Parameter identification in models with sharp phase transitions

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The aim of this project is the analysis of and development of algorithms for parameter identification problems where the solution of the forward problem exhibits multiple structurally different phases. Such problems occur in climatology (cloud nucleation, glacial melting), material science (crystal growth, steel annealing) and mechanics (contact, defect mechanics) and can be modeled as variational inequalities or partial differential equations involving non-smooth nonlinearities. A common feature of such models is that the solution operator is nonlinear and non-differentiable, making classical analytical and numerical approaches that rely on local linearization inapplicable. Our goal is therefore to derive error estimates and to develop new iterative algorithms for parameter estimation problems for a class of partial differential equations involving non-smooth but Lipschitz continuous and directionally differentiable nonlinearities such as those appearing in the two-phase Stefan problem.