Optimal multilevel randomized quasi-Monte-Carlo method for the stochastic drift-diffusion-Poisson system

In this paper, an optimal multilevel randomized quasi-Monte-Carlo method to solve the stationary stochastic drift-diffusion-Poisson system is developed. We calculate the optimal values of the parameters of the numerical method such as the mesh sizes of the spatial discretization and the numbers of quasi-points in order to minimize the overall computational cost for solving this system of stochastic partial differential equations. This system has a number of applications in various fields, wherever charged particles move in a random environment. It is shown that the computational cost of the optimal multilevel randomized quasi-Monte-Carlo method, which uses randomly shifted low-discrepancy sequences, is one order of magnitude smaller than that of the optimal multilevel Monte-Carlo method and five orders of magnitude smaller than that of the standard Monte-Carlo method. The method developed here is applied to a realistic transport problem, namely the calculation of random-dopant effects in nanoscale field-effect transistors.