From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEs

TL;DR

This paper introduces tensor train-based methods combined with robust regression for high-dimensional PDEs and BSDEs, offering a new approach that balances accuracy and computational efficiency.

Contribution

It develops continuous-time iterative schemes using tensor trains for PDEs and BSDEs, leveraging low-rank structures for improved compression and efficiency.

Findings

Methods achieve a good trade-off between accuracy and efficiency

Tensor trains enable effective low-rank approximations in high dimensions

Numerical results demonstrate robustness and computational advantages

Abstract

The numerical approximation of partial differential equations (PDEs) poses formidable challenges in high dimensions since classical grid-based methods suffer from the so-called curse of dimensionality. Recent attempts rely on a combination of Monte Carlo methods and variational formulations, using neural networks for function approximation. Extending previous work (Richter et al., 2021), we argue that tensor trains provide an appealing framework for parabolic PDEs: The combination of reformulations in terms of backward stochastic differential equations and regression-type methods holds the promise of leveraging latent low-rank structures, enabling both compression and efficient computation. Emphasizing a continuous-time viewpoint, we develop iterative schemes, which differ in terms of computational efficiency and robustness. We demonstrate both theoretically and numerically that our methods can achieve a favorable trade-off between accuracy and computational efficiency. While previous methods have been either accurate or fast, we have identified a novel numerical strategy that can often combine both of these aspects.