On the curse of dimensionality in the Fokker-Planck equation
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The curse of dimensionality associated with numerical solution of Fokker-Planck equation (FPE) is addressed in this paper. Two versions of the meshless, nodebased partition of unity finite element method (PUFEM), namely, standard-PUFEM and particle-PUFEM are discussed. Both methods formulate the problem as weak form equations of FPE using local shape functions over a meshless cover of the solution domain. The variational (i.e. weak form) integrals are evaluted using quasi Monte-Carlo methods to handle the curse of dimensionality in numerical integration. The particle-PUFEM approach is presented as a generalization of standard-PUFEM and shown to provide flexibility in domain construction in high dimensional spaces and better handle the curse than the standard approach. In the current paper, it is used to solve FPE numerically for systems having up to five dimensional state-space on a small computing workstation, a result hence-far absent from numerical FPE literature. Coupled with local enrichment of the approximation space, it is shown that the particle-PUFEM approach can be used to immensely curb the curse of dimensionality in numerical solution of FPE, thus opening avenues for use of FPE in nonlinear filtering problems with long propagation times between measurements for space applications.
author list (cited authors)
Kumar, M., Chakravorty, S., & Junkins, J. L.