Projections, Deflation, and Multigrid for Nonsymmetric Matrices
Luis García Ramos, René Kehl, Reinhard Nabben · SIAM Journal on Matrix Analysis and Applications · 2020
Deflation is a well-known technique to accelerate Krylov subspace methods for solving linear systems of equations. In contrast to preconditioning, in deflation methods singular systems have to be solved. The original system is multiplied by a projection which leads to a singular linear system which can be more favorable for a Krylov subspace method. Deflation methods are also closely related to multigrid or multilevel and domain decomposition methods. Here, we continue the analysis of deflation methods with arbitrary deflation subspaces applied to nonsymmetric matrices. We give some characterizations of the spectrum of the deflated matrix, and we prove an equivalence theorem for two types of deflation methods. New results on projections, established here, allow simple proofs for the spectral relation between deflation and multigrid methods.