## 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 $A x = b$ 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 $X$.

Full Text (PDF) [115 KB], BibTeX

### Key words

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

65F10, 65F08