Parameter identification in models with sharp phase transitions

该项目的目的是分析和开发参数识别问题的算法，其中正向问题的解决方案表现出多个结构上不同的阶段。这些问题发生在气候学（云成核，冰川融化），材料科学（晶体生长，钢退火）和力学（接触，缺陷力学）中，可以用涉及非光滑非线性的变分不等式或偏微分方程来建模。这些模型的一个共同特点是解算子是非线性和不可微的，使得依赖局部线性化的经典解析和数值方法不适用。因此，我们的目标是推导误差估计，并为一类涉及非光滑但Lipschitz连续和方向可微非线性的偏微分方程的参数估计问题开发新的迭代算法，例如在两相Stefan问题中出现的那些。

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.