Solution of high dimensional transient Fokker-Planck equations by tensor decomposition

A tensor decomposition approach combined with Chebyshev spectral differentiation is developed to solve the transient Fokker-Planck equation (FPE) in high dimensional cases. This method drastically reduces the degrees of freedom required to maintain accuracy in the approximation as dimensionality increases. The transient solution is sought in a single CANDECOMP/PARAFAC decomposition form for all times by the alternating least squares algorithm. This is accomplished by decoupling the spatial dimensions as well as the separation of temporal domain from spatial domain. As a result, the transient solution is obtained without resorting to expensive time-stepping schemes. In addition, a new approach for imposing the vanishing boundary condition in the tensor product framework is proposed, improving the quality of the approximation. Numerical transient solutions for systems up to 14 dimensional state space are successfully obtained on a regular personal computer to demonstrate the advantages of the proposed method.