PRECONDITIONING ITERATIVE METHODS FOR THE OPTIMAL CONTROL OF THE STOKES EQUATIONS

Journal article

Solving problems regarding the optimal control of partial differential equations (PDEs)-also known as PDE-constrained optimization-is a frontier area of numerical analysis. Of particular interest is the problem of flow control, where one would like to effect some desired flow by exerting, for example, an external force. The bottleneck in many current algorithms is the solution of the optimality system-a system of equations in saddle point form that is usually very large and ill conditioned. In this paper we describe two preconditioners-a block diagonal preconditioner for the minimal residual method and a block lower-triangular preconditioner for a nonstandard conjugate gradient method-which can be effective when applied to such problems where the PDEs are the Stokes equations. We consider only distributed control here, although we believe other problems could be treated in the same way. We give numerical results, and we compare these with those obtained by solving the equivalent forward problem using similar techniques. © 2011 Society for Industrial and Applied Mathematics.

Authors: Rees, T; Wathen, A

• Publication status: Published
• Journal: SIAM JOURNAL ON SCIENTIFIC COMPUTING
• Volume: 33
• Issue: 5
• Pages: 2903-2926
• Publication date: 2011-01-01
• DOI: 10.1137/100798491
• EISSN: 1095-7197
• ISSN: 1064-8275
• Language: English
• Keywords: saddle-point problems; preconditioning; Stokes equations; PDE-constrained optimization; linear systems; all-at-once methods; flow control; optimal control