Computational Mathematics and Applications Seminar

I will start with two very brief surveys.  First is a class of problems, namely variational inequalities (VIs), which generalize PDE problems, and second is a class of solver algorithms, namely full approximation storage (FAS) nonlinear multigrid for PDEs.  Motivation for applying FAS to VIs is demonstrated in the standard mathematical model for glacier surface evolution, a very general VI problem relevant to climate modeling.  (Residuals for this nonlinear and non-local VI problem are computed by solving a Stokes model.)  Some existing nonlinear multilevel VI schemes, based on global (Newton) linearization would seem to be less suited to such general VI problems.  From this context I will sketch some work-in-progress toward the scalable solutions of nonlinear and nonlocal VIs by an FAS-type multilevel method.

The neutron transport equation is a linear Boltzmann-type PDE that models radiative transfer processes, and fission nuclear reactions. The computation of the largest eigenvalue of this Boltzmann operator is crucial in nuclear safety studies but it has classically been formulated only at a discretized level, so the predictive capabilities of such computations are fairly limited. In this talk, I will give an overview of the modeling for this equation, as well as recent analysis that leads to an infinite dimensional formulation of the eigenvalue problem. We leverage this point of view to build a numerical scheme that comes with a rigorous, a posteriori estimation of the error between the exact, infinite-dimensional solution, and the computed one.

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