Colloquium

Applied Mathematics Department Colloquium - Anna-Karin Tornberg

Joshua Jeng — Thu, 03 Apr 2025 19:19:14 +0000

Applied Mathematics Department Colloquium - Anna-Karin Tornberg Joshua Jeng Thu, 04/03/2025 - 13:19

Anna-Karin Tornberg, Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden

Layer potentials - quadrature error estimates and approximation with error control

When numerically solving PDEs reformulated as integral equations, so called layer potentials must be evaluated. The quadrature error associated with a regular quadrature rule for evaluation of such integrals increases rapidly when the evaluation point approaches the surface and the integrand becomes sharply peaked. Error estimates are needed to determine when the accuracy becomes insufficient, and then, a sufficiently accurate special quadrature method needs to be employed.

In this talk, we motivate the use of integral equations for some problems in microfluidics. We then discuss how to estimate quadrature errors, building up from simple integrals in one dimension to layer potentials over smooth surfaces in three dimensions. We also discuss a new special quadrature technique for axisymmetric surfaces with error control. The underlying technique is so-called interpolatory semi-analytical quadrature in conjunction with a singularity swap technique. Here, adaptive discretizations and parameters are set automatically given an error tolerance, utilizing further quadrature and interpolation error estimates derived for this purpose. Several different examples are shown, including examples with rigid particles in Stokes flow.

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Applied Mathematics Department Colloquium - Michelle Girvan

Joshua Jeng — Thu, 06 Mar 2025 23:16:00 +0000

Applied Mathematics Department Colloquium - Michelle Girvan Joshua Jeng Thu, 03/06/2025 - 16:16

Michelle Girvan, Department of Physics, University of Maryland

Tailored Forecasts from Short Time Series Using Meta-Learning and Reservoir Computing

Machine learning (ML) models can be effective for forecasting the dynamics of unknown systems from time-series data, but they often require large datasets and struggle to generalize—that is, they fail when applied to systems with dynamics different from those seen during training. Combined, these challenges make forecasting from short time series particularly difficult. To address this, we introduce Meta-learning for Tailored Forecasting from Related Time Series (METAFORS), which supplements limited data from the system of interest with longer time series from systems that are suspected to be related. By leveraging a library of models trained on these potentially related systems, METAFORS builds tailored models to forecast system evolution with limited data. Using a reservoir computing implementation and testing on simulated chaotic systems, we demonstrate METAFORS’ ability to predict both short-term dynamics and long-term statistics, even when test and related systems exhibit significantly different behaviors, highlighting its strengths in data-limited scenarios.

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Applied Mathematics Department Colloquium - Vladimir Rokhlin

Joshua Jeng — Thu, 06 Feb 2025 22:02:18 +0000

Applied Mathematics Department Colloquium - Vladimir Rokhlin Joshua Jeng Thu, 02/06/2025 - 15:02

Vladimir Rokhlin, Arthur K Watson Professor of Computer Science & Mathematics, Yale University

Finding scattering resonances via generalized colleague matrices

Locating scattering resonances is a standard task in certain areas of physics and engineering. This often can be reduced to finding zeros of complex analytic functions. In this talk, I will discuss a scheme for finding all roots of