Block Krylov Subspace Recycling for Shifted Systems with Unrelated Right-Hand Sides

Many Krylov subspace methods for shifted linear systems take advantage of the invariance of the Krylov subspace under a shift of the matrix. However, exploiting this fact in the non-Hermitian case introduces restrictions; e.g., initial residuals must be collinear and this collinearity must be maintained at restart. Thus we cannot simultaneously solve shifted systems with unrelated right-hand sides using this strategy, and all shifted residuals cannot be simultaneously minimized over a Krylov subspace such that collinearity is maintained. It has been shown that this renders them generally incompatible with techniques of subspace recycling [K. M. Soodhalter, D. B. Szyld, and F. Xue, Appl. Numer. Math., 81C (2014), pp. 105--118]. This problem, however, can be overcome. By interpreting a family of shifted systems as one Sylvester equation, we can take advantage of the known “shift invariance” of the Krylov subspace generated by the Sylvester operator. Thus we can simultaneously solve all systems over one bloc...

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