A Comparative Study of Iterative Solvers Exploiting Spectral Information for SPD Systems

When solving the Symmetric Positive Definite (SPD) linear system Ax = b with the conjugate gradient method, the smallest eigenvalues in the matrix A often slow down the convergence. Consequently if the smallest eigenvalues in A could be somehow “removed”, the convergence may be improved. This observation is of importance even when a preconditioner is used, and some extra techniques might be investigated to futher improve the convergence rate of the conjugate gradient on the given preconditioned system. Several techniques have been proposed in the literature that either consist of updating the preconditioner or enforcing conjugate gradient to work in the orthogonal complement of an invariant subspace associated with smallest eigenvalues. Among these approaches we consider first a two-phase algorithm using a deflation-type idea. In a first stage this algorithm computes a partial spectral decomposition simply using matrix-vector products. More precisely it combines Chebyshev iterations with a block Lanczos procedure to accurately compute an orthogonal basis of the invariant subspace associated with the smallest eigenvalues. Then, the solution on this subspace is computed using a projector while the solution in the orthogonal complement is obtained with Chebyshev iterations that benefit from the reduced condition number. For sake of comparison, this eigen-information is used in combination with other techniques. In particular we consider the deflated version of conjugate gradient. As representative of techniques exploiting the spectral information to update the preconditioner we consider also the approaches that attempt to shift the smallest eigenvalues close to one where most of the eigenvalues of the preconditioned matrix should be located. Finally we consider an algebraic two-grid scheme inspired by ideas from the multigrid philosophy. In this paper, we describe these various variants as well as the observed numerical behavior on a set of model problems from Matrix Market or arising from the discretization via finite element technique of some 2D heterogeneous diffusion PDE problems. We discuss their numerical efficiency, computational complexity and sensitivity to the accuracy of the eigencalculation.