Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

Poisson and convection-diffusion control

Abstract

Saddle point matrices of a special structure arise in optimal control problems. In this paper we consider distributed optimal control for various types of scalar stationary partial differential equations and compare the efficiency of several numerical solution methods. We test the particular case when the arising linear system can be compressed after eliminating the control function. In that case, a system arises in a form which enables application of an efficient block matrix preconditioner that previously has been applied to solve complex-valued systems in real arithmetic. Under certain assumptions the condition number of the so preconditioned matrix is bounded by 2. The numerical and computational efficiency of the method in terms of number of iterations and elapsed time is favourably compared with other published methods.

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Correspondence to Maya Neytcheva.

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Axelsson, O., Farouq, S. & Neytcheva, M. Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems. Numer Algor 73, 631–663 (2016). https://doi.org/10.1007/s11075-016-0111-1

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Keywords

  • PDE-constrained optimization problems
  • Finite elements
  • Iterative solution methods
  • Preconditioning