ABSTRACT. In this talk I'll describe a few challenging problems in modeling the flow of glaciers and ice sheets and how I think they could be solved using Firedrake.

1. Ice flow has a nonlinear constitutive law that makes the stress balance equations explode around zero strain rate. A stable velocity-stress mixed formulation could transform the problem in a way that would be much more tractable. This has implications for modeling Raymond arches, which occur at ice divides and domes.

2. The terminus of a glacier advances and retreats; how should we track this dynamic geometry? I'll show some preliminary work I've done using the phase field method.

3. Many geophysical inverse problems use data defined on a lower-dimensional submanifold of the spatial domain. While expressing the model-data misfit functional in UFL might not be possible, we may be able to steer around this problem using the alternating direction method of multipliers.

ABSTRACT. Nonsmooth functions are difficult to minimize due to the lack of gradient and higher-order information. Such restrictions often arise naturally in the context of regularization, especially when solutions to systems are ill-conditioned and necessitate elements of sparsity. A variety of nonsmooth optimization techniques have been developed and extended upon in recent years, many of which have provided first order techniques to efficiently enforce these nonsmooth regularizers. The purpose of this talk is to touch on the theoretical fundamentals behind some of these algorithms, and demonstrate their effectiveness in the realm of PDE optimization. Specifically, we derive the proximal-gradient algorithm and the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), and show the progress of these algorithms on the Obstacle Problem.

ABSTRACT. Numerical resolution of exterior Helmholtz problems requires some approach to domain truncation. As an alternative to approximate nonreflecting boundary conditions and invocation of the Dirichlet-to-Neumann map, we introduce a new, nonlocal boundary condition. This condition is exact and requires the evaluation of layer potentials involving the free space Green's function. However, it seems to work in general unstructured geometry, and Galerkin finite element discretization leads to convergence under some mesh constraints imposed by Garding-type inequalities. We will sketch this method, show how we have integrated Firedrake with pytential (a fast-multipole code), and discuss what issues this raises for extending Firedrake to work with more general nonlocal operators.