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Block-triangular preconditioners for PDE-constrained optimization. (English) Zbl 1240.65097
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:
65F08Preconditioners for iterative methods
65F10Iterative methods for linear systems
65K10Optimization techniques (numerical methods)
49J20Optimal control problems with PDE (existence)