Posted January 25, 2024
Colloquium Questions or comments?
3:30 pm – 4:30 pm Lockett 232
Changhong Mou, University of Wisconsin-Madison
Reduced Order Modeling in the Age of Data
Abstract: Data-driven modeling of complex dynamical systems is becoming increasingly popular across various domains of science and engineering. In this talk, I will introduce a systematic multiscale data-driven closure reduced order model (ROM) framework for complex systems with strong chaotic or turbulent behavior. I will utilize available data to construct novel ROM closure terms, thereby capturing the interaction between resolved and unresolved modes. Next, I will explain how the new data-driven closure ROM can be integrated with a conditional Gaussian data assimilation framework that employs cost-effective, conditionally linear functions to capture the statistical features of the closure terms. This leads to the stochastic data-driven closure ROM that facilitates an efficient and accurate scheme for nonlinear data assimilation (DA), the solution of which is provided by closed analytic formulae that do not require ensemble methods. It also allows the ROM to avoid many potential numerical and sampling issues in recovering the unobserved states from partial observations. Furthermore, I will introduce a hybrid DA algorithm for complex dynamical systems with partial observations. The method exploits cheap stochastic parameterized ROMs for filtering the observed state variables, significantly reducing the computational cost. It also uses machine learning to build a nonlinear map between observed and unobserved state variables, which enables the efficient computation of the ensemble members of the unobserved states. The hybrid DA algorithm is successfully applied to a precipitating quasi-geostrophic (PQG) model, which includes the effects of water vapor, clouds, and rainfall beyond the classical two-level QG model.
Posted January 25, 2024
Colloquium Questions or comments?
3:30 pm – 4:30 pm Lockett 232
Michael Novack, Carnegie Mellon
Soap films, Plateau's laws, and the Allen-Cahn equation
Abstract: Plateau's problem of minimizing area among surfaces with a common boundary is the basic model for soap films and leads to the theory of minimal surfaces. In this talk we will discuss a modification of Plateau's problem in which surfaces are replaced with regions of small but positive volume. The model captures features of real soap films that cannot be described by minimal surfaces, and the corresponding analysis requires the development of new ideas in geometric measure theory. We will also discuss the PDE approximation of this problem via the Allen-Cahn equation and its relation to Plateau's laws, which govern singularities in soap films.
Posted January 31, 2024
Colloquium Questions or comments?
2:30 pm – 3:30 pm Zoom
Bruno Poggi, Universitat Autònoma de Barcelona
Two problems in the mathematical physics of the magnetic Schrödinger operator and their solutions via the landscape function.
Abstract. In two papers in the 90's, Zhongwei Shen studied non-asymptotic bounds for the eigenvalue counting function of the magnetic Schrödinger operator, as well as the localization of eigenfunctions. But in dimensions 3 or above, his methods required a strong scale-invariant quantitative assumption on the gradient of the magnetic field; in particular, this excludes many singular or irregular magnetic fields, and the questions of treating these later cases had remained open, giving rise to a problem and a conjecture. This strong assumption on the gradient of the magnetic field has appeared also in the harmonic analysis related to the magnetic Schrödinger operator. In this talk, we present our solutions to these questions, and other new results on the exponential decay of solutions (eigenfunctions, integral kernels, resolvents) to Schrödinger operators. We will introduce the Filoche-Mayboroda landcape function for the (non-magnetic) Schrödinger operator, present its pointwise equivalence to the classical Fefferman-Phong-Shen maximal function (also known as the critical radius function in harmonic analysis literature), and then show how one may use directionality assumptions on the magnetic field to construct a new landscape function in the magnetic case. We resolve the problem and the conjecture of Z. Shen (and recover other results in the irregular setting) by putting all these observations together.
Posted February 1, 2024
Last modified February 2, 2024
Colloquium Questions or comments?
3:30 pm – 4:30 pm Lockett 232
Narek Hovsepyan, Rutgers University
On the lack of external response of a nonlinear medium in the second-harmonic generation process.
