# Deflated Krylov Subspace Methods for Nonlinear Schrödinger Problems

André Gaul^{*}

Sequences of linear algebraic systems frequently occur in the numerical solution process of various kinds of problems, e.g. time stepping schemes or Newton's method. Often, the operators in subsequent linear algebraic systems have similar spectral properties. When Krylov subspace methods like CG, MINRES or GMRES are used for solving these sequences, we would like to improve the convergence by recycling valuable information from previous linear systems. This can be achieved by employing a deflated Krylov subspace method. In a deflated method the operator is projected in order to eliminate components that are suspected to hamper convergence, e.g. a part of the spectrum. As an added benefit, deflation can also incorporate spectral information that is explicitly provided. For example, in nonlinear Schrödinger problems an approximation to an eigenvector corresponding to an eigenvalue close to zero can be derived from the theoretical properties of the problem. We present and analyze common deflation strategies and compare it to other approaches.

Mathematics Subject Classification: 65F10 65F08

Keywords: Deflation, Krylov subspace methods, Recycling, nonlinear Schrödinger

Minisymposion: Iterative Methods for Ill-Posed Problems