Adaptive Gaussian particle method for the solution of the Fokker-Planck equation

• Published in: Zeitschrift für angewandte Mathematik und Mechanik Vol. 92; no. 10; pp. 770 - 781
• Main Authors: Scharpenberg, M.D., Lukáčová-Medviová, M.
• Format: Journal Article
• Language: English
• Published: Berlin WILEY-VCH Verlag 01.10.2012; WILEY‐VCH Verlag
• Subjects: dynamic systems; Fokker-Planck equation; particle methods; Uncertainty quantification
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.201100088

The Fokker‐Planck equation describes the evolution of the probability density for a stochastic ordinary differential equation (SODE) or a deterministic ordinary differential equation (ODE) with stochastic initial values. A solution strategy for this partial differential equation (PDE) up to a relatively large number of dimensions is based on particle methods using Gaussians as basis functions. An initial probability density is decomposed into a sum of multivariate normal distributions and these are propagated according to the ODE. The decomposition as well as the propagation is subject to possibly large numerical errors due to the difficulty to control the spatial residual over the whole domain. In this paper a new particle method is derived, which allows a deterministic error control for the resulting probability density. It is based on global optimization and allows an adaption of an efficient surrogate model for the residual estimation. The Fokker‐Planck equation describes the evolution of the probability density for a stochastic ordinary differential equation (SODE) or a deterministic ordinary differential equation (ODE) with stochastic initial values. A solution strategy for this partial differential equation (PDE) up to a relatively large number of dimensions is based on particle methods using Gaussians as basis functions. An initial probability density is decomposed into a sum of multivariate normal distributions and these are propagated according to the ODE. The decomposition as well as the propagation is subject to possibly large numerical errors due to the difficulty to control the spatial residual over the whole domain. In this paper a new particle method is derived, which allows a deterministic error control for the resulting probability density. It is based on global optimization and allows an adaption of an efficient surrogate model for the residual estimation.