A Spectral Preconditioner for the Conjugate Gradient Method with Iteration Budget
This work provides new insight and practical strategies for improving the efficiency of spectral preconditioners in PCG when iterations are limited, which is relevant for large-scale matrix-free linear system solvers.
The paper addresses the problem of solving large symmetric positive-definite linear systems with a limited iteration budget using preconditioned conjugate gradient (PCG) with spectral preconditioning. It formulates the design of spectral preconditioners as a constrained optimization problem to minimize the error in energy norm at a fixed iteration, and proposes practical strategies for selecting the scaling parameter that yield significant improvements in error during initial iterations.
We study the solution of large symmetric positive-definite linear systems in a matrix-free setting with a limited iteration budget. We focus on the preconditioned conjugate gradient (PCG) method with spectral preconditioning. Spectral preconditioners map a subset of eigenvalues to a positive cluster via a scaling parameter, and leave the remainder of the spectrum unchanged, in hopes to reduce the number of iterations to convergence. We formulate the design of the spectral preconditioners as a constrained optimization problem. The optimal cluster placement is defined to minimize the error in energy norm at a fixed iteration. This optimality criterion provides new insight into the design of efficient spectral preconditioners when PCG is stopped short of convergence. We propose practical strategies for selecting the scaling parameter, hence the cluster position, that incur negligible computational cost. Numerical experiments highlight the importance of cluster placement and demonstrate significant improvements in terms of error in energy norm, particularly during the initial iterations.