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- ItemDynamical low-rank approximations of solutions to the Hamilton--Jacobi--Bellman equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eigel, Martin; Schneider, Reinhold; Sommer, DavidWe present a novel method to approximate optimal feedback laws for nonlinar optimal control basedon low-rank tensor train (TT) decompositions. The approach is based on the Dirac-Frenkel variationalprinciple with the modification that the optimisation uses an empirical risk. Compared to currentstate-of-the-art TT methods, our approach exhibits a greatly reduced computational burden whileachieving comparable results. A rigorous description of the numerical scheme and demonstrations ofits performance are provided.
- ItemApproximation of solutions to multidimensional parabolic equations by approximate approximations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Lanzara, Flavia; Mazya, Vladimir; Schmidt, GuntherWe propose a fast method for high order approximations of the solution of n-dimensional parabolic problems over hyper-rectangular domains in the framework of the method of approximate approximations. This approach, combined with separated representations, makes our method effective also in very high dimensions.We report on numerical results illustrating that our formulas are accurate and provide the predicted approximation rate 6 also in high dimensions.