Geodesic
A note on preconditioners and scalar products in Krylov subspace methods for self-adjoint problems in Hilbert space
Electronic transactions on numerical analysis, Tome 41 (2014), pp. 13-20
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 $X$.
Classification : 65F10, 65F08
Keywords: Krylov subspace methods, preconditioners, scalar products, Hilbert spaces, Riesz isomorphism 
@article{ETNA_2014__41__a24,
     author = {G\"unnel,  Andreas and Herzog,  Roland and Sachs,  Ekkehard},
     title = {A note on preconditioners and scalar products in {Krylov} subspace methods for self-adjoint problems in {Hilbert} space},
     journal = {Electronic transactions on numerical analysis},
     pages = {13--20},
     year = {2014},
     volume = {41},
     zbl = {1295.65062},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2014__41__a24/}
}
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%A Sachs,  Ekkehard
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%J Electronic transactions on numerical analysis
%D 2014
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%F ETNA_2014__41__a24
Günnel,  Andreas; Herzog,  Roland; Sachs,  Ekkehard. A note on preconditioners and scalar products in Krylov subspace methods for self-adjoint problems in Hilbert space. Electronic transactions on numerical analysis, Tome 41 (2014), pp. 13-20. http://geodesic.mathdoc.fr/item/ETNA_2014__41__a24/