Mathematical Sciences Colloquium

Computing the polar decomposition and the related matrix sign function has been a well-studied problem in numerical analysis for decades. Recently, it has emerged as an important subroutine within the Muon algorithm for training deep neural networks. However, the requirements of this application differ sharply from classical settings: deep learning demands GPU-friendly algorithms that prioritize high throughput over high precision. We introduce Polar Express, a new method for computing the polar decomposition.

Algorithms for the numerical solution of partial differential equations often come in two flavors: iterative methods (which build successively better approximations of the solution until convergence to some tolerance is reached) and direct methods (which construct the solution all at once). Historically, iterative methods have been the norm in many settings, as naive direct solvers can be prohibitively expensive for large-scale problems.

Camouflaging is a widely used concealment tactic across the animal kingdom, but when is it actually beneficial for an organism to use? In this talk, we focus on analyzing when it is worthwhile for a pursuer to utilize motion camouflage (MC) amidst uncertainty in when an evader will feel threatened and attempt to escape. Using MC movement techniques to trick an evader's visual system into believing that a pursuer is less threatening than they actually are has been observed in hover flies during mating rituals and in dragonflies during territorial disputes.

Given an unlabeled data set (n data points in d dimensions), we address the task of performing near-optimal machine learning whilst labeling as few data points as possible, denoted the label complexity of the problem. Our focus is linear models and tight approximation guarantees. Existing theory identifies the data to label from the bottom up, and this approach does not work for tight approximation guarantees. We give algorithms that achieve tight approximation by choosing the data to label from the top down, via rejection.

Once upon time, Henry Stommel was tasked by Harold Sverdrup to provide an explanation for why western boundary currents such as the Gulf Stream are found on the western boundary, not the eastern boundary.  The resolution to the problem, appearing in Stommel 1948, was to add a frictional operator into a barotopic vorticity equation.  The interplay of friction with asymmetry encoded within the planetary dispersion relation, i.e.

I will present the CoQui [1] software package developed at the Flatiron Institute’s Center for Computational Quantum Physics. The overarching goal of CoQui is the development and implementation of perturbative, low-scaling approaches to the many-body problem for first-principles electronic structure. CoQui relies on the use of Interpolative Separable Density Fitting to generate compact factorizations of the coulomb interaction, achieving an efficient framework for ab-initio many-body perturbation theory (MBPT).

In this talk, we present the first set of algorithms that can compute the Kreiss constant of a square matrix to arbitrary accuracy under reasonable assumptions.  As famously introduced by H.-O. Kreiss over six decades ago, Kreiss constants inform us about how severely stable systems of ordinary differential (or difference) equations will exhibit transient behavior before they settle down.  Specifically, the Kreiss Matrix Theorem provides theoretically tight upper and lower bounds on the largest magnitude of transient behavior that a given system will attain.

Machine learning (ML) has shown great promise for accelerating numerical simulation, with examples such as numerical weather prediction and molecular dynamics with MLpredicted potentials. At the same time, many data-driven ML approaches face wellknown challenges, including limited interpretability, reduced reliability, and substantial demands on training data and computational resources such as GPUs.

The use of neural networks for solving partial differential equations (PDEs) has attracted considerable attention in recent years. In this talk, I will first highlight their advantages over traditional numerical methods, including improved approximation rates and the poential to overcome the curse of dimensionality. I will then discuss the challenges that arise when applying neural networks to PDEs, particularly in training. Because training is inherently a highly nonconvex optimization problem, it can lead to poor local minma with large training errors, especially in complex PDE settings.