Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A parallel-in-time Newton's method-based ODE solver

Casian Iacob, Hassan Razavi, Simo Särkkä

TL;DR

The paper develops a parallel-in-time framework for nonlinear ODE solvers by reframing the time-stepping as a root-finding problem and solving it with Newton's method. It exploits parallel prefix-sums to compute the affine components of the Newton step in logarithmic span, achieving significant speedups over traditional Parareal, especially on GPUs. Theoretical results establish quadratic convergence of both explicit and implicit parallel Newton iterations under standard regularity assumptions. Numerical experiments on a suite of explicit and stiff problems demonstrate faster runtimes and favorable scaling compared to sequential solvers and Parareal, indicating practical potential for real-time applications.

Abstract

In this article, we introduce a novel parallel-in-time solver for nonlinear ordinary differential equations (ODEs). We state the numerical solution of an ODE as a root-finding problem that we solve using Newton's method. The affine recursive operations arising in Newton's step are parallelized in time by using parallel prefix sums, that is, parallel scan operations, which leads to a logarithmic span complexity. This yields an improved runtime compared to the previously proposed Parareal method. We demonstrate the computational advantage through numerical simulations of various systems of ODEs.

A parallel-in-time Newton's method-based ODE solver

TL;DR

The paper develops a parallel-in-time framework for nonlinear ODE solvers by reframing the time-stepping as a root-finding problem and solving it with Newton's method. It exploits parallel prefix-sums to compute the affine components of the Newton step in logarithmic span, achieving significant speedups over traditional Parareal, especially on GPUs. Theoretical results establish quadratic convergence of both explicit and implicit parallel Newton iterations under standard regularity assumptions. Numerical experiments on a suite of explicit and stiff problems demonstrate faster runtimes and favorable scaling compared to sequential solvers and Parareal, indicating practical potential for real-time applications.

Abstract

In this article, we introduce a novel parallel-in-time solver for nonlinear ordinary differential equations (ODEs). We state the numerical solution of an ODE as a root-finding problem that we solve using Newton's method. The affine recursive operations arising in Newton's step are parallelized in time by using parallel prefix sums, that is, parallel scan operations, which leads to a logarithmic span complexity. This yields an improved runtime compared to the previously proposed Parareal method. We demonstrate the computational advantage through numerical simulations of various systems of ODEs.

Paper Structure

This paper contains 15 sections, 2 theorems, 65 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Problem formulation
  4. Parallel-in-time solution
  5. Temporal parallelization of Newton's method for ODE integration
  6. Prefix-sum operations
  7. Parallel-in-time explicit ODE solvers
  8. Theoretical results for the parallel-explicit Newton's method
  9. Parallel-in-time implicit ODE solvers
  10. Theoretical results for the parallel-implicit Newton's method
  11. Numerical results
  12. Explicit ODE solution approximation
  13. Implicit ODE solution approximation
  14. Conclusion
  15. Review of the Parareal method

Key Result

Theorem 3.2

Suppose Assumption assum2 holds. Let the sequence of Newton iterates $\left\{\xi^{(k)}\right\}$ be defined by eq:newton-iterates, and let the Newton step $u^{(k)}$ be defined by eq:newton-step. If the initial guess $\xi^{(0)}$ is sufficiently close to $\xi^*$, the sequence $\{\xi^{(k)}\}$ converges

Figures (4)

  • Figure 1: Average runtime comparison of explicit parallel Newton-based method, Parareal, and sequential integration for the logistic equation, the van der Pol oscillator, and the cart-pole system over a fixed time interval with varying numbers of intermediary steps.
  • Figure 2: Evolution of the residual norms over Newton iterations for the logistic equation, van der Pol oscillator, and cart-pole.
  • Figure 3: Average runtime comparison of implicit parallel Newton-based method and the sequential method for the Dahlquist test problem and the Robertson chemical reaction system over a fixed time interval with varying numbers of intermediary steps.
  • Figure 4: Evolution of the residual norms over Newton iterations for the Dahlquist test and Robertson's chemical reaction system.

Theorems & Definitions (4)

  • Theorem 3.2
  • Proof 1
  • Theorem 3.4
  • Proof 2