Hitting time distributions for efficient simulations of drift‐diffusion processes

Abstract

Numerical solutions to partial differential equations in anisotropic, heterogeneous media obtained by their probabilistic representations are useful for a number of purposes, including our own interests in biomedical simulations. These solutions are obtained by a walk with both random and deterministic components hitting a boundary. Hitting time distributions are required to efficiently simulate such processes. The distributions for hitting time T and place XT on a surface for a particle undergoing both diffusion and drift are presented here, generalizing the previous work. Importantly, the distributions directly obtained are, on the whole, not useful for numerical work, being expressed as very poorly and nonuniformly convergent series. Resummed series with rapid convergence for probabilities over a useful range of the hitting time to the boundary are here obtained in different ways. A numerical inverse is constructed to implement the random walk. We have used these hitting time distributions, combined with efficient inverse function construction, to obtain speed‐up of simulations of fluid and particle transport by permitting the steps allowed for a given accuracy to be as large as possible.

Addendum to “Hitting time distributions for efficient simulations of drift‐diffusion processes”

“…We would like to restate here that none of the results reported in Reference 1 are affected by this omission. Series like equation (4) of the paper, that are directly obtained from the moment generating functions are indeed not useful for nonzero times other than “large” times (ie, well away from time zero) where the probability of hitting the sphere is close to unity.…”

“…Moreover, the point T = 0 is excluded in any algorithms or numerical methods for simulation, since the sphere is of radius unity (nonzero radius) and hence a zero hitting time has zero probability. In Reference 1, we emphasize the need for a Poisson resummation of such series for numerical evaluation for smaller, indeed for most times that occur in a simulation. These resummed series converge very fast near T = 0, and their T = 0 limit is also zero, though the series is not the expansion of any analytic function at time zero since all derivatives vanish.…”

“…In the paper “Hitting time distributions for efficient simulations of drift‐diffusion processes”, 1 we exhibited density functions and cumulative distributions for diffusion processes with and without drift. In particular, the previously well‐known cumulative distribution for the hitting time T to a sphere of radius R of a particle released at the center of the sphere, and undergoing only diffusive transfer was shown in equation (4) loc.…”

Efficient numerical solutions of Neumann problems in inhomogeneous media from their probabilistic representations and applications

“…(It can be more efficient to fix the step length of the walker to be D∕2 and to instead compute the random time required to take this step. We have provided new formulas and methods for executing such a simulation 8 and implemented them in software. 4 ) We presuppose such simulations and describe here the additional steps needed to implement the reflection process required for the Neumann problem.…”

“…(It can be more efficient to fix the step length of the walker to be D /2 and to instead compute the random time required to take this step. We have provided new formulas and methods for executing such a simulation and implemented them in software …”