Iterative Solutions of Some Scattering Problems I

Gene H. Golub
Computer Science
SCCM
GATES 2B MC 9025
Stanford, CA 94305-9025
golub@sccm.stan.ed
http://www-sccm.stanford.edu/Faculty/Golub.html

Gerald N. Minerbo
Schlumberger-Sugar Land Product Center
P.O. Box 2175,
Houston, TX 77252-2175
GMINERBO@sugar-land.anadrill.slb.com

Paul Saylor
NCSA
Dept. of Computer Science
University of Illinois, Urbana-Champaign
Urbana, IL 61801
saylor@cs.uiuc.edu
http://www.uiuc.edu/ph/www/p-saylor

Abstract:
In many situations, scattering problems require computing c^T A^{-1} b, where the vectors b and c represent the transmitter and receiver, respectively. Problems of this type arise in acoustics, electromagnetics, radar, optics, and quantum mechanics. Often A is a complex symmetric matrix, whose imaginary part is a positive definite matrix. Solving Ax = b corresponds to computing the field everywhere. Computing c^T A^{-1} b corresponds to estimating the receiver signal, which should be easier than finding x accurately everywhere. If x_k is an iterative approximation, c^T x_k is a straightforward estimate. We shall look at other forms with which to estimate the received signal and look at formulating c^T x as a constraint on the solution of Ax = b.

Keywords:
Constraint, Bi-Conjugate Gradient Method, inner products, scatterer