Department Mathematik 
University of Hamburg
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Oberwolfach Seminar

Mathematics of PDE constrained optimization

Organisers: Michael Hintermüller, Michael Hinze, Ronald H.W. Hoppe

Optimization problems subject to constraints given by partial differential equations (PDEs) with additional constraints on the control and/or state variables belong to the most challenging problem classes in industrial, medical and economical applications, where the transition from model-based numerical simulation to model-based design and optimal control is crucial. In the mathematical treatment of such optimization problems the interaction of mathematical modeling, analysis, optimization techniques and numerical simulation plays a central role. For instance, after defining an appropriate PDE model of the underlying phenomenon the choice of a suitable objective functional to achieve a desired output is paramount. Its form influences the so-called adjoint equation emerging in the optimization theoretic characterization (Karush-Kuhn-Tucker-KKT-conditions) of optimal solutions. The investigation of regularity properties of the corresponding adjoint state requires an in-depth analysis combing the underlying PDE (or state system) and the potential additional constraints on controls and states. These existence and regularity results are then relevant for the numerical treatment of the problem, for instance with respect to a suitable discretization of the various quantities involved in the KKT conditions. Typically, after proper discretization the number of optimization variables in associated numerical schemes varies between 1.000 and It is only very recently that the enormous advances in computing power have made it possible to handle problems of this size. For the success of the overall task it is crucial to utilize and further explore the specific mathematical structure of the underlying optimization problems with PDE constraints and to develop new approaches concerning mathematical analysis and modeling, structure exploiting algorithmic concepts, and tailored discretization, with a special focus on prototype applications governed by nonlinear PDE systems. In view of the recent research advances and future challenges, it appears now timely to present state-of-the-art mathematical concepts in the field of PDE constrained optimization to a considerable number of interested young researchers. With the proposed seminar we intend to provide a concise introduction to the state of the art in this prosperous research field. The target group of attendees contains researchers on advanced graduate and early post doc level. The topics of the seminar include

  • Modeling: PDE constrained optimization is often driven by applications, where mathematical modeling plays a central role.Within this seminar the derivation of the PDE system, the incorporation of realistic so-called hard constraints on controls and/or states, and the proper choice of objective functionals are considered as central building blocks of the modeling part related to the optimization problem.

  • Analysis: PDE constrained optimization problems give rise to optimality conditions which contain nonlinear systems of coupled PDEs, and which lack classical differentiability if hard constraints on controls and/or states are present.Within this seminar we discuss tailored analytical techniques for the mathematical treatment of such systems.

  • Algorithmic concepts: PDE constrained optimization problems in general are large scale, where a function evaluation is associated to a PDE system solve. For this reason structure exploiting algorithmic concepts currently are developed whose numerical effort is in the range of only a small multiple of the numerical effort needed for a function evaluation.Within this seminar we show that it is crucial to derive algorithms which make it possible to achieve mesh-independence and, consequently, uniform convergence behavior. We discuss recent research which shows that semi-smooth Newton concepts as well as interior point and penalization methods in optimization combined with multigrid methods from PDEs bear the potential to yield highly efficient algorithms for the numerical solution of complex optimization problems including PDEs with state and control constraints.

  • Interaction of optimization and discretization, adaptivity: To achieve solution concepts with optimal complexity necessitates the development of discrete concepts which allow to conserve as much as possible the structure of the infinite-dimensional optimization problem on the discrete level. Within this seminar, tailored discretization approaches are discussed with respect to their impact on optimization algorithms. Special focus is taken here on the handling of hard constraints on the controls and/or states, on aspects of tuning relaxation parameters and mesh sizes of discretization, and on mesh adaption using residual-type and goal-oriented a posteriori concepts.

    In addition to the topic areas highlighted above, we plan to offer special sessions and round table discussions on various case studies, where participants exchange their reflections on questions and problems posed by the lecturers in earlier sessions. At the end of the seminar week open problems and future research directions will be discussed as well.


  • Mo 09-10.30: Hintermüller/Hoppe/Hinze: Introduction
    11-12.30: Hoppe: Analysis (basic concepts) Slides
    16-18.00: Discussions/round tables of the attendees

  • Di 09-10.30: Hintermüller: Algorithmic concepts (basis algorithms) Slides
    11-12.30: Hinze: Discrete concepts (common approaches) Slides
    16-18.00: Discussions, case studies

  • Mi 09-10.30: Hintermüller: Analysis (concepts of non-smooth functional analysis)
    11-12.30: Hintermüller: Algorithmic concepts (treatment of constraints)

  • Do 09-10.30: Hinze: Discrete concepts (appropriate Ansätze for the variables involved) Slides
    11-12.30: Hinze: Algorithmic concepts and their discrete realization (interplay of optimization and discretization) Slides
    16-18.00: Discussions, case studies

  • Fr 09-10.30: Hoppe: Adaptive concepts Slides
    11-12.30: Hoppe: Adaptivity in the presence of pointwise constraints
    14-16 Current research topics in the field, open questions, outlook, future research directions

  •  Seitenanfang  Impressum 2006-07-31, Michael Hinze