A well-conditioned direct PinT algorithm for first-and second-order evolutionary equations

TL;DR

This paper introduces a direct parallel-in-time algorithm for first- and second-order evolutionary equations that is well-conditioned, enabling high parallel efficiency and large-scale computations with minimal roundoff error.

Contribution

The authors develop a novel direct PinT algorithm with explicit formulas for eigenvector matrices, ensuring a well-conditioned diagonalization process for large numbers of time levels.

Findings

Achieves over 60 times speedup with 256 cores.

Proves the condition number of eigenvector matrix grows as O(n^2).

Demonstrates the method's stability and efficiency through numerical experiments.

Abstract

In this paper, we propose a direct parallel-in-time (PinT) algorithm for time-dependent problems with first- or second-order derivative. We use a second-order boundary value method as the time integrator that leads to a tridiagonal time discretization matrix. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix, which yields a direct parallel implementation across all time levels. A crucial issue on this methodology is how the condition number of the eigenvector matrix $V$ grows as $n$ is increased, where $n$ is the number of time levels. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for computing $V$ and $V^{-1}$, by which we prove that $\mathrm{Cond}_2(V)=\mathcal{O}(n^{2})$. This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as $n$ grows and thus, compared to other direct PinT algorithms, a much larger $n$ can be used to yield satisfactory parallelism. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.