We consider the Hamiltonian perturbed by a potential on an infinite periodic lattice. We are interested in the behavior of the solution for the lattice model to that for the equation for the continuous model as the mesh size tends to 0. For some lattices such as square, triangular and hexagonal lattices, we show that the scattering solutions (i.e. the soultions associated with the continuous spectrum) converge to the solution to the Shrodedinger equation in the continuous model. For the case of the hexagonal lattice, we can also derive the convergence to the massless Dirac equation. The idea of the proof relies on the micro-local calculus for lattice Schroedinger operators and the classical method of the limiting absorption principle.

Quantum computing is a technology that utilizes quantum mechanical phenomena to do computation faster than is believed to be possible with classical computers. It is a rapidly developing and interdisciplinary field comprising of physics, computer science, and mathematics. It is predicted that in the future quantum computers will enable scientists to solve problems outside the capabilities of classical computers in many fields such as molecular simulations in drug discovery and complex combinatorics problems.

In this talk, we consider a quantum algorithm for an inverse travel time problem on a graph. This problem is a discrete version of the inverse travel time problem encountered in seismic and medical imaging and the boundary rigidity problem studied in Riemannian geometry. We also consider the computational complexity of the inverse problem, and show that the quantum algorithm has a quadratic improvement in computational cost when compared to the standard classical algorithm.

We study the discrete version of Gel'fand's inverse spectral problem, formulated as follows for a finite weighted graph and the graph Laplacian on it. Suppose we are given a subset $B$ of vertices and the spectral data $(\lambda_j,\phi_j|_B)$, where $\lambda_j$ are the eigenvalues of the graph Laplacian and $\phi_j|_B$ are the values of the corresponding eigenfunctions on $B$. We consider if these data uniquely determine the graph structure and the weights. In general, this problem is not uniquely solvable without assumptions on the graph or the set $B$ due to counterexamples. We introduce a so-called Two-Points Condition on graphs (with respect to $B$), and prove that the inverse spectral problem is uniquely solvable under this condition. We also consider inverse problems for random walks on finite graphs. We show that under the Two-Points Condition, the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on $B$.

In this talk I will introduce graphical representations of stochastic partial differential equations with the goals of approximating Matérn Gaussian fields on manifolds and generalizing the Matérn model to abstract point clouds. I will show that these graph-based prior models can give optimal posterior contraction in semi-supervised learning, and illustrate their use in various inverse problems on manifolds.