Graduate Students Seminar

Speaker 1: Ryan Lafferty

Title

Nonlocal Priors for Bayesian Model Selection

Abstract

We introduce the theory of Bayesian hypothesis testing and model selection.  Classically, so-called local priors have been used for the prior distribution under the alternative hypothesis. However, these suffer from both practical and philosophical limitations. To address these limitations, we discuss the theory of non-local priors, initiated by V.E. Johnson and David Rossell. We review recent results and literature relating to nonlocal priors for Bayesian model selection, particularly in the case of high-dimensional regression. Finally, we define a family of non-local priors for model selection of Gaussian directed acyclic graph models and discuss basic properties and theoretical results.

Speaker 2: Michael Retzlaff

Title

Boundary Control of Elliptic PDEs with Efficient Solvers and Applications

Abstract

Optimal control of PDEs serves a variety of interesting applications across multiple disciplines including fluid flow, electronics, medicine, image processing, and many more.  The crux of the discipline is that the user determines certain initial or forcing conditions in order to achieve a desired end state of some physical process constrained by a system of PDEs.  In this talk we consider boundary control, an optimal control problem where we solve for the boundary conditions of an elliptic PDE so that the state variable aligns with given data.  After deriving a specific control problem, we explain the use of an efficient solver called Multigrid Preconditioning (PCMG), first introduced by Draganescu and Dupont.  We show for the boundary control problem at-hand, that PCMG attains a quasi-optimal convergence rate with respect to the discretization, meaning significant computational cost savings can be achieved at higher resolutions.  We conclude by exhibiting the performance of PCMG on a specific implementation of the aforementioned boundary control problem.