Georgia Institute of Technology College of Sciences

Abstract:
This talk explores the Uzawa method, tracing its development from early applications in partial differential equations (PDEs) to modern advancements in optimization, image processing, and scientific computing. We will examine recent refinements for developing GPU-adaptive solvers for huge-scale linear programming and its extension to semidefinite programming arising in quantum information science. The discussion will also highlight the method's integration with deep learning and unrolling techniques for optimal control problems of PDEs, as well as its applications in industry.

Bio:

Xiaoming Yuan is a Professor in the Department of Mathematics at The University of Hong Kong. His research spans optimization, optimal control, scientific machine computing, and artificial intelligence. He is well recognized for his fundamental contributions to first-order optimization algorithms, including the Alternating Direction Method of Multipliers (ADMM), primal-dual methods, and proximal point algorithms. He also collaborates extensively with the AI and cloud computing industries. He led the development of the first automatic bandwidth allocation system for the cloud computing sector. His team was honored as a Franz Edelman Award Finalist in 2023.

Let us discuss how to use generative AI to help with math and coding.

My presentation features two scenarios:

Coding in LaTeX. Suppose you have a raw draft of what potentially could be a math paper. We will consider and apply simple AI tools that may help realizing the potential.

Coding in CAS. Suppose you have a raw idea for a package in a Computer Algebra System; your raw idea may be limited to a rough description of the input/output of a method you would like to implement. How far can an AI assistant take you? Can it autonomously code a working software package?   

The Reissner–Nordström spacetime models a spherically symmetric and time-independent charged black hole in general relativity. The Cauchy horizon in the interior of such a black hole is subject to an infinite blueshift instability. In 1989, Poisson and Israel discovered a nonlinear manifestation of this instability in the spherically symmetric setting called "mass inflation," where the Hawking mass becomes identically infinite at the Cauchy horizon. 

We complete the first proof of mass inflation for a wave-type matter model, namely the spherically symmetric Einstein–Maxwell–scalar field system. This result follows from a large-data decay result for the scalar field in the black hole exterior combined with works of Dafermos, Luk–Oh, and Luk–Oh–Shlapentokh-Rothman.

Particle physics research relies on making statistical statements about Nature. The field is one of the last bastions of classical statistics and certainly among its most rigorous users, relying on a worldwide computing grid to process zettabyte-scale data. Recent AI-enabled developments have reinvigorated research in classical statistics, particularly by removing the need for asymptotic approximations in many calculations.

In this talk, I will discuss how AI has allowed us to question core assumptions in our statistical inference techniques. Neural networks enable high-dimensional statistical inference, avoiding aggressive data reduction or the use of unnecessary assumptions. However, they also introduce new sources of systematic uncertainty that require novel uncertainty quantification tools. AI further enables more robust statistical inference by accelerating Neyman inversion and confidence-interval calibration. These advances allow the design of new test statistics that leverage Bayesian mathematical tools while still guaranteeing frequentist coverage, an approach that was previously considered computationally infeasible. These new techniques raise questions about practical methods for handling nuisance parameters, the definition of point estimators, and the computationally efficient implementation of mathematical solutions. If time permits, I will also introduce the emerging challenge of non-nestable hypothesis testing in particle physics.

My group is among the teams leading this revitalization of classical statistical research in particle physics, and I look forward to connecting with students and senior colleagues at Georgia Tech who are interested in contributing to this emerging field.

Bio: Aishik Ghosh is an assistant professor in the School of Physics at Georgia Tech with a focus on developing AI methods to accelerate fundamental physics and astrophysics. His group works on theoretical physics, statistical methods, and experiment design. For robust scientific applications, Dr. Ghosh focuses on uncertainty quantification, interpretability, and verifiability of AI algorithms, targeting publications in physics journals and ML conferences.