Spring 2022

Abstract: We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with random weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. A law of large numbers result is established as the system size increases and the underlying graphons converge. Under suitable convexity assumptions, we show the exponential ergodicity for the system, establish the uniform in time law of large numbers and concentration bounds, and analyze the uniform in time Euler approximation. The precise rate of convergence of the Euler approximation is provided. Based on joint works with Erhan Bayraktar and Suman Chakraborty.

Host person: Ruimeng Hu

Abstract: Analyzing contemporary high-dimensional datasets often leads to extremely large-scale interaction modeling problems, where the challenge is posed to identify important interactions among billions of candidate pairwise interactions. While several methods have recently been proposed to tackle this challenge, they are mostly designed by (1) focusing on linear models with interactions and (or) (2) assuming the hierarchy assumption. In practice, however, neither of these two building blocks has to hold. We propose an interaction modeling framework in generalized linear models (GLMs) which is free of any assumptions on hierarchy. The basic premise is a non-trivial extension of the reluctant interaction modeling framework in linear models (Yu, et al, 2019), where main effects are preferred over interactions if all else is equal, to the GLMs setting. The proposed method is easy to implement, and is highly scalable to large-scale datasets. Theoretically, we show that the proposed method successfully recovers all the important interactions with high probability. Both the favorable computational and statistical properties are demonstrated through comprehensive empirical studies.

Host person: Sui Tang

Abstract: Mean field game (MFG) theory has been very successful in the past two decades, and its analysis usually deals with a large population of comparably small players. When a major player (such as an institutional trader against a large number of small traders) is introduced, an interesting theory can be developed due to the rich interaction structure among the players, and different equilibrium notions can be adopted. We will briefly overview the past developments by the MFG community, and clarify the relation of different results.

We will further present a recent work when a mean field Stackelberg equilibrium is considered. A crucial issue with this solution concept is time consistency. We will address this issue using a notion of t-selves proposed in the economic literature. (This part is based on work with Xuwei Yang)

Host person: Ruimeng Hu

Abstract: For cells to divide, they must undergo the process of mitosis: the process of spatially organizing their copied genetic material to precise locations. Underlying stochastic molecular components accomplish this task with puzzling speed and accuracy. To understand this, we propose viewing mitosis as collective motion akin to schooling fish or insect swarms. I will discuss two related projects in this pursuit. The first is a classical modeling study, exploring how mutant cancer cells can mimic healthy cells in mitosis to avoid death. The second project focuses on incorporating 3D experimental spatial trajectories directly into the modeling process to learn unknown interactions. This work is with Alex Mogilner (NYU) and Alexey Khodjakov (Wadsworth Center; New York Dept. of Health).

Host person: Sui Tang

Abstract: Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, for example, the peridynamics model of fracture mechanics, they also come with increased difficulty in computation with nonlocality involved. A central idea to design effective numerical approximations to nonlocal models is the so-called asymptotic compatible schemes that are robust under the change of the nonlocal length parameter. In this talk, we are also going to address nonlocal problems in heterogeneous media with coefficients depending on finitely many parameters. The parameters are realizations of random variables which could come from a truncated Karhunen-Loeve decomposition of a random field. The analyticity of the parameter to solution map is shown for a class of nonlocal elliptic equations. Spatial discretization of the nonlocal problem is done with an asymptotically compatible meshfree method while a probabilistic collocation method is used for discretization in the random parameter space. The efficiency of the method is demonstrated for nonlocal diffusion and nonlocal mechanics problems for fracture modeling. 

Host person: Sui Tang

Abstract: The MBO scheme is an efficient algorithm for approximating mean curvature flow. Recently, Bertozzi et al. adapted it to the setting of graphs, where it is used for data clustering. Given some data, one constructs the similarity graph associated to the data points. The goal is to split the data in two meaningful clusters. The algorithm produces the clusters by alternating between diffusion on the graph and pointwise thresholding. After a suitable stopping criterion, one is left with one cluster (and its complement). It turns out that this can be thought of as a local minimizer of the thresholding energy on the graph - which is defined by averaging the diffusion of the first step over points belonging to the complement of the diffused cluster. It is then natural to ask the following question: assume that the data points are sampled from a weighted Riemannian submanifold of some Euclidean space, what can we say about the outcomes of MBO as the number of data points goes to infinity? We will give partial answers to the question. This is based on the paperhttps://arxiv.org/abs/2112.06737 and some ongoing work with Tim Laux.

Host person: Katy Craig

Abstract: In this talk, we consider problems involving fractional-order operators on bounded Lipschitz domains. After introducing the definitions of fractional Laplacian, we prove Besov regularity results for fractional p-Laplacian, which could also be extended to a wide class of fractional quasi-linear problems. Next, we discuss the numerical methods for the fractional p-Laplacian problem and obtain convergence rates based on our regularity results. Finally, we introduce a DG method for the linear fractional Laplacian motivated by an integration by parts formula. We prove that the bilinear form of this DG formulation satisfies the desired properties and prove error estimates.

Host person: Jea-Hyun Park

Abstract: Motivated by re-weighted L1 approaches for sparse recovery, we propose a lifted L1 (LL1) regularization that can be generalized to several popular regularizations in the literature. During the course of reformulating the existing methods into our framework, we discover two types of lifting functions that can guarantee that the proposed approach is equivalent to the L0 minimization. Computationally, we design an efficient algorithm via the alternating direction method of multiplier (ADMM) and establish the convergence for an unconstrained formulation. Experimental results are presented to demonstrate how this generalization improves sparse recovery over the state-of-the-art. This is a joint work with Yaghoub Rahimi and Sung Ha Kang from the Georgia Institute of Technology. 

Host person: Sui Tang