Abstract:
This paper presents a block extension of the generalized minimum residual method (GMRES) with a new block reduction technology. Unlike the currently known methods, a block can be reduced not only when it has degenerated, but also when a part of the residuals converges with the required accuracy or when the residuals become linearly dependent with a given accuracy. In addition, the method makes it possible to continue the process when adding new right-hand sides. At the same time, after reducing the block and adding new right-hand sides, the method retains its compact form and low complexity. This makes it possible to use it in cases where not all the right-hand sides are known in advance. It also makes it possible to limit the maximum block size, thus balancing between performance and the final dimension of the space, i.e. the required memory. Numerical experiments confirm the high efficiency of the method in comparison with the non-block extension of GMRES and its naive block generalization.
Key words:linear systems with many right-hand sides, block krylov methods, generalized minimum residual method (GMRES), block reduction.
