Generalized minimal residual method for systems with multiple right-hand sides
S. Sukmanyuk, D. Zheltkov, B. Valiakhmetov
TL;DR
This work addresses solving $A\mathbf{x}^{(s)}=\mathbf{b}^{(s)}$ for $s=1,\dots,M$ with a fixed nonsingular $A$ when right-hand sides are not available simultaneously. It extends GMRES by constructing a shared search space $\\mathcal{L}_k$ of dimension $k$ and updating $\\mathbf{x}_k^{(m)}=\\mathbf{x}_0^{(m)}+\\mathbf{y}_k^{(m)}$, with $\\mathbf{r}_k^{(m)}=\\mathbf{b}^{(m)}-A\\mathbf{x}_k^{(m)}$ minimized over $\\mathcal{L}_k$, while representing both $\\mathcal{L}_k$ and $A\\mathcal{L}_k$ in a common orthonormal basis. The main contributions show that $\\dim\\mathcal{L}_k=k$, establish theoretical equivalence to the generalized GCR for multiple RHS, and demonstrate improved robustness and memory efficiency, validated on wave-scattering PDE problems with up to hundreds of RHS. The numerical results indicate the method can achieve tighter tolerances than GCR and seed GMRES, while maintaining a compact, orthonormal framework. This provides a practical, robust approach for sequential right-hand sides in large-scale PDE discretizations and related problems.
Abstract
A new variant of the GMRES method is presented for solving linear systems with the same matrix and subsequently obtained multiple right-hand sides. The new method keeps such properties of the classical GMRES algorithm as follows. Both bases of the search space and its image are maintained orthonormal that increases the robustness of the method. Moreover there is no need to store both bases since they are effectively represented within a common basis. Along with it our method is theoretically equivalent to the GCR method extended for a case of multiple right-hand sides but is more numerically robust and requires less memory. The main result of the paper is a mechanism of adding an arbitrary direction vector to the search space that can be easily adopted for flexible GMRES or GMRES with deflated restarting.