The optimal multilevel Monte-Carlo approximation of the stochastic drift-diffusion-Poisson system

TL;DR

This paper develops an optimal multilevel Monte Carlo finite-element method for approximating solutions of stochastic drift-diffusion-Poisson systems, with applications in nanoscale device modeling.

Contribution

It introduces a new MLMC-FEM approach for stochastic drift-diffusion-Poisson systems, analyzing convergence and complexity for the first time.

Findings

The method achieves optimal convergence rates.

Numerical results confirm efficiency and accuracy.

Applications include modeling noise in nanoscale devices.

Abstract

Existence and local-uniqueness theorems for weak solutions of a system consisting of the drift-diffusion-Poisson equations and the Poisson-Boltzmann equation, all with stochastic coefficients, are presented. For the numerical approximation of the expected value of the solution of the system, we develop a multi-level Monte-Carlo (MLMC) finite-element method (FEM) and we analyze its rate of convergence and its computational complexity. This allows to find the optimal choice of discretization parameters. Finally, numerical results show the efficiency of the method. Applications are, among others, noise and fluctuations in nanoscale transistors, in field-effect bio- and gas sensors, and in nanopores.