We will present a set up for simulations of the so-called glymphatic system, which is believed to
be an important mechanism for clearance of waste from the brain. Accumulation of waste can be coupled to
brain shrinkage and conditions like Alzheimers disease. An approach to design mathematical models for the
glymphatic system is to consider the brain as a poroelastic material, with complex geometry, bathed in water, and where various parameters of the model will vary over extreme ranges. A challenge in discretizing such systems in to design methods which behave well with respect variations of the parameters. In particular, we will explain how parameter robust stability estimates for the PDE systems can be used as a key tool to design effective algorithms for the corresponding discrete systems.

A nematic liquid crystal is essentially a liquid that exhibits some orientational order. The liquid crystalline state is typically observed in materials composed of highly asymmetric molecules for temperatures in the intermediate range when the material is neither a classical liquid nor a solid. In this talk we will be interested in describing a phase transition between the nematic and the isotopic state when the nematic order melts and the liquid crystal turns into a regular isotopic fluid. The experimental observations of this process indicate that the transition proceeds via evolution of interfaces separating different phases where the interfaces are generally not smooth. Our goal in this talk is to explain presence of these phase boundary singularities from a mathematical point of view .

The nematic-to-isotopic phase transition can be described within the so called Landau-de Genes variational theory for a tensor-valued order parameter. Because this theory is rather complex, in order to develop an initial understanding of transitions between the ordered and disordered states, we propose a simpler toy model based on the modified Ginzburg-Landau-type energy defined over vector fields on the plane. The corresponding variational model consists of anisotropic gradient terms and a potential that vanishes on two disconnected sets. While this model may not quantitatively describe the nematic state, the topology of the simplified target set retains the crucial features of the full problem. The principal observation from the study of the simplified model is that the phase boundary singularities can be explained by large disparity between the elastic constants that specify the gradient contribution to the energy. In the talk we will present a combination of rigorous analysis and numerics that leads to this conclusion.

In the absence of analytical reconstruction methods, numerical solutions of nonlinear inverse problems have been mostly based on least-square formulations where solutions are sought by minimizing the $L^2$ difference between model predictions and measured data. We present here recent computational studies of some nonlinear inverse problems where quadratic Wasserstein distances are used to measure the discrepancy between model predictions and measured data. Related numerical and theoretical issues will be discussed.