离散随机扩散模型中跳跃系数的分析与设计
Analysis and design of jump coefficients in discrete stochastic diffusion models
AI总结：
本文研究了离散随机扩散模型中跳跃系数的非负性问题，提出了一种通过后向分析算法计算扩散系数的方法，以最小化误差。
摘要：
在计算系统生物学中，反应-扩散动力学的mesoscopic模型由连续时间、离散空间的马尔可夫过程描述。为随机模拟扩散，跳跃系数通过扩散方程的离散化获得。使用无结构网格表示复杂几何可能导致使用线性有限元时出现负系数。已有方法试图修改系数以保证非负性。本文提出一种量化该修改引入误差的方法，将修改后的离散化矩阵解释为扰动方程的精确有限元离散化。正向误差（原方程与扰动方程解析解间的误差）被后向误差（两个方程扩散间的误差）所限制。本文提出一种后向分析算法，从给定的离散化矩阵计算扩散系数。理论表明，新方法能以最小化后向误差的方式推导非负跳跃系数。数值实验表明，新方法优于现有方法，并且也最小化正向误差。
英文摘要：
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.
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