Volume 41, pp. 13-20, 2014.

A note on preconditioners and scalar products in Krylov subspace methods for self-adjoint problems in Hilbert space

Andreas Günnel, Roland Herzog, and Ekkehard Sachs

Abstract

The conjugate gradient and minimal residual methods for the solution of linear systems Ax=b are considered. The operator A is bounded and self-adjoint and maps a Hilbert space X into its dual X. This setting is natural for variational problems such as those involving linear partial differential equations. The derivation of the two methods in Hilbert spaces shows that the choice of a preconditioner is equivalent to the choice of the scalar product in X.

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Key words

Krylov subspace methods, preconditioners, scalar products, Hilbert spaces, Riesz isomorphism

AMS subject classifications

65F10, 65F08

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