Monte Carlo tracking drift-diffusion trajectories algorithm for solving narrow escape problems

Karl K. Sabelfeld, Nikita Popov

Research output: Contribution to journalArticlepeer-review


This study deals with a narrow escape problem, a well-know difficult problem of evaluating the probability for a diffusing particle to reach a small part of a boundary far away from the starting position of the particle. A direct simulation of the diffusion trajectories would take an enormous computer simulation time. Instead, we use a different approach which drastically improves the efficiency of the diffusion trajectory tracking algorithm by introducing an artificial drift velocity directed to the target position. The method can be efficiently applied to solve narrow escape problems for domains of long extension in one direction which is the case in many practical problems in biology and chemistry. The algorithm is meshless both in space and time, and is well applied to solve high-dimensional problems in complicated domains. We present in this paper a detailed numerical analysis of the method for the case of a rectangular parallelepiped. Both stationary and transient diffusion problems are handled.

Original languageEnglish
Pages (from-to)177-191
Number of pages15
JournalMonte Carlo Methods and Applications
Issue number3
Publication statusPublished - 1 Sep 2020


  • drift-diffusion trajectory
  • first passage time
  • Narrow escape problem
  • random walk on spheres

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