About: On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
Description	Let $A$ be a nonsingular complex matrix and $b$ and $c$ be complex vectors. We investigates approaches for efficient approximations of the bilinear form $c^A^{-1}b$. Equivalently, we wish to approximate the scalar value $c^x$, where $x$ solves the linear system $Ax = b$. Here the matrix $A$ can be very large or its elements can be too costly to compute so that $A$ is not explicitly available and it is used only in the form of the matrix-vector product. Therefore a direct method is not an option. For $A$ Hermitian positive definite, $b^A^{-1}b$ can be efficiently approximated as a by-product of the conjugate-gradient iterations, which is mathematically equivalent to the matching moment approximations computed via the Gauss–Christoffel quadrature. We propose a new method using the biconjugate gradient iterations which is applicable to the general complex case. The proposed approach is compared with existing ones using analytic arguments and numerical experiments. Let $A$ be a nonsingular complex matrix and $b$ and $c$ be complex vectors. We investigates approaches for efficient approximations of the bilinear form $c^A^{-1}b$. Equivalently, we wish to approximate the scalar value $c^x$, where $x$ solves the linear system $Ax = b$. Here the matrix $A$ can be very large or its elements can be too costly to compute so that $A$ is not explicitly available and it is used only in the form of the matrix-vector product. Therefore a direct method is not an option. For $A$ Hermitian positive definite, $b^A^{-1}b$ can be efficiently approximated as a by-product of the conjugate-gradient iterations, which is mathematically equivalent to the matching moment approximations computed via the Gauss–Christoffel quadrature. We propose a new method using the biconjugate gradient iterations which is applicable to the general complex case. The proposed approach is compared with existing ones using analytic arguments and numerical experiments. (en)
Title	On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b (en)
skos:prefLabel	On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b (en)
skos:notation	RIV/67985807:_____/11:00358802!RIV12-AV0-67985807
http://linked.open...avai/predkladatel	Ústav informatiky AV ČR, v.v.i.
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(GA201/09/0917), P(IAA100300802), Z(AV0Z10300504), Z(MSM0021620839)
http://linked.open...iv/cisloPeriodika	2
http://linked.open...vai/riv/dodaniDat	2012
http://linked.open...aciTvurceVysledku	Tichý, Petr
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Ústav informatiky AV ČR, v. v. i.
http://linked.open...dnocenehoVysledku	218006
http://linked.open...ai/riv/idVysledku	RIV/67985807:_____/11:00358802
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	bilinear forms; scattering amplitude; method of moments; Krylov subspace methods; conjugate gradient method; biconjugate gradient method; Lanczos algorithm; Arnoldi algorithm; Gauss-Christoffel quadrature; model reduction (en)
http://linked.open.../riv/klicoveSlovo	Krylov subspace methods Arnoldi algorithm Gauss-Christoffel quadrature Lanczos algorithm biconjugate gradient method bilinear forms model reduction scattering amplitude conjugate gradient method method of moments
http://linked.open...odStatuVydavatele	US - Spojené státy americké
http://linked.open...ontrolniKodProRIV	[B5714B90A55B]
http://linked.open...i/riv/nazevZdroje	SIAM Journal on Scientific Computing
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	2 (xsd:int)
http://linked.open...vavai/riv/projekt	Theory of Krylov subspace methods and its relationship to other mathematical disciplines Mathematical and computer analysis of the evolution processes in nonlinear viscoelastic fluid-like materials
http://linked.open...UplatneniVysledku	2011
http://linked.open...v/svazekPeriodika	33
http://linked.open...iv/tvurceVysledku	Tichý, Petr Strakoš, Z.
http://linked.open...ain/vavai/riv/wos	000289973500005
http://linked.open...n/vavai/riv/zamer	Informatika pro informační společnost: modely, algoritmy, aplikace Metody moderní matematiky a jejich aplikace
issn	1064-8275
number of pages	23 (xsd:int)
http://bibframe.org/vocab/doi	10.1137/090753723
is http://linked.open...avai/riv/vysledek of	On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b

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