Tyrone Rees

Optimal Solvers for PDE-Constrained Optimization

Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, particularly in problems of design. The solution of such PDE-constrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2- and 3-dimensional Poisson problem is the PDE. The large-dimensional linear systems which result from discretization and which need to be solved are of saddle-point type. We introduce two optimal preconditioners for these systems, which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other PDEs.

Block-triangular preconditioners for PDE-constrained optimization

In this paper we investigate the possibility of using a block triangular preconditioner for saddle point problems arising in PDE constrained optimization. In particular we focus on a conjugate gradient-type method introduced by Bramble and Pasciak which uses self adjointness of the preconditioned system in a non-standard inner product. We show when the Chebyshev semi-iteration is used as a preconditioner for the relevant matrix blocks involving the finite element mass matrix that the main drawback of the Bramble-Pasciak method – the appropriate scaling of the preconditioners – is easily overcome. We present an eigenvalue analysis for the block triangular preconditioners which gives convergence bounds in the non-standard inner product and illustrate their competitiveness on a number of computed examples.

Preconditioning Iterative Methods for the Optimal Control of the Stokes Equations

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.

Additive Schwarz with Variable Weights

For Additive Schwarz preconditioning of nonsymmetric systems, it is proposed to use weights that change from one iteration to the next. At each iteration, weights for all earlier iterations are implicitly chosen to minimize the current residual. This strategy fits the paradigm of the recently proposed multipreconditioned GMRES. Numerical experiments illustrating the potential of the proposed method are presented.

A comparative study of null‐space factorizations for sparse symmetric saddle point systems

Summary Null-space methods for solving saddle point systems of equations have long been used to transform an indefinite system into a symmetric positive definite one of smaller dimension. A number of independent works in the literature have identified that we can interpret a null-space method as a matrix factorization. We review these findings, highlight links between them, and bring them into a unified framework. We also investigate the suitability of using null-space factorizations to derive sparse direct methods and present numerical results for both practical and academic problems.