  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.
