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Department of Mathematics

Spring School 2019

View the recordings of all the lectures on our YouTube Playlist, or select a specific presentation below.



Dates: March 17 – March 20 (2019)
Location: University of South Carolina
Topic: Models and Data

Sponsored by the National Science Foundation

The 2019 Spring School has been funded by the National Science Foundation to support all participants. It features five lecturers who are considered high-caliber representatives of their respective areas of expertise. The lectures are tutorial in nature, interactive, and are interlaced with open, group discussions.

The interactive lectures cover a diversity of topics organized under our main them, namely to foster synergetic synthesis of on the one hand classical "model-based" and, on the other hand "data-driven" methodologies. The topics addressed in the Spring Schools pertain to ongoing vibrant developments in areas like machine learning, notable deep learning, computational harmonic analysis, as a pillar of data science, or uncertainty quantification and related modelling or numerical analysis aspects.

The topics concern several relevant areas in Data Science which is an emerging transdisciplinary scientific field or research. The mathematical concepts treated in lectures and discussions are therefore relevant for a large range of disciplines where data in combination with physical/mathematical models provide major sources of information.

Invited Speakers

Profile picture of Albert Cohen

Albert Cohen

Least-Squares Methods for High Dimensional Problems

Abstract: Various mathematical problems are challenged by the fact they involve functions of a very large number of variables. Such problems arise naturally in learning theory, partial differential equations or numerical models depending on parametric or stochastic variables. They typically result in numerical difficulties due to the so-called "curse of dimensionality". We shall first discuss how these difficulties may be theoretically handled in the context of stochastic-parametric PDE’s through the concept of sparse polynomial approximation. We shall then focus on a class of concrete algorithms based on least-squares fitting that provably achieve the convergence properties of these approximations.

Profile image of Ronald DeVore

Ronald DeVore

An Overview of Approximation of Functions of Many Variables

Abstract: We discuss Approximation in High Dimensions, focusing on entropy widths, optimal recovery, model classes, etc. We will round out the discussion with polynomial approximation in high dimensions and neural networks, which is only peripherally high dimensional.

Profile image of Gitta Kutyniok

Gitta Kutyniok

Theory of Deep Learning

Abstract: Dr. Kutyniok’s main research topics include applied harmonic analysis, compressed sensing, data science, deep learning, frame theory, high dimensional data analysis, imaging science, inverse problems, machine learning, numerical analysis of partial differential equations, sparse approximation.

Profile image of Eitan Tadmore

Eitan Tadmore

Collective Dynamics: Emergent Behavior with Long-Range and Short-Range Interactions

Abstract: Collective dynamics is driven by alignment that tend to self-organize the crowd, and by different external forces that keep the crowd together. Prototype models based on environmental averaging are found in opinion dynamics of human networks, self-organization of biological organisms, and rendezvous of mobile systems. Different emerging equilibria are self-organized into parties, flocks, tissues, etc. I will overview recent results of collective dynamics driven by different "rules of engagement". I begin with a survey of several classical models of agent-based systems, and follow with two fundamental questions that arise in the context of such systems, namely --- their large time behavior and their large crowd dynamics. In particular, I address the question how short-range interactions lead, over time, to the emergence of long-range patterns, comparing geometric vs. topological interactions. I conclude with a general framework which describes the competition between pairwise alignment with external forcing.

Profile image of Changui Tan

Changui Tan

Asymptotic Preserving Schemes on Kinetic Models with Singular Limits

Abstract: We will discuss kinetic models with singular hydrodynamic limits. The asymptotic preserving (AP) schemes aim to provide a universal solver for both the full system and the limit system, in the sense that the stability does not depend on the parameter. For systems that have mono-kinetic singular limits, standard AP schemes lose accuracy when the parameter is close to the limit. To overcome such difficulty, we introduce a velocity scaling method that transforms the singular limit to a non-singular one, and build AP schemes on the transformed systems.

Qingguang Guan

Poor Global Optima for Fully Connected Deep ReLU Neural Networks - Special Examples for HD Approximation

Majid Noroozi

Clustering in Popularity Adjusted Stochastic Block Model

Simon Brugiapaglia

Compressive Sensing Approaches for High-Dimensional Function Approximation

Challenge the conventional. Create the exceptional. No Limits.