Galerkin Projection Methods for Solving Multiple Linear Systems

In this paper, we consider using Galerkin projection methods for solving multiple linear systems A (i) x (i) = b (i) , for 1 i s, where the coeecient matrices A (i) and the right-hand sides b (i) are diierent 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 conjugate gradient (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 6, 19, 21, 23, 24] considered only the case where the coeecient matrices A (i) are the same but the right-hand sides are diierent. We extend the method to solve multiple linear systems with varying coeecient matrices and right-hand sides. We also analyze the method and extend the theoretical result of the projection method for solving linear systems with multiple right-hand sides given in Chan and Wan 6]. 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. Applications of the method to multiple linear systems arising from image restorations and recursive least squares computations are considered. In particular, we show that the the theoretical error bound of the method can be applied to these applications. Finally, numerical results are reported to illustrate the eeectiveness of the Galerkin projection method.