Abstract: Second Harmonic Generation (SHG) is a process in which the input wave (e.g. laser beam) interacts with a nonlinear medium and generates a new wave, called the second harmonic, at double the frequency of the original input wave. We investigate whether there are situations in which the generated second harmonic wave does not scatter and is localized inside the medium, i.e., the nonlinear interaction of the medium with the probing wave is invisible to an outside observer. This leads to the analysis of a semilinear elliptic system formulated inside the medium with non-standard boundary conditions. More generally, we set up a mathematical framework needed to investigate a multitude of questions related to the nonlinear scattering problem associated with SHG (or other similar multi-frequency optical phenomena). This is based on a joint work with F. Cakoni, M. Lassas and M. Vogelius.
Posted January 25, 2024
Last modified January 31, 2024
Colloquium Questions or comments?
3:30 pm Lockett 232
Ali Kara, University of Michigan
Reinforcement Learning in Non-Markovian Environments under General Information Structures
Abstract: For decision-making under uncertainty, typically only an ideal model is assumed, and the control design is based on this given model. However, in reality, the assumed model may not perfectly reflect the underlying dynamics, or there might not be an available mathematical model. To overcome this issue, one approach is to use the past data of perceived state, cost and control trajectories to learn the model or the optimal control functions directly, a method also known as reinforcement learning. The majority of the existing literature has focused on methods structured for systems where the underlying state process is Markovian and the state is fully observed. However, there are many practical settings where one works with data and does not know the possibly very complex structure under which the data is generated and tries to respond to the environment. In this talk, I will present a convergence theorem for stochastic iterations, particularly focusing on Q-learning iterates, under a general, possibly non-Markovian, stochastic environment. I will then discuss applications of this result to the decision making problems where the agent's perceived state is a noisy version of some hidden Markov state process, i.e. partially observed MDPs, and when the agent keeps track of a finite memory of the perceived data. I will also discuss applications for a class of continuous-time controlled diffusion problems.
Posted February 12, 2024
Colloquium Questions or comments?
3:30 pm – 4:30 pm 232 Lockett Hall
Ke Chen, University of Maryland
Towards efficient deep operator learning for forward and inverse PDEs: theory and algorithms
Abstract: Deep neural networks (DNNs) have been a successful model across diverse machine learning tasks, increasingly capturing the interest for their potential in scientific computing. This talk delves into efficient training for PDE operator learning in both the forward and inverse PDE settings. Firstly, we address the curse of dimensionality in PDE operator learning, demonstrating that certain PDE structures require fewer training samples through an analysis of learning error estimates. Secondly, we introduce an innovative DNN, the pseudo-differential auto-encoder integral network (pd-IAE net), and compare its numerical performance with baseline models on several inverse problems, including optical tomography and inverse scattering. We will briefly mention some future works at the end, focusing on the regularization of inverse problems in the context of operator learning.
Posted September 29, 2023
Last modified January 29, 2024
Colloquium Questions or comments?
3:30 pm – 4:20 pm Lockett 232
Jacob Rasmussen, University of Illinois Urbana-Champaign
The L-space conjecture for 3-manifolds
The L-space conjecture of Boyer-Gordon-Watson and Juhasz relates three very different properties that a closed 3-manifold M can possess. One of these properties is algebraic: is \pi_1(M) left orderable? The second is geometric: does the M admit a coorientable taut foliation? The third is analytic: is the Heegaard Floer homology M as simple as it can be, given the size of H_1(M). If the conjecture is true, it would reveal the existence of a striking dichotomy for rational homology 3-spheres. In this talk, I'll explain what each of the three conditions appearing in the L-space conjecture mean, and then discuss efforts to prove and disprove it, and why we should care.
Posted January 20, 2024
Last modified February 5, 2024
Colloquium Questions or comments?
3:30 pm – 4:20 pm Lockett 232
Chongying Dong, UC Santa Cruz
Monstrous moonshine and orbifold theory
This introductory talk will survey the recent development of the monstrous moonshine. Conjectured by McKay-Thompson-Conway-Norton and proved by Borcherds, the moonshine conjecture reveals a deep connection between the largest sporadic finite simple group Monster and genus zero functions. From the point of view of vertex operator algebra, moonshine is a connection among finite groups, vertex operator algebras and modular forms. This talk will explain how the moonshine phenomenon can be understood in terms of orbifold theory.