A spectrally preconditioned and initially deflated variant of the restarted block GMRES method for solving multiple right-hand sides linear systems

The solution of large linear systems with multiple right-hand sides given simultaneously is required in many large-scale scientific and engineering applications modelled by either partial differential or boundary integral equations. Block Krylov subspace methods are attractive to use for this problem class as they can overcome the memory bottleneck of direct methods and they perform block matrix-vector products, achieving high computational efficiency on modern cache-based computer architectures. In this paper we introduce variants of the block GMRES method that combine initial deflation and eigenvalue recycling strategies to remedy some typical convergence problems of block Krylov solvers. The new class of block iterative solvers has the ability to handle the approximate linear dependence of the block of right-hand sides and exploits approximate invariant subspaces recycled over the iterations to mitigate the bad effects that small eigenvalues can have on the convergence, by adapting an existing preconditioner. We illustrate the numerical behavior of the spectrally updated and initially deflated block GMRES method on a set of linear systems arising from the discretization of the Dirac equation and of boundary integral equations in electromagnetics scattering.