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Efficient preconditioning for sequences of parametric complex symmetric linear systems. (English) Zbl 1066.65048

Summary: The solution of sequences of complex symmetric linear systems of the form \(A_jx_j=b_j\), \(j=0,\dots,s\), \(A_j=A+\alpha_jE_j\), \(A\) Hermitian, \(E_0,\dots,E_s\) complex diagonal matrices and \(\alpha_0,\dots,\alpha_s\) scalar complex parameters arise in a variety of challenging problems. This is the case of time dependent partial differential equations; 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 \(A_j\) is complex symmetric. The case \(A\) Hermitian positive semidefinite, \(\text{Re} (\alpha_j)\geq 0\) and such that the diagonal entries of \(E_j\) \(j=0,\dots,s\) 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.

MSC:

65F35 Numerical computation of matrix norms, conditioning, scaling
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35K05 Heat equation
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