2021/02/23 by Lorenz Richter, Richter, Lorenz, Leon Sallandt +3 · 1 voice · 5 citations
#stat.ML #cs.LG #math.NA #math.PR
paper · pdf · doi:10.48550/arxiv.2102.11830
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.