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

0358802 - ÚI 2012 RIV US eng J - Journal Article
Strakoš, Z. - Tichý, Petr
On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b.
SIAM Journal on Scientific Computing. Roč. 33, č. 2 (2011), s. 565-587. ISSN 1064-8275. E-ISSN 1095-7197
R&D Projects: GA AV ČR IAA100300802
Grant - others:GA ČR(CZ) GA201/09/0917; GA AV ČR(CZ) M100300901
Program: GA
Institutional research plan: CEZ:AV0Z10300504
Keywords : 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
Subject RIV: BA - General Mathematics
Impact factor: 1.569, year: 2011

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.
Permanent Link: http://hdl.handle.net/11104/0196736