An efficient Monte Carlo-SubSampling approach to first passage problems

Abstract

In the realm of the Continuous Time Random Walk (CTRW) and in conjunction with the Monte Carlo (MC) approach, we consider the transport of a chemical or radioactive pollutant in a 3D heterogeneous medium, focusing on the first passage time (FPT), defined as the time required by the walkers representative of the dangerous particles to travel within the medium until crossing a target disk, thus entering another medium which should instead remain clean, such as a water well or an aquifer. The advantage of the MC approach is the possibility of simulating different features of the travel such as different waiting-time probabilities, space dependent jump lengths, absorption and desorption phenomena, Galilei invariant and variant velocities driven by external forces. When the computer time required for collecting a suitable number of target crossings is excessive, we propound to hybridize the MC approach with the recent SubSampling computational procedure, usefully applied in the engineering reliability field to computing very small failure probabilities in short computer time. To tackle the FPT problem we iteratively consider groups of a few thousands of walkers: in each iteration we select a fraction of them closer to the target, ignoring the remaining ones, and then restore the group by creating with the MC technique new walkers even more close to the target. The successive groups have large conditional probabilities which can be estimated one after the other in short time and whose product yields the total probability. By so doing the FPT can be actually computed in much shorter times: we report examples of Gaussian and anomalous distributions in which the reduction with respect to a pure MC computation is of orders of magnitude.