Inner iterations in eigenvalue solvers

We consider inverse iteration-based eigensolvers, which require at each step solving an “inner” linear system. We assume that this linear system is solved by some (preconditioned) Krylov subspace method. In this framework, several approaches are possible, which differ by the linear system to be solved and/or the way the preconditioner is used. This includes methods such as inexact shift-and-invert, inexact Rayleigh quotient iteration, Jacobi–Davidson, generalized Davidson and generalized preconditioned inverse iteration. In this paper, we discuss and compare them, focusing on the evolution of the “outer” convergence (towards the desired eigenpair) according to the numerical effort spent in inner iterations. This gives some advantage to Jacobi–Davidson, and we further introduce simplified variants of this method that perform similarly while being somewhat cheaper. We also introduce a new variant of the shift-and-invert and Rayleigh quotient iteration methods, which cures their main weakness by making a cleverer use of the preconditioner. For each of these approaches, we show how the evolution of the outer convergence may be estimated with parameters readily computable during inner iterations. We also propose a simple test that allows to detect the point from where the outer convergence stagnates because one is close enough to the exact solution of the linear system. We discuss how a practical stopping test may be build on this basis, and we conclude with some numerical experiments.