In this talk, we take a data-driven approach and apply machine learning to the moment closure problem for the kinetic equations, including radiative transfer equations and Boltzmann BGK equations. Traditional closures often rely on empirical assumptions, while naive machine learning closures can violate structural properties of the PDEs, leading to ill-posedness and numerical instability. To address these challenges, we propose a gradient-based moment closure, where neural networks directly learn the gradient of the high-order moment. Furthermore, we develop two strategies to enforce hyperbolicity, ensuring well-posed and stable evolution of the machine learning model. Extensive benchmark tests demonstrate that our hyperbolic ML closures achieve high accuracy, robust stability, and good generalization beyond training regimes.

The sketch-and-project is a unifying framework for many well-known projective iterative methods for solving linear systems, as well as their extensions to nonlinear problems. It generalizes popular algorithms such as Randomized Kaczmarz and Gauss-Seidel, as well as their block variants. In this talk, I will present our recent work that develops better ways to quantify the convergence of the sketch-and-project methods and concludes that — if properly designed —  these methods quickly capture the large outlying singular values of the linear system, implicitly preconditioning it. It creates new particularly efficient solvers for approximately low-rank systems, such as those commonly arising in machine learning (e.g., kernel matrices, signal-plus-noise models). Our approach combines novel spectral analysis of randomly sketched projection matrices with classical numerical analysis techniques, such as including momentum, adaptive regularization, and memoization.

Abstract: I will describe an interesting question of Alex Wright about fixed-point-free homeomorphisms of surfaces, as well as the question's implications for surface bundles over surfaces. Then I'll describe a complete answer for periodic maps on a surface, and say a few words about the main case of interest (pseudo-Anosov maps).

The Putnam Competition is a particularly challenging annual competition for undergraduates. Come learn a bit about the competition, and, of course, enjoy free pizza!

Among the many concepts in numerical analysis for differential equations, "stiffness" appears one of the least straightforward ones to define and characterize. This talk contrasts various forms of stiff limits and associated model problems, and establishes how these different shapes of stiffness manifest in the numerical solution of both ordinary and partial differential equations. Then it is presented which hierarchies of additional order conditions can be formulated that ensure that various classes of Runge-Kutta methods retain their desired convergence rates in the eye of different forms of stiffness.

Traditionally, computational fluid dynamics (CFD) is based on direct discretizations of the Navier-Stokes equations. This traditional approach of CFD is now being challenged as new multi-scale and multi-physics problems have begun to emerge in many fields -- in nanoscale systems, the scale separation assumption does not hold; macroscopic theory is therefore inadequate, yet microscopic theory may be impractical because it requires computational capabilities far beyond our present reach. Methods based on mesoscopic theories, which connect the microscopic and macroscopic descriptions of the dynamics, provide a promising approach.  Besides their connection to microscopic physics, kinetic methods also have certain numerical advantages due to the linearity of the advection term in the Boltzmann equation. We will discuss two mesoscopic methods: the lattice Boltzmann equation and the gas kinetic scheme, their mathematical theory and their applications to simulate various complex flows. Examples include incompressible homogeneous isotropic turbulence, hypersonic flows, and micro-flows.

Multiphysics models couple two or more physical processes together in a single simulation. These combinations may include systems of differential equations with different type (parabolic, hyperbolic, etc.), with different degrees of nonlinearity, and that evolve on disparate time scales. As a result, such simulations prove challenging for "monolithic" time integration methods that treat all processes using a single approach.

In this talk, I will discuss recent work on time integration methods that allow the flexibility to apply different techniques to distinct physical processes. While such techniques have existed for some time, including additive Runge-Kutta implicit-explicit (ImEx), multirate (a.k.a., multiple time stepping), and operator-splitting methods, there have been comparably few that combine these types of flexibility into a single family, while also supporting high orders of accuracy and temporal adaptivity. In this talk, I focus on newly developed implicit-explicit families of methods for multirate problems, along with novel techniques for time adaptivity in multirate infinitesimal time integration methods.