Time-parallel iterative solvers for parabolic evolution equations

TL;DR

This paper introduces novel time-parallel algorithms for solving linear parabolic evolution equations, reformulating the problem as a symmetric saddle-point system and proposing a robust, scalable preconditioner for efficient parallel computation.

Contribution

The paper develops a new symmetric saddle-point reformulation and a parallel-in-time preconditioner for implicit Euler discretizations of parabolic equations, enhancing scalability and robustness.

Findings

Preconditioner achieves convergence rates independent of time-steps and mesh sizes.

Method demonstrates excellent weak and strong scaling in large-scale parallel experiments.

Reformulation ensures inf-sup stability and robustness of the solution process.

Abstract

We present original time-parallel algorithms for the solution of the implicit Euler discretization of general linear parabolic evolution equations with time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory of parabolic problems, we show that the standard nonsymmetric time-global system can be equivalently reformulated as an original symmetric saddle-point system that remains inf-sup stable with respect to the same natural parabolic norms. We then propose and analyse an efficient and readily implementable parallel-in-time preconditioner to be used with an inexact Uzawa method. The proposed preconditioner is non-intrusive and easy to implement in practice, and also features the key theoretical advantages of robust spectral bounds, leading to convergence rates that are independent of the number of time-steps, final time, or spatial mesh sizes, and also a theoretical parallel complexity that grows only logarithmically with respect to the number of time-steps. Numerical experiments with large-scale parallel computations show the effectiveness of the method, along with its good weak and strong scaling properties.