Florida State University

We first consider multilevel Monte Carlo and stochastic collocation methods for determining statistical information about an output of interest that depends on the solution of a PDE with inputs that depend on random parameters. In our context, these methods connect a hierarchy of spatial grids to the amount of sampling done for a given grid, resulting in dramatic acceleration in the convergence of approximations. We then consider multifidelity methods for the same purpose which feature a variety of models that have different fidelities. For example, we could have coarser grid discretizations, reduced-order models, simplified physics, surrogates such as interpolants, and, in principle, even experimental data. No assumptions are made about the fidelity of the models relative to the “truth” model of interest so that unlike multilevel methods, there is no a priori model hierarchy available. However, our approach can still greatly accelerate the convergence of approximations.

The melting of ice in Greenland and Antarctica would, of course, be by far the major contributor any possible sea level rise. Thus, to make science-based predictions about sea-level rise, it is crucial that the ice sheets covering those land masses be accurately mathematically modeled and computationally simulated. In fact, the 2007 IPCC report on the state of the climate did not include predictions about sea level rise because it was concluded there that the science of ice sheets was not developed to a sufficient degree. As a result, predictions could not be rationally and

confidently made. In recent years, there has been much activity in trying to improve the state-of-the-art of ice sheet modeling and simulation. In

this lecture, we review a hierarchy of mathematical models for the flow of ice, pointing out the relative merits and demerits of each, showing how

they are coupled to other climate system components (ocean and atmosphere), and discussing where further modeling work is needed. We then discuss algorithmic approaches for the approximate solution of ice sheet flow models and present and compare results obtained from simulations using the different mathematical models.

We define a set of nonlocal operators and develop a nonlocal vector calculus that mimics the classical differential vector calculus. Included are the definitions of nonlocal divergence, gradient, and curl operators and the derivation of nonlocal integral theorems and identities. We indicate how, through certain limiting processes, the nonlocal operators are connected to their differential counterparts. The nonlocal operators are shown to appear in nonlocal models for diffusion and in the nonlocal, spatial derivative free, peridynamics continuum model for solid mechanics. We show, for example, that unlike elliptic partial differential equations, steady state versions of the nonlocal models do not necessary result in the smoothing of data. We also briefly consider finite element methods for nonlocal problems, focusing on solutions containing jump discontinuities; in this setting, nonlocal models can lead to optimally accurate approximations.