Georgia Institute of Technology College of Sciences

Over the last 40 years there have been great advances in computer hardware, solvers (methods for solving Ax=b and F(x)=0), meshing algorithms, time stepping methods, adaptivity and so on. Yet accurate prediction of fluid motion (for settings where this is needed) is still elusive. This talk will review three major hurdles that remain: ensemble simulations, time accuracy and model stagnation. Three recent ideas where numerical analysis can help push forward the boundary between what can be done and what can't be done will be described. This talk is based on joint work with many. It should be completely understandable by grad students with a basic PDE class.

Motivated by applications to group synchronization and quadratic assignment on random data, we study a general problem of Bayesian inference of an unknown “signal” belonging to a high-dimensional compact group, given noisy pairwise observations of a featurization of this signal.

We establish a quantitative comparison between the signal-observation mutual information in any such problem with that in a simpler model with linear observations, using interpolation methods. For group synchronization, our result proves a replica formula for the asymptotic mutual information and Bayes-optimal mean-squared error. Via analyses of this replica formula, we show that the conjectural phase transition threshold for computationally-efficient weak recovery of the signal is determined by a classification of the real-irreducible components of the observed group representation(s), and we fully characterize the information-theoretic limits of estimation in the example of angular/phase synchronization over SO(2)/U(1). For quadratic assignment, we study observations given by a kernel matrix of pairwise similarities and a randomly permuted and noisy counterpart, and we show in a bounded signal-to-noise regime that the asymptotic mutual information coincides with that in a Bayesian spiked model with i.i.d. signal prior.

This is based on joint work with Kaylee Yang and Zhou Fan.

A Latin square is an nxn grid filled with n symbols such that each symbol appears exactly once in each row and column. A transversal in a Latin square is a collection of n cells such that each row, column and symbol appears exactly once in the collection.

Latin squares were introduced by Euler in the 1700s and he was interested in the question of when a Latin square decomposes fully into transversals. Equivalently, when does a Latin square have an 'orthogonal mate'?

We'll discuss the history of this question, and some upcoming joint work with Richard Montgomery.