A Progressive Statistical Method for Preconditioning Matrix-Free Solution of High-Order Discretization of Linear Time-Dependent Problems

TL;DR

This paper introduces a progressive, statistical preconditioning method for matrix-free solutions of high-order discretized time-dependent PDEs, improving preconditioner quality iteratively during Krylov subspace solves.

Contribution

It presents a novel statistical approach to build a preconditioner matrix on the fly using matrix-vector products, avoiding explicit matrix formation.

Findings

Preconditioner quality improves with more data collected during iterations.

The method produces a banded diagonal preconditioner that enhances convergence.

Validated on a sample implementation demonstrating effectiveness.

Abstract

Preconditioning of a linear system obtained from spectral discretization of time-dependent PDEs often results in a full matrix which is expensive to compute and store specially when the problem size increases. A matrix-free implementation is usually applied to resolve this issue. In this framework, preconditioning is typically challenging since the entries of the matrix are not explicitly available. In this short note, we propose a statistical approach to gradually create a preconditioner matrix by collecting the information obtained from matrix-vector product in the Arnoldi loop of an unpreconditioned Krylov subspace algorithm. The gathered information are then correlated using a multiple regressors estimate where the error is assumed to be normally distributed. This procedure yields a banded diagonal matrix which is then used as a preconditioner in the next iterative solve. This is repeated between iterative solves until a good preconditioner is constructed on fly. This statistically iterative procedure is progressive since the fidelity of the preconditioning matrix improves by adding more data obtained from matrix-vector product during the entire solution procedure. The proposed algorithm is validated for a sample implementation.