@article{FackeldeyOsterSallandtetal.2022, author = {Fackeldey, Konstantin and Oster, Mathias and Sallandt, Leon and Schneider, Reinhold}, title = {Approximative Policy Iteration for Exit Time Feedback Control Problems driven by Stochastic Differential Equations using Tensor Train format}, volume = {20}, journal = {SIAM Journal on Multiscale Modeling and Simulation}, number = {1}, arxiv = {http://arxiv.org/abs/2010.04465}, doi = {10.1137/20M1372500}, pages = {379 -- 403}, year = {2022}, abstract = {We consider a stochastic optimal exit time feedback control problem. The Bellman equation is solved approximatively via the Policy Iteration algorithm on a polynomial ansatz space by a sequence of linear equations. As high degree multi-polynomials are needed, the corresponding equations suffer from the curse of dimensionality even in moderate dimensions. We employ tensor-train methods to account for this problem. The approximation process within the Policy Iteration is done via a Least-Squares ansatz and the integration is done via Monte-Carlo methods. Numerical evidences are given for the (multi dimensional) double well potential and a three-hole potential.}, language = {en} } @article{RichterSallandtNuesken2024, author = {Richter, Lorenz and Sallandt, Leon and N{\"u}sken, Nikolas}, title = {From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEs}, volume = {25}, journal = {Journal of Machine Learning Research}, arxiv = {http://arxiv.org/abs/2307.15496}, pages = {248}, year = {2024}, 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 romise 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.}, language = {en} }