Galerkin projection methods for solving multiple linear systems

In this paper, we consider using conjugate gradient (CG) methods for solvin g multiple linear systems A((i)) x((i)) = b((i)), for 1 less than or equal to i less than or equal to s, where the coefficient matrices A((i)) and the right-hand sides b (i) are different in general. In particular, we focus o n the seed projection method which generates a Krylov subspace from a set o f 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 wh ole process is repeated until all the systems are solved. Most papers in th e literature [T. F. Chan and W. L. Wan, SIAM J. Sci. Comput., 18 (1997), pp . 1698-1721; B. Parlett Linear Algebra Appl., 29 (1980), pp. 323-346; Y. Sa ad, Math. Comp., 48 (1987), pp. 651-662; V. Simoncini and E. Gallopoulos, S IAM J. Sci. Comput., 16 (1995), pp. 917-933; C. Smith, A. Peterson, and R. Mittra, IEEE Trans. Antennas and Propagation, 37 (1989), pp. 1490-1493] con sidered 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 so lve multiple linear systems with varying coefficient matrices and right-han d sides. A theoretical error bound is given for the approximation obtained from a projection process onto a Krylov subspace generated from solving a p revious linear system. Finally, numerical results for multiple linear syste ms arising from image restorations and recursive least squares computations are reported to illustrate the effectiveness of the method.