Christina Frederick: Determining microstructure using multiscale methods

The talk focuses on inverse problems for elliptic partial differential equations with highly oscillating coefficients, where the goal is to determine coefficients in the equation by matching predictions with observed data. Such inverse problems pose a challenge in both the mathematical formulation and the numerical modeling, which is hard even for forward computations. In addition, the problem in its full generality is typically ill-posed; different sequences of coefficients produce the same limiting equations. It can be shown that, in certain cases where prior knowledge of microscale features is given, the inverse problem is well-posed. Furthermore, multiscale numerical techniques can be used to simplify computations. Numerical examples are chosen to be relevant for applications in composite design and seismic inversion.