Invited Session Fri.3.H 0111

Friday, 15:15 - 16:45 h, Room: H 0111

Cluster 19: PDE-constrained optimization & multi-level/multi-grid methods [...]

Preconditioning in PDE constrained optimization


Chair: Roland Herzog



Friday, 15:15 - 15:40 h, Room: H 0111, Talk 1

Markus Kollmann
Robust iterative solvers for a class of PDE-constrained optimization problems

Coauthor: Walter Zulehner


In this talk we discuss the construction and analysis of robust solution techniques for saddle point problems with a natural block 2-by-2 structure where the left upper and the right lower block are different mass matrices. For these systems, solvers are discussed. Saddle point systems of this structure are, e.g., optimality systems of optimal control problems where the observation domain differs from the control domain or the linearized systems resulting after applying a semi-smooth Newton method to the nonlinear optimality systems of optimal control problems with inequality constraints on the control or the state. As examples we discuss the distributed elliptic optimal control problem and the distributed optimal control problem for the Stokes equations.
Numerical examples are given which illustrate the theoretical results.



Friday, 15:45 - 16:10 h, Room: H 0111, Talk 2

John Pearson
Iterative solution techniques for Stokes and Navier-Stokes control problems


The development of efficient iterative methods for the solution of PDE-constrained optimization problems is an area of much recent interest in computational mathematics. In this talk, we discuss preconditioned iterative methods for the Stokes and Navier-Stokes control problems, two of the most important problems of this type in fluid dynamics. We detail the Krylov subspace methods used to solve the matrix systems involved, develop the relevant preconditioners using the theory of saddle point matrices, and present analytical and numerical results to demonstrate the effectiveness of our proposed preconditioners in theory and practice.



Friday, 16:15 - 16:40 h, Room: H 0111, Talk 3

Ekkehard Sachs
Reduced order models in preconditioning techniques

Coauthor: Xuancan Ye


The main effort of solving a PDE constrained optimization problem is devoted to solving the corresponding large scale linear system, which is usually sparse and ill conditioned. As a result, a suitable Krylov subspace solver is favourable, if a proper preconditioner is embedded. Other than the commonly used block preconditioners, we exploit knowledge of proper orthogonal decomposition (POD) for preconditioning and achieve some interesting features. Numerical results on nonlinear test problems are presented.


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