Applications such as electrical-impedance tomography, nanoelectrode sensors, and nanowire sensors lead to deterministic and stochastic partial differential equations that model electrostatics and charge transport. The main model equations are the nonlinear Poisson-Boltzmann equation and the stochastic drift-diffusion-Poisson-Boltzmann system. After a discussion of the model equations, theoretic results as well as a numerical method, namely optimal multi-level Monte Carlo, are presented.

Knowing these model equations, the question how as much information as possible can be extracted from measurements arises next. We use computational Bayesian PDE inversion to reconstruct physical and geometric parameters of the body interior in electrical-impedance tomography and of target molecules in the two nanoscale sensors considered here. Computational Bayesian estimation provides us with the ability not only to estimate unknown parameter values but also their probability distributions and hence the uncertainties in reconstructions, which is important in the case of ill-posed inverse problems. In addition to theoretic results, numerical results for the three applications such as multifrequency reconstruction for nanoelectrode sensors are shown.