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


The conjugate gradient and minimal residual methods for the solution of linear systems $A x = b$ 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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