Galerkin Projection Methods for Solving Multiple Linear Systems

In this paper, we consider using conjugate gradient (CG) methods for solving multiple linear systems $A^{(i)} x^{(i)} = b^{(i)},$ for $1 \le i \le s,$ where the coefficient matrices $A^{(i)}$ and the right-hand sides $b^{(i)}$ are different in general.\ In particular, we focus on the seed projection method which generates a Krylov subspace from a set of direction vectors obtained by solving one of the systems, called the seed system, by the CG method and then projects the residuals of other systems onto the generated Krylov subspace to get the approximate solutions.\ The whole process is repeated until all the systems are solved.\ Most papers in the literature [T.\ F.\ Chan and W.\ L.\ Wan, {\it SIAM J.\ Sci.\ Comput.}, 18 (1997), pp.\ 1698--1721; B.\ Parlett {\it Linear Algebra Appl.}, 29 (1980), pp.\ 323--346; Y.\ Saad, {\it Math.\ Comp.}, 48 (1987), pp.\ 651--662; V.\ Simoncini and E.\ Gallopoulos, {\it SIAM J.\ Sci.\ Comput.}, 16 (1995), pp.\ 917--933; C.\ Smith, A.\ Peterson, and R.\ Mittra, {\it IEEE Trans.\ Antennas and Propagation}, 37 (1989), pp. 1490--1493] considered only the case where the coefficient matrices $A^{(i)}$ are the same but the right-hand sides are different.\ We extend and analyze the method to solve multiple linear systems with varying coefficient matrices and right-hand sides. A theoretical error bound is given for the approximation obtained from a projection process onto a Krylov subspace generated from solving a previous linear system. Finally, numerical results for multiple linear systems arising from image restorations and recursive least squares computations are reported to illustrate the effectiveness of the method.