Efficient preconditioning for sequences of parametric complex symmetric linear systems

Abstract

Solution of sequences of complex symmetric linear systems of the form Ajxj = bj, j = 0,..., s, Aj = A + αjEj, A Hermitian, E0, ..., E a complex diagonal matrices and α0, ..., αa scalar complex parameters arise in a variety of challenging problems. This is the case of time dependent PDEs; lattice gauge computations in quantum chromodynamics; the Helmholtz equation; shift-and-invert and Jacobi-Davidson algorithms for large-scale eigenvalue calculations; problems in control theory and many others. If A is symmetric and has real entries then Aj is complex symmetric. The case A Hermitian positive semideflnite, Re(αj) ≥ 0 and such that the diagonal entries of E j, j = 0,..., s have nonnegative real part is considered here. Some strategies based on the update of incomplete factorizations of the matrix A and A-1 are introduced and analyzed. The numerical solution of sequences of algebraic linear systems from the discretization of the real and complex Helmholtz equation and of the diffusion equation in a rectangle illustrate the performance of the proposed approaches