Analysis and design of jump coefficients in discrete stochastic diffusion models

TL;DR

This paper introduces a new method for designing jump coefficients in stochastic diffusion models that minimizes errors caused by coefficient modifications, improving accuracy in biological simulations on complex geometries.

Contribution

The paper proposes a novel backward analysis approach to derive non-negative jump coefficients that reduce both backward and forward errors in stochastic diffusion models.

Findings

The new method outperforms existing approaches in numerical experiments.

It effectively minimizes the forward error between original and perturbed equations.

The approach ensures non-negativity of coefficients on unstructured meshes.

Abstract

In computational system biology, the mesoscopic model of reaction-diffusion kinetics is described by a continuous time, discrete space Markov process. To simulate diffusion stochastically, the jump coefficients are obtained by a discretization of the diffusion equation. Using unstructured meshes to represent complicated geometries may lead to negative coefficients when using piecewise linear finite elements. Several methods have been proposed to modify the coefficients to enforce the non-negativity needed in the stochastic setting. In this paper, we present a method to quantify the error introduced by that change. We interpret the modified discretization matrix as the exact finite element discretization of a perturbed equation. The forward error, the error between the analytical solutions to the original and the perturbed equations, is bounded by the backward error, the error between the diffusion of the two equations. We present a backward analysis algorithm to compute the diffusion coefficient from a given discretization matrix. The analysis suggests a new way of deriving non-negative jump coefficients that minimizes the backward error. The theory is tested in numerical experiments indicating that the new method is superior and minimizes also the forward error.