Krylov methods for solving a sequence of large systems of linear equations.


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Consider solving a sequence of linear systems A_{(i)}x^{(i)}=b^{(i)}, i=1, 2, ... where A₍ᵢ₎ ϵℂⁿᵡⁿ and b⁽ⁱ⁾ϵℂⁿ using some variations of Krylov subspace methods, like GMRES. For a single system Ax=b, it is well-known that the eigenvectors of the coefficient matrix A can be used to speed up the convergence of GMRES by deflating the corresponding eigenvalues. In this dissertation, we propose a deflation-based algorithm that utilizes the eigenvalue and eigenvector information obtained from one system to improve the convergence of GMRES for solving the subsequent systems. When the change in the system is small enough, the algorithm will REUSE the eigenvectors from the previous system to deflate the small eigenvalues from the new system via a projection to speed up convergence. When the change is significant enough that projection loses effectiveness, the algorithm will RECYCLE the eigenvectors from the previous system by adding them to the new Krylov subspace, thus improving them so that they can be suitable candidates for deflation once again. If the system has changed too much, or the new system is completely unrelated to the previous system, the algorithm will REGENERATE a new set of eigenvectors to help with deflation.



GMRES. Krylov subspace. Deflation. GMRES-DR. GMRES-E. Subspace recycling.