In this talk, based on joint work with Plamen Stefanov and Gunther Uhlmann, I discuss the boundary rigidity problem on manifolds with boundary (for instance, a domain in Euclidean space with a perturbed metric), i.e. determining a Riemannian metric from the restriction of its distance  function to the boundary. This corresponds to travel time tomography, i.e. finding the Riemannian metric from the time it takes for solutions of the corresponding wave equation to travel between boundary points. A version of this relates to finding the speed of seismic waves inside the Earth from travel time data, which in turn permits a study of the structure of the inside of the Earth.

This non-linear problem in turn builds on the geodesic X-ray transform on such a Riemannian manifold with boundary. The geodesic X-ray transform on functions associates to a function its integral along geodesic curves, so for instance in domains in Euclidean space along straight lines. The X-ray transform on symmetric tensors is similar, but one integrates the tensor contracted with the tangent vector of the geodesics. I will explain how, under suitable convexity assumptions, one can invert the geodesic X-ray transform on functions, i.e. determine the function from its X-ray transform, in a stable manner, as well as the analogous tensor result, and the connection to the full boundary rigidity problem.

Heterogeneous problems with high contrast, multiscale and possibly also random coefficients arise frequently in practice, e.g., reservoir simulation and material sciences. However, due to the disparity of scales, their efficient and accurate simulation is notorious challenging. First, I will describe some impor- tant applications, and review several state-of-the-art multiscale model reduction algorithms, especially the Generalized Multiscale Finite Element Method (GMsFEM). Then I will describe recent efforts on developing a mathematical theory for GMsFEM, and ongoing works on algorithmic developments and novel applications.

References

[1] Guanglian Li, On the Convergence Rates of GMsFEMs for Heterogeneous Elliptic Problems without Oversampling Techniques, submitted to Multiscale Modeling & Simulation, 2018.

[2] Shubin Fu, Eric Chung and Guanglian Li, Edge Multiscale Methods for elliptic problems with hetero- geneous coefficients, submitted to J. Comput. Phys, 2018.

Let G(D) be the set of all graphs with degree sequence d. The Erdos-Gallai conditions give the necessary and sufficient conditions for the existence of a graph with degree sequence d. If G(D) is nonempty, how do we approximate the size of all such graphs? I will discuss a maximum entropy approach to this problem. It involves considering G(D) as the set of integer points of a certain polytope in R^(n choose 2) and constructing a probability distribution that is constant on this set of points. Using a concentration result, we can use this distribution to approximate the size of G(D).

We present yet another proof of Anderson localization for the Anderson model.