Analysis and design of jump coefficients in discrete stochastic diffusion models. (English) Zbl 1330.65017
Summary: In computational systems 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 nonnegativity 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 nonnegative jump coefficients that minimizes the backward error. The theory is tested in numerical experiments indicating that the new method is superior and also minimizes the forward error.
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 nonnegative jump coefficients that minimizes the backward error. The theory is tested in numerical experiments indicating that the new method is superior and also minimizes the forward error.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 65C40 | Numerical analysis or methods applied to Markov chains |
| 60J60 | Diffusion processes |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
stochastic simulation; diffusion; unstructured mesh; finite element method; error bound; reaction-diffusion kinetics; continuous time, discrete space Markov process; algorithm; numerical experimentReferences:
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