Randomized time stepping of nonlinearly parametrized solutions of evolution problems

TL;DR

This paper introduces a randomized time stepping method using sketching for nonlinear parametrizations in evolution problems, improving conditioning and efficiency in model reduction and PDE solutions.

Contribution

It proposes a novel randomized scheme that regularizes and reduces the complexity of time-stepping in nonlinear parametrized models.

Findings

Improved conditioning of least-squares problems.

Reduced computational complexity per time step.

Competitive accuracy with enhanced runtime efficiency.

Abstract

The Dirac-Frenkel variational principle is a widely used building block for using nonlinear parametrizations in the context of model reduction and numerically solving partial differential equations; however, it typically leads to time-dependent least-squares problems that are poorly conditioned. This work introduces a randomized time stepping scheme that solves at each time step a low-dimensional, random projection of the parameter vector via sketching. The sketching has a regularization effect that leads to better conditioned least-squares problems and at the same time reduces the number of unknowns that need to be solved for at each time step. Numerical experiments with benchmark examples demonstrate that randomized time stepping via sketching achieves competitive accuracy and outperforms standard regularization in terms of runtime efficiency.