Stochastic partial differential equations are becoming increasingly important to quantify uncertainties, noise, fluctuations, and process variations. Various results regarding multiscale problems, stochastic multiscale problems, existence, uniqueness, and regularity of solutions are presented for important model equation such as the stochastic Poisson equation, the stochastic nonlinear Poisson-Boltzmann equation, and the stochastic drift-diffusion-Poisson system.

The development of efficient numerical methods will also be discussed: In particular, we present a multi-level Monte-Carlo method for the stochastic drift-diffusion-Poisson, basis adaptation for the Poisson-Boltzmann equation, and a non-intrusive stochastic Galerkin method for the Poisson-Boltzmann equation.

Finally, simulation results for realistic structures are shown. The model equations and stochastic processes are motivated by applications in nanotechnology (such as nanowire bio- and gas sensors as well as nanoscale transistors) and materials science. These simulations are important for the rational design of nanoscale sensors and other devices.