Numerical approximation of high-dimensional Fokker–Planck equations with polynomial coefficients

Abstract

This paper is concerned with the numerical solution of high-dimensional Fokker–Planck equations related to multi-dimensional diffusion with polynomial coefficients or Pearson diffusions. Classification of multi-dimensional Pearson diffusion follows from the classification of one-dimensional Pearson diffusion. There are six important classes of Pearson diffusion—three of them possess an infinite system of moments (Gaussian, Gamma, Beta) while the other three possess a finite number of moments (inverted Gamma, Student and Fisher–Snedecor). Numerical approximations to the solution of the Fokker–Planck equation are generated using the spectral method. The use of an adaptive reduced basis technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition (SVD). The basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite-dimensional subspace of the solution space. This is achieved through basis enrichment and projection stages. Numerical results with different boundary conditions are presented to demonstrate the accuracy and efficiency of the numerical scheme.