Solving high-dimensional parabolic PDEs using the tensor train format
This addresses a fundamental problem in economics, science, and engineering by enabling more efficient numerical solutions for high-dimensional PDEs, though it is incremental as it builds on existing tensor train and regression techniques.
The paper tackles the challenge of solving high-dimensional parabolic PDEs, which are difficult due to the curse of dimensionality, by proposing tensor train-based methods that achieve a favorable trade-off between accuracy and computational efficiency compared to neural network approaches.
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.