HDG methods and data-driven techniques for the van Roosbroeck model and its applications

Noninvasive estimation of doping inhomogeneities in semiconductors is relevant for many industrial applications. The goal is to estimate experimentally the unknown doping profile of a semiconductor by means of reproducible, indirect and non--destructive measurements. A number of technologies (such as LBIC, EBIC and LPS) have been developed which allow the indirect detection of doping variations via photovoltaic effects. The idea is to illuminate the sample at several positions while measuring the resulting voltage drop or current at the contacts. These technologies lead to inverse problems for which we still do not have a complete theoretical framework. In this thesis, we present three different data-driven approaches based on least squares, multilayer perceptrons, and residual neural networks. We compare the three strategies after having optimized the relevant hyperparameters and we measure the robustness of our approaches with respect to noise. The methods are trained on synthetic data sets (pairs of discrete doping profiles and corresponding photovoltage signals at different illumination positions) which are generated by a numerical solution of the forward problem using a physics-preserving finite volume method stabilized using the Scharfetter--Gummel scheme. In view of the need of generating larger datasets for trainings, we study the possibility to apply high-order Discontinuous Galerkin methods to the forward problem, preserving the stability properties of the Scharfetter--Gummel scheme. We prove that the Hybridizable Discontinuous Galerkin methods (HDG), a family of high-order DG methods, are equivalent to the Scharfetter--Gummel scheme on uniform unidimensional grids for a specific choice of the HDG stabilization parameter. This result is generalized to two and three dimensions using an approach based on weighted scalar products, and on local Slotboom changes of variables. We show that the proposed numerical scheme is well-posed, and numerically validate that it has the same properties of classical HDG methods, including optimal convergence and superconvergence of postprocessed solutions. For polynomial degree zero, dimension one, and vanishing HDG stabilization parameter, W-HDG coincides with the Scharfetter-Gummel stabilized finite volume scheme (i.e., it produces the same system matrix).