Volume 22, Issue 1

In this paper, we obtain an explicit representation of the transition density of the one-dimensional skew Brownian motion with (a constant drift and) two semipermeable barriers. Moreover, we propose a rejection sampling method to simulate this density in an exact way.

We adapt and study a variance reduction approach for the homogenization of elliptic equations in divergence form. The approach, borrowed from atomistic simulations and solid-state science [23], [24], [25], consists in selecting random realizations that best satisfy some statistical properties (such as the volume fraction of each phase in a composite material) usually only obtained asymptotically. We study the approach theoretically in some simplified settings (one-dimensional setting, perturbative setting in higher dimensions), and numerically demonstrate its efficiency in more general cases.

We suggest a series of extremely fast stochastic algorithms based on exact representations we derive in this paper for the first passage time and exit point probability densities, splitting and survival probabilities. We apply the developed algorithms to the following three classes of problems: (1) simulation of epitaxial nanowire growth, (2) diffusion imaging of microstructures, in particular, cathodoluminescence imaging for threading dislocations, and (3) simulation of the annihilation of electrons and holes in vicinity of nonradiative centers and quantum efficiency evaluation. In the last example the Random Walk on Spheres method is used to solve nonlinear diffusion equations, and to more general systems of nonlinear Smoluchowski equations combined with the kinetic Monte Carlo method.

The background of the Monte Carlo method is connected with two very important events in the modern human history: the World War Two and the building of first electronic computers. For that reason, we will try to synthesize the most important facts of those events that have relation to the development of the method in order to produce a clearer picture of the origination of the Monte Carlo method of stochastic sampling. The research covers the period from 1930 to 1959.