Schedule of Talks. Monday, Nov :10-15:50 Martin Burger Optimization Problems in Semiconductor Dopant Profiling.

Transcription

1 Schedule of Talks Monday, Nov 21 9:15-9:30 Opening 9:30-10:30 John A. Burns Sensitivity Analysis, Transition to Turbulence and Control 10:30-11:00 Coffee Break 11:00-12:00 Rainer Tichatschke Bregman-function-based regularization of variational inequalities with monotone operators 12:00-13:00 Lunch Break 13:00-14:00 Stefan Ulbrich Parallel All-at-Once methods with Parareal time-domain decomposition for time-dependent PDE-constrained optimization 14:00-14:30 Coffee Break 14:30-15:10 Michael Ulbrich Semismooth Newton methods for optimal control problems with parabolic PDEs and mixed control state constraints 15:10-15:50 Martin Burger Optimization Problems in Semiconductor Dopant Profiling Tuesday, Nov 22 9:30-10:30 Andy Wathen Preconditioning for saddle-point problems 10:30-11:00 Coffee Break 11:00-12:00 Angela Kunoth Multiscale Methods for PDE-Constrained Control Problems: Optimal Preconditioners and Fast Iterative Solvers 12:00-13:00 Lunch Break 13:00-14:00 George Biros A survey of iterative solvers for inverse problems for systems governed by parabolic PDEs 14:00-14:30 Coffee Break 14:30-15:30 Walter Zulehner Uzawa-type and Projection-type Methods for Block-structured Indefinite Linear Systems 1

2 Wednesday, Nov 23 9:30-10:30 Karen E. Willcox An Optimization Framework for Goal-Oriented, Model-Based Reduction of Large-Scale Systems 10:30-11:00 Coffee Break 11:00-12:00 Peter Benner Model reduction for parabolic control systems based on balanced truncation 12:00-13:00 Lunch Break 13:00-14:00 Stefan Volkwein Error estimates for abstract linear-quadratic optimal control problems using POD 14:00-14:30 Coffee Break 14:30-15:10 Arnd Rösch On the quality of numerical solutions for semilinear optimal control problems 15:10-15:50 Boris Vexler Adaptive Space-Time Finite Element Methods for Parabolic Optimization Problems Thursday, Nov 24 9:30-10:30 Michael Hinze Discrete concepts for optimal control problems with time-dependent PDEs 10:30-11:00 Coffee Break 11:00-12:00 Lars Grasedyck Existence of structured solutions of large scale matrix equations and how to compute them in (almost) optimal complexity 12:00-12:30 Friedemann Leibfritz Output feedback controller design for PDE systems 2

3 List of Abstracts Model reduction for parabolic control systems based on balanced truncation Peter Benner Chemnitz University of Technology Mathematics in Industry and Technology Department of Mathematics benner/ We will discuss model reduction techniques for the control of dynamical processes described by parabolic partial differential equations from a systemtheoretic point of view. The methods considered here are based on spatial semi-discretization of the PDE followed by balanced truncation techniques applied to the resulting large-scale system of ordinary differential equations. Several choices of the system Gramians that are used for balancing will be presented. We will discuss open-loop and closed-loop techniques that allow to preserve system properties important for controller design. Furthermore we will discuss an error estimate based on a combination of FEM and model reduction error bounds. We will also discuss how the state of the full-order system can be recovered from the reduced-order model. Several numerical examples will be used to demonstrate the proposed model reduction techniques. A survey of iterative solvers for inverse problems for systems governed by parabolic PDEs George Biros Department of Mechanical Engineering and Applied Mechanics Department of Computer and Information Science University of Pennsylvania biros/ Convection-reaction-diffusion can model many different physical systems in science and engineering. We are interested in devising efficient numerical schemes for inverse and control problems of such systems. In this talk I will analyze a very simple problem: distributed control for the 1D time dependent heat equation. I will derive the optimality conditions, and discuss solution algorithms. The focus point of my presentation will be 3

4 spectral analysis of the reduced Hessian, and of preconditioner variants that can be used as smoothers in a multigrid scheme. Optimization Problems in Semiconductor Dopant Profiling Martin Burger Industrial Mathematics Institute Johannes Kepler University Linz In this talk we discuss some problems related to the identification and design of semiconductor devices. All technological tasks in this context can be formulated as optimization problems with nonlinear systems of partial differential equations. We start from the stationary case, where efficient and surprisingly simple one shot approaches can be constructed by carefully inspecting the structure of the optimality system. Then we proceed to instationary problems and identification problems with a very high number of state equations (of similar complexity as instationary problems) and discuss some solution strategies. Finally, we present a challenging problem that has not yet been attacked by mathematical optimization techniques, namely the optimal control of doping by ion implantation Joint work with H.W.Engl, M.Wolfram (Linz), P.Markowich (Vienna), P.Pietra (Pavia), R.Pinnau (Kaiserslautern), B.Kaltenbacher (Erlangen), H.Ceric, H.Kosina (TU Wien) Sensitivity Analysis, Transition to Turbulence and Control John A. Burns Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University The problem of predicting transition to turbulence in shear flows is a problem with considerable implications in modern fluid dynamics and flow control. Classical linear stability analysis fails to predict the correct transition even for simple Poiseuille flows. During the past ten years several new theories have emerged to explain this age old problem. In addition, some of these theories have been successful in predicting the correct critical Reynolds number for certain flow problems where classical approaches fail. Unlike the classical approaches to hydrodynamic stability, the new theories are based on concepts with roots in modern robust control theory. The basic idea behind this 4

5 body of work is that non-normal systems can be extremely sensitive to small perturbations in initial data, boundary conditions, inputs and parameters and these sensitivities can lead to large transient growth even for exponentially stable systems. It has been suggested that this sensitivity might be used to explain the onset of turbulence. This is a mostly lineartheory. However, even if the linear part of the system is self-adjoint certain non-linear systems can be infinitely sensitive to parameters and boundary conditions. Although the basic idea behind all these theories is extreme sensitivity, it is not yet clear that any single theory will be able to capture the correct physics. In this presentation we discuss several of these scenarios and present examples to suggest how control theory and bifurcation under uncertainty can be employed to address some of the unresolved issues. We illustrate the basic ideas on simple low dimensional model problems and use Burgers equation to demonstrate the importance of sensitive boundary conditions. References [1] E. Allen, J. Burns, D. Gilliam, J. Hill and V. Shubov, The Impact of Finite Precision Arithmetic and Sensitivity on the Numerical Solution of Partial Differential Equations, Math. and Computer Modeling 35 (2002), [2] J. S. Baggett, T. A. Driscoll, and L. N. Trefethen, A Mostly Linear Model of Transition to Turbulence, Physics of Fluids 7 (1995), [3] J. Burns and J. Singler, Feedback Control of Low Dimensional Models of Transition to Turbulence, 44th IEEE Conference on Decision and Control, Seville, Spain, December 2005, accepted. [4] J. Burns and J. Singler, Modeling Transition: New Scenarios, System Sensitivity and Feedback Control, Transition and Turbulence Control, M. Gad-el-Hak and H. M. Tsai, Eds., World Scientific Publishing, 2005, [5] B. Bamieh and M. Dahleh, Energy Amplification in Channel Flows with Stochastic Excitation, Physics of Fluids 13 (2001), [6] Peter J. Schmid and Dan S. Henningson, Stability and Transition in Shear Flows, Springer-Verlag, New York, [7] L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, Hydrodynamic Stability Without Eigenvalues, Science 261 (1993),

6 Existence of structured solutions of large scale matrix equations and how to compute them in (almost) optimal complexity Lars Grasedyck Max-Planck-Institut für Mathematik in den Naturwissenschaften d.html In the first part of the talk we derive representation formulae for the N by N solution matrix of a matrix equation of type Lyapunov, Sylvester and Riccati. These representations allow an efficient approximation by data-sparse matrices of type low-rank or hierarchical matrix. The approximations use only O(N log N) data. In the second part of the talk we will present algorithms that are able to compute the solution of a matrix equation directly in the data-sparse format in O(N log N) complexity. The most efficient method is the multigrid method which relies on a hierarchy of matrix equations that stem from the discretisation of an elliptic partial differential operator. Discrete concepts for optimal control problems with time-dependent PDEs Michael Hinze Fachrichtung Mathematik, Institut für Numerische Mathematik Technische Universität Dresden hinze/ I discuss a variational discretization concept for abstract time-dependent control problems with control constraints. Discretization only is applied to the state variables, which in turn implicitly yields a discretization of the control variables by means of the first order optimality condition. For discrete controls obtained in this way I prove optimal error estimates. As numerical solution algorithms I consider primal-dual active set strategies and semismooth Newton methods, and I also provide numerical examples. 6

7 Multiscale Methods for PDE-Constrained Control Problems: Optimal Preconditioners and Fast Iterative Solvers Angela Kunoth Institut für Angewandte Mathematik und Institut für Numerische Simulation Universität Bonn kunoth/ The fast numerical solution of PDE-constrained control problems provides a formidable challenge, in particular, for problems constrained by time-dependent PDEs. For a single elliptic PDE which may be considered as the core system to be solved in any case, there exist by now three classes of preconditioners, all of which are of multiscale stucture (multigrid, BPX and wavelet preconditioners), which assure uniformly bounded spectral condition numbers of the system matrix independent of the discretization. In my talk I wish to address fast multiscale solvers for control problems constrained by elliptic PDEs with distributed as well as Dirichlet boundary control. I will present the main ingredients for obtaining theoretical estimates which guarantee optimality of the multilevel preconditioners. Together with employing iterative solvers and a nested iteration scheme, I will show that they therefore provide the solution ingredients of the control problem (state, costate and control) up to discretization error accuracy in optimal linear complexity. Corresponding numerical results confirming these estimates will be shown. Finally, I would like to address optimal preconditioners for control problems constrained by parabolic PDEs. Output feedback controller design for PDE systems Friedemann Leibfritz FB IV - Department of Mathematics University of Trier leibfritz/ We consider static output feedback (SOF) control design problems, e. g. SOF H synthesis, and focus the discussion on the numerical solution of SOF problems if the control system is described by partial differential equations (PDEs). The discretization of those problems leads to very large scale non convex and nonlinear semidefinite programs (NSDPs), e. g. the so called H NSDP. We discuss some theoretical and practical difficulties which arise in the solution of such problems. Moreover, we consider some algorithmic strategies for solving the non convex NSDPs and discuss some unstable 7

8 PDE based models which are currently implemented in COMPl e ib 1.1: the COnstrained Matrix-optimization Problem library which contains more than 170 test examples drawn from a variety of control systems engineering applications. In particular, COMPl e ib may serve as a useful benchmark tool for NSDP (including BMI ) and other matrix optimization problem (including linear SDP) solvers. As a byproduct, COMPl e ib can be used as a test environment for parts of control design procedures, e. g. model reduction algorithms and large scale Riccati equation solvers. On the quality of numerical solutions for semilinear optimal control problems Arnd Rösch RICAM, Linz Austrian Academy of Sciences The most numerical methods for optimal control problems base on necessary optimality conditions. Sufficient second-order optimality conditions ensure stability of solutions and local quadratic convergence for Newton type algorthms. However, the verification of sufficient second-order optimality conditions for a specific problem (with unknown solution) is a unsolved problem. In this talk, we will shed light on the numerical verification of optimality conditions. We will present a new condition for the solution of the full discretized problem which guarantees the existence of a local minimizer of the undiscretized problem in a well determined neighborhood of the numerical solution. Morover, it can be shown that this local minimizer of the undiscretized problem fulfills the sufficient second-order optimality conditions. This talk is a joint work with D. Wachsmuth (TU Berlin). Bregman-function-based regularization of variational inequalities with monotone operators Rainer Tichatschke FB IV - Department of Mathematics University of Trier tichatschke/ Some ideas for improvements, extensions and applications of proximal point methods (with non-quadratic regularization functionals) to variational 8

9 inequalities in Hilbert spaces will be considered. These methods are closely related and will be joined in a general framework, which admits a consecutive approximation of the problem data including applications of finite element techniques and the ε-enlargement of set-valued monotone operators. With the use of a reserve of monotonicity of the operator in the variational inequality, the concept of weak regularization by means of Bregmanfunction-based proximal methods is developed. The use of a Bregman function as a regularizator transforms a constrained variational inequality into an unconstrained one, which effects an interior-point algorithm. We analyze convergence under a relaxed error tolerance criterion in the subproblems. Numerical examples, like Signorini problems without and with friction and contact problems will be presented. Semismooth Newton methods for optimal control problems with parabolic PDEs and mixed control state constraints Michael Ulbrich Fachbereich Mathematik, Schwerpunkt Optimierung und Approximation Universität Hamburg We develop and analyze semismooth Newton methods for optimal control problems with parabolic PDEs and mixed control state constraints. This type of constraints can be used to regularize state constraints. Theory as well as numerical results are presented. We plan to have first numerical tests completed to show results for optimal flow control problems. Parallel All-at-Once methods with Parareal time-domain decomposition for time-dependent PDE-constrained optimization Stefan Ulbrich Fachbereich Mathematik, AG10 Technische Universitt Darmstadt The application of classical All-at-Once methods like SQP to the optimal control of nonlinear time-dependent PDEs leads to severe difficulties, since the storage of the state can be too costly or even impossible. 9

10 In this talk we propose a parallel All-at-Once method for the solution of time-dependent PDE-constrained optimization problems. The algorithm is based on a generalized SQP-framework with inexact iterative solvers using the parallel Parareal time-decomposition algorithm. The resulting All-at- Once method avoids the storage problems caused by classical All-at-Once methods and leads to an efficient parallel solver for complex optimal control problems governed by instationary PDEs. The efficiency of the approach is demonstrated by numerical results for the optimal control of parabolic PDEs in 2-D. Adaptive Space-Time Finite Element Methods for Parabolic Optimization Problems Boris Vexler RICAM, Linz Austrian Academy of Sciences In this talk we present an adaptive algorithm for efficient solution of optimization problems governed by parabolic partial differential equations. The discretization of the state equation is based on the space-time finite element method. We derive a posteriori error estimates which assess the error between the solution of the continuous and the discrete optimization problem with respect to a given quantity of interest. In order to set up an efficient adaptive algorithm we separate the influence of the time discretization, the space discretization and the discretization of the control variable. This allows to balance these types of error and successively to improve the accuracy by construction of locally refined meshes for time, space and the control discretizations. We discuss numerial examples illustrating the behaviour of our method. 10

11 Error estimates for abstract linear-quadratic optimal control problems using POD Stefan Volkwein Institute of Mathematics and Scientific Computing University of Graz Michael Hinze Fachrichtung Mathematik, Institut für Numerische Mathematik Technische Universität Dresden hinze/ In this paper we investigate POD discretizations of abstract linear- quadratic optimal control problems with control constraints. We apply the discrete technique developed in [1] and prove error estimates for the corresponding discrete controls, where we combine error estimates for the state and the adjoint system from [2, 3]. Finally, we present numerical examples that illustrate the theoretical results. References [1] Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Computational Optimization and Applications, 30, (2005). [2] Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numerische Mathematik, 90, (2001). [3] Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal., 40, (2002). Preconditioning for saddle-point problems Andy Wathen Oxford University Computing Laboratory Saddle-point problems arise in almost all eqilibrium or extremal problems with constraints. In fluid mechanics/dynamics the conservation of mass equation in incompressible flow is precicely such a constraint. In Optimization, constraints are ubiquitous. Solution algorithms for linear(ized) systems of saddle-point form are therefore important in a number of contexts. For large scale problems, iterative methods are usually required and as is commonly 11

12 the case, the choice of good preconditioning is vital to get acceptably fast convergence. In this talk we will discuss the general structure and look at some of the possible approaches for preconditioned iterative solution of saddle-point problems both in the context of some knowledge of the problem (as in fluids) and also when only the algebraic problem is known (as can often be the case in Optimization) An Optimization Framework for Goal-Oriented, Model-Based Reduction of Large-Scale Systems Karen E. Willcox Department of Aeronautics & Astronautics Massachusetts Institute of Technology Optimization-ready reduced-order models should target an output functional, span a range of dynamic and parametric inputs, and respect the underlying governing equations. Our approach uses a goal-oriented, model-based optimization formulation to determine a projection basis. We propose an efficient solution strategy that borrows concepts from the proper orthogonal decomposition and employs recent methods for optimization of systems governed by partial differential equations to make the approach tractable for large-scale problems. Uzawa-type and Projection-type Methods for Block-structured Indefinite Linear Systems Walter Zulehner Institute of Computational Mathematics Johannes Kepler University Linz Mixed finite element methods for boundary value problems or the discretized KKT conditions for optimization problems with pde constraints typically lead to symmetric indefinite systems of linear equations with a natural block structure. In this talk we will discuss efficient iterative methods for solving such systems. Two classes of methods are considered, which can be viewed as generalizations of the classical Uzawa method and the projection method. These methods can be used either for preconditioning the indefinite system or as smoothers in a multigrid approach. Convergence properties for preconditioned iterative methods as well as smoothing properties for a multigrid approach are discussed. 12