Problem adapted Hierarchical Model Reduction for the Fokker-Planck equation

TL;DR

This paper introduces a hierarchical model reduction framework for the Fokker-Planck equation that constructs a problem-dependent basis in velocity space using reduced basis methods, improving solution accuracy.

Contribution

It develops a novel reduction approach that adapts basis functions to the solution shape, unlike traditional fixed basis methods like Legendre moments.

Findings

The new method produces a problem-dependent basis that better captures the solution shape.

Numerical experiments show improved accuracy over traditional methods.

The framework effectively reduces the dimensionality of the Fokker-Planck equation.

Abstract

In this paper we introduce a new hierarchical model reduction framework for the Fokker-Planck equation. We reduce the dimension of the equation by a truncated basis expansion in the velocity variable, obtaining a hyperbolic system of equations in space and time. Unlike former methods like the Legendre moment models, the new framework generates a suitable problem-dependent basis of the reduced velocity space that mimics the shape of the solution in the velocity variable. To that end, we adapt the framework of [M. Ohlberger and K. Smetana. A dimensional reduction approach based on the application of reduced basis methods in the framework of hierarchical model reduction. SIAM J. Sci. Comput., 36(2):A714-A736, 2014] and derive initially a parametrized elliptic partial differential equation (PDE) in the velocity variable. Then, we apply ideas of the Reduced Basis method to develop a greedy-algorithm that selects the basis from solutions of the parametrized PDE. Numerical experiments demonstrate the potential of this new method.