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

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