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Zbl 1240.65097
Rees, Tyrone; Stoll, Martin
Block-triangular preconditioners for PDE-constrained optimization. (English)
[J] Numer. Linear Algebra Appl. 17, No. 6, 977-996 (2010). ISSN 1070-5325; ISSN 1099-1506/e

Summary: 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 that uses self-adjointness of the preconditioned system in a non-standard inner product. We show that, when the Chebyshev semi-iteration is used as a preconditioner for the relevant matrix blocks involving the finite element mass matrix, 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 that gives convergence bounds in the nonstandard inner product and illustrates their competitiveness on a number of computed examples.

MSC 2000:: *65F08
65F10 Iterative methods for linear systems
65K10 Optimization techniques (numerical methods)
49J20 Optimal control problems with PDE (existence)

Keywords: saddle point problems; linear systems; Krylov subspace method; preconditioning; PDE-constrained optimization; numerical examples; conjugate gradient-type method; Bramble-Pasciak method; Chebyshev semi-iteration

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