Efficient numerical approximation of a non-regular Fokker--Planck equation associated with first-passage time distributions

TL;DR

This paper presents a novel transformation and numerical approach for efficiently approximating solutions to a singular Fokker-Planck equation in neuroscience, enabling faster and more accurate model evaluations.

Contribution

It introduces a transformation that regularizes the PDE, allowing the use of a space-time minimal residual method and sparse tensor interpolation for efficient parameter-dependent solutions.

Findings

The transformed PDE is more regular, improving convergence.

Numerical simulations confirm predicted convergence rates.

The method effectively handles parameter variations in boundary conditions.

Abstract

In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker--Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations.