Hierarchical Interpolative Factorization Preconditioner for Parabolic Equations

TL;DR

This paper introduces a hierarchical interpolative factorization preconditioner that significantly accelerates the solution of linear systems in parabolic equations, reducing iteration counts and computational complexity.

Contribution

The paper presents a novel preconditioning approach using hierarchical interpolative factorization for efficient solving of parabolic equations with large condition numbers.

Findings

Reduces conjugate gradient iterations needed for convergence.

Computes the preconditioner in linear time.

Demonstrates improved performance in numerical experiments.

Abstract

This note proposes an efficient preconditioner for solving linear and semi-linear parabolic equations. With the Crank-Nicholson time stepping method, the algebraic system of equations at each time step is solved with the conjugate gradient method, preconditioned with hierarchical interpolative factorization. Stiffness matrices arising in the discretization of parabolic equations typically have large condition numbers, and therefore preconditioning becomes essential, especially for large time steps. We propose to use the hierarchical interpolative factorization as the preconditioning for the conjugate gradient iteration. Computed only once, the hierarchical interpolative factorization offers an efficient and accurate approximate inverse of the linear system. As a result, the preconditioned conjugate gradient iteration converges in a small number of iterations. Compared to other classical exact and approximate factorizations such as Cholesky or incomplete Cholesky, the hierarchical interpolative factorization can be computed in linear time and the application of its inverse has linear complexity. Numerical experiments demonstrate the performance of the method and the reduction of conjugate gradient iterations.