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

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