@article{FackeldeyOsterSallandtetal., 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}, series = {SIAM Journal on Multiscale Modeling and Simulation}, volume = {20}, journal = {SIAM Journal on Multiscale Modeling and Simulation}, number = {1}, doi = {10.1137/20M1372500}, pages = {379 -- 403}, 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} }