Repository: Freie Universität Berlin, Math Department

Solving high-dimensional parabolic PDEs using the tensor train format

Richter, Lorenz and Sallandt, Leon and Nüsken, Nikolas (2021) Solving high-dimensional parabolic PDEs using the tensor train format. Proceedings of the 38th International Conferenceon Machine Learning, 139 . pp. 8998-9009.

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High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional gridbased 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.

Item Type:Article
Additional Information:
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics
ID Code:2502
Deposited By: Monika Drueck
Deposited On:02 Mar 2021 10:55
Last Modified:11 Feb 2022 14:23

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