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

Leonenko, Ganna M. ORCID: https://orcid.org/0000-0001-8025-661X and Phillips, Timothy Nigel ORCID: https://orcid.org/0000-0001-6455-1205 2015. Numerical approximation of high-dimensional Fokker-Planck equations with polynomial coefficients. Journal of computational and applied mathematics 273 , pp. 296-312. 10.1016/j.cam.2014.05.024

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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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics Medicine MRC Centre for Neuropsychiatric Genetics and Genomics (CNGG)
Subjects:	Q Science > QA Mathematics
Additional Information:	Pdf uploaded in accordance with publisher's policy at http://www.sherpa.ac.uk/romeo/issn/0377-0427/ (accessed 20/08/14).
Publisher:	Elsevier
ISSN:	0377-0427
Date of First Compliant Deposit:	30 March 2016
Date of Acceptance:	30 May 2014
Last Modified:	19 Nov 2024 15:00
URI:	https://orca.cardiff.ac.uk/id/eprint/63542

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