
Oberwolfach Seminar
Mathematics of PDE constrained optimization
Abstract:
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 modelbased numerical simulation to modelbased 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 socalled adjoint equation emerging in the optimization theoretic characterization
(KarushKuhnTuckerKKTconditions) of optimal solutions. The investigation of
regularity properties of the corresponding adjoint state requires an indepth 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 10.000.000.000. 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
stateoftheart 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 socalled 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 meshindependence and, consequently, uniform convergence
behavior. We discuss recent research which shows that semismooth 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 infinitedimensional 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 residualtype and goaloriented 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.
Schedule:
Mo 0910.30: Hintermüller/Hoppe/Hinze: Introduction
1112.30: Hoppe: Analysis (basic concepts) Slides
1618.00: Discussions/round tables of the attendees
Di 0910.30: Hintermüller: Algorithmic concepts (basis algorithms) Slides
1112.30: Hinze: Discrete concepts (common approaches) Slides
1618.00: Discussions, case studies
Mi 0910.30: Hintermüller: Analysis (concepts of nonsmooth functional analysis)
1112.30: Hintermüller: Algorithmic concepts (treatment of constraints)
Do 0910.30: Hinze: Discrete concepts (appropriate Ansätze for the variables involved) Slides
1112.30: Hinze: Algorithmic concepts and their discrete realization (interplay of optimization and discretization) Slides
1618.00: Discussions, case studies
Fr 0910.30: Hoppe: Adaptive concepts Slides
1112.30: Hoppe: Adaptivity in the presence of pointwise constraints
1416 Current research topics in the field, open questions, outlook, future research directions
 