem governed by space-time measure controls. Twoapproaches to discretize this problem will be compared. The rst approach has been consideredby Eduardo Casas and Karl Kunisch and employs a discontinuous Galerkin method for the statediscretization where controls are discretized piecewise constant in time and by Dirac measuresconcentrated in the nite element nodes in space. In the second approach we use variationaldiscretization of the control problem utilizing a Petrov-Galerkin approximation of the state whichinduces controls that are composed of Dirac measures in space and time, i.e. variational discretecontrols that are Dirac measures concentrated in nite element nodes with respect to space, andconcentrated on the grid points of the time integration scheme with respect to time. The latterapproach then yields maximal sparsity in space-time on the discrete level. Numerical experimentsshow the differences of the two approaches.92
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CONTRIBUTED TALKSAn inexact bundle algorithm for nonconvex nondifferentiable functions in Hilbert spaceLukas HertleinTechnische Universität MünchenMotivated by optimal control problems for elliptic variational inequalities we develop an inexactbundle method for nonsmooth nonconvex optimization subject to general convex constraints.The proposed method requires only approximate (i.e., inexact) evaluations of the cost functionand of an element of Clarke’s subdifferential. The algorithm allows for incorporating curvatureinformation while aggregation techniques ensure that the piecewise quadratic subproblem canbe solved eciently. A global convergence theory in a suitable innite-dimensional Hilbert spacesetting is presented. We discuss the application of our framework to optimal control of the obstacleproblem and present preliminary numerical results.A Lagrange multiplier method for semilinear elliptic state constrained optimal control prob-lemsVeronika KarlJulius-Maximilians-Universität WürzburgIn this paper we apply an augmented Lagrange method to a class of non-convex optimal controlproblems with pointwise state constraints. These control problems are non-convex due to thenonlinearity of the state equation. We show strong convergence on subsequences of the primalvariables to a KKT point of the original problem as well as weak convergence of the adjoint statesand weak* convergence of the multipliers associated to the state constraint. Under second orderconditions, we reach convergence towards local solutions. Further we are able to show that for everyiteration of our algorithm there exist KKT points of the arising augmented Lagrange sub-problem inarbitrary small neighborhoods of local solutions of the original problem, provided that the penaltyparameter of the applied augmented Lagrange term is suciently large. We present some numericalexamples to illustrate our results.External polyhedral estimates of reachable sets of discrete-time systems with integral boundson additive termsElena K. KostousovaKrasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sci-encesThe reachability problem is one of the fundamental problems of the mathematical control theory.Since exact construction of reachable sets is usually a very complicated problem, different numericalmethods were devised for approximations. In particular, different techniques were developedfor estimating reachable sets by domains of some xed shape such as ellipsoids, parallelepipedsand some others. Most of the results in this direction were obtained for linear systems with hard93
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CONTRIBUTED TALKSbounds on additive input terms. Here we study linear and bilinear discrete-time systems for the casewhen additive input terms are restricted by integral nonquadratic constraints and initial states arerestricted by parallelepiped-valued constraints. We consider such a class of bilinear systems wherehard interval bounds on the coecients (in other terms, on the matrices) of the system are imposed.The main attention is paid to time-invariant systems. We construct external parallelepiped-valued(shorter, polyhedral) estimates of reachable sets of the considered systems. First, algorithms forconstructing families of touching external estimates with constant orientation matrices for reachablesets of linear time-invariant systems are developed. Then techniques for constructing polyhedralestimates for the case of bilinear systems are proposed. The research is supported by RFBR, Project18-01-00544a.Generalized solution concepts to the Ericksen–Leslie equations modeling liquid crystal owRobert LasarzikWIAS BerlinThis talk focuses on a system of nonlinear partial differential equations, the so-called Ericksen–Leslie equations, describing the ow of liquid crystals. Whether solutions to such a system exist,is a challenging question by itself. Different partial answers are presented in the form of differentsolvability concepts. The plethora of different solution concepts ranges from strong over weak andmeasure-valued solutions to dissipative solutions. These concepts are introduced and importantproperties are reviewed. Additionally, the numerical approximation is discussed and ideas for anoptimal control scheme for the Ericksen–Leslie system are presented.Regularity assumptions for partial outer convexication of semilinear-constrained MIOCPsPaul MannsTU BraunschweigMixed-Integer-Optimal-Control-Problems (MIOCPs) being constrained by time-dependent differentialequations can be relaxed by continuous OCPs by means of partial outer convexication. Then, so-called Sum-Up-Rounding algorithms can be used to approximate feasible points of the relaxed,convexied continuous problems with binary ones that are feasible up to some arbitrarily smallconstant. This approximation property of the relaxed problem has been known to hold for ODEs anda class of semilinear PDEs with its linear part being the generator of a strongly continuous semigroupunder some uniform estimates on the derivative of a term arising in the variation of constantsformula giving mild (and classical) solutions of the inhomogeneous abstract Cauchy problem. Weare going to show that one can relax these requirements from differentiability to uniform continuityfor both cases.94
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CONTRIBUTED TALKSLimit analysis shape optimization for von Mises criterionAymeric MauryUniversité LavalConsidering a structure made of a perfectly plastic material it is known that the stress problem iswell posed as long as an admissible stress exists for the given loads F . The goal of limit analysis is tocompute λ > 0 such that λF is the threshold load between the existence and the non-existence of astress eld.From a mathematical point of view, the problem takes the form of a saddle point with respect todisplacement u and stress. For the von Mises yield function it is possible to write this problem as aninmum problem with respect to u taken in BD since it can present surface discontinuities.Our goal is to perform a sensitivity analysis of λ and compute the shape gradient. The non-uniquessof u and its intrinsic lack of smoothness urge the use of a regularization. We choose the Norton-Hoff-Friaa regularisation which admits a unique solution up ∈ W 1,p(Ω)d . We discuss the convergenceof this model as p → 1. Then, reformulating the problem with respect to up into a saddle pointproblem, we compute the shape derivative and present some numerical experiments.Exploring sparsity in image and data domains in photoacoustic tomographyBolin PanUCLIn photoacoustic tomography, the acoustic propagation time across the specimen is the ultimate limiton sequential sampling frequency. Any further speed-up can only be obtained by parallel acquisitionand subsampling/compressed sensing. In this talk, we consider the photoacoustic reconstructionproblem from compressed/subsampled measurements utilizing the sparsity of photoacoustic dataor photoacoustic image in the Curvelet frame. We discuss the relative merits of the two approachesand demonstrate the results on 3D simulated and real data.On nding of initial conditions of equations of exural-torsional vibrations of a barAysel RamazanovaUni Duisburg-EssenThe problem of nding the initial conditions in the boundary-value problem for the system ofexural-torsional vibrations of a bar with additional conditions on the straight line is brought to theoptimal control problem and is studied by the methods of optimal control theory. We show that thefunctional (10) is differentiable. The gradient of the functional is calculated and using the gradientexpression a necessary and sucient optimality condition were proved.95
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CONTRIBUTED TALKSUniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equa-tionJose UrquizaUniversité LavalFor a Legendre-Galerkin semi-discretization of the 1-D wave equation, the high frequency compo-nents of the numerical solution prevent us from obtaining the boundary observability (inequality),uniformly with regard to the discretization parameter. A classical Fourier ltering that lters outthe high frequencies is sucient to recover the uniform observability. Unfortunately, this remedyneeds to compute all the frequencies of the underlying system. We will present several cheaperalternative remedies. Among them: a spectral ltering technique, a mixed formulation of the 1-Dwave equation, and Nitsche’s method to append Dirichlet type boundary conditions. Numericalresults will be presented in order to show the effectiveness of these remedies. This is a joint workwith Ludovick Gagnon (Université de Nice Sophia-Antipolis).Fully discrete scheme for Bean’s critical-state model with temperature effects in supercon-ductivityMalte WincklerUniversität Duisburg-EssenThis talk is devoted to the electromagnetic phenomenon in type-II superconductivity, which occursin many technological applications nowadays. Focusing on the original formulation of Maxwelltogether with Bean’s critical-state constitutive law, we obtain a non-smooth hyperbolic Maxwellsystem. After deriving a suitable formulation, we address the numerical analysis of hyperbolic mixedvariational inequalities of the second kind for the governing evolutionary Maxwell’s equations. Atrst, we propose a fully discrete scheme based on the implicit Euler in time and a mixed FEM inspace consisting of Nédélec’s edge elements for the electric eld and piecewise constant elementsfor the magnetic induction. As a main result, we prove strong convergence for the fully discretescheme under physically reasonable regularity assumptions on the initial data. In particular, thisresult yields a solution to the variational inequality satisfying the physical Gauss law. After presentinga priori error estimates, we close our talk by demonstrating some numerical results, which conrmnot only our theoretical ndings but also the physical effects in type-II superconductivity