Tuesday, March 26, 2024, 2:45 PM, Room 214 Mudd

Speaker: Andrew Stuart, Caltech

Title: Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance  

Abstract: Sampling a probability distribution with an unknown  normalization constant is a fundamental problem  in computational science and engineering. This task  may be cast as an optimization problem over all  probability measures, by choice of a suitable energy  function. Then an initial distribution can be evolved  to the desired minimizer (the target distribution) via  a gradient flow with respect to a chosen metric. The choice of the energy and the metric lead to different approaches and it is of interest to understand  their role. We provide theoretical insights into these choices.  Having chosen an energy and a metric the next task is to choose an algorithm to approximate the gradient flow. Mean-field models, whose law is governed by the gradient  flow in the space of probability measures, may be identified;  particle approximations of these mean-field models form  the basis of algorithms. The gradient flow approach is  also the basis of algorithms for variational inference,  in which the optimization is performed over a parameterized  family of probability distributions such as Gaussians or Gaussian mixtures; the underlying gradient flow is  restricted to the parameterized family. Numerical results are presented to illustrate the resulting methodologies.    

Joint work with Y. CHEN (NYU), D.Z. HUANG (PKU),  J. HUANG (U Penn) and S. REICH (Potsdam).

Biography: Professor Andrew Stuart has research interests in applied and computational mathematics, and is interested in particular in the question of how to optimally combine complex mechanistic models with data. He joined Caltech in 2016 as Bren Professor of Computing and Mathematical Sciences, after 17 years as Professor of Mathematics at the University of Warwick (1999--2016). Prior to that he was on the faculty in The Departments of Computer Science and Mechanical Engineering at Stanford University (1992--1999), and in the Mathematics Department at Bath University (1989--1992). He obtained his PhD from the Oxford University Computing Laboratory in 1986, and held postdoctoral positions in Mathematics at Oxford University and at MIT in the period 1986--1989