A BLOCK PRECONDITIONED HARMONIC PROJECTION METHOD FOR LARGE-SCALE NONLINEAR EIGENVALUE PROBLEMS

We propose a block preconditioned harmonic projection (BPHP) method for solving large-scale nonlinear eigenproblems of the form T (lambda) v = 0. Similar to classical preconditioned eigensolvers such as the locally optimal block preconditioned conjugate gradient method and preconditioned Lanczos, BPHP aims at computing a few eigenvalues of the nonlinear problem close to a specified shift, using preconditioners that enhance the local spectrum, without the need for exact solution of large shifted linear systems. We explore the development of search subspaces, stabilized preconditioning, nonlinear harmonic Rayleigh-Ritz projections, thick restart, and soft deflation capable of resolving linearly dependent eigenvectors. Numerical experiments show that BPHP with a good preconditioner is storage efficient, and it exhibits robust convergence. A moving-window-style partial deflation enables BPHP to reliably compute a large number of eigenvalues.