Solving high-dimensional parabolic PDEs using the tensor train format

Lorenz Richter; Leon Sallandt; Nikolas Nüsken

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arXiv · 2102.11830

Solving high-dimensional parabolic PDEs using the tensor train format

Lorenz Richter·Leon Sallandt·Nikolas Nüsken

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Abstract

High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the curse of dimensionality. In this paper, we argue that tensor trains provide an appealing approximation framework for parabolic PDEs: the combination of reformulations in terms of backward stochastic differential equations and regression-type methods in the tensor format holds the promise of leveraging latent low-rank structures enabling both compression and efficient computation. Following this paradigm, we develop novel iterative schemes, involving either explicit and fast or implicit and accurate updates. We demonstrate in a number of examples that our methods achieve a favorable trade-off between accuracy and computational efficiency in comparison with state-of-the-art neural network based approaches.

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Lorenz Richter, Leon Sallandt, Nikolas Nüsken. 2021-02-23. Solving high-dimensional parabolic PDEs using the tensor train format. https://arxiv.org/abs/2102.11830

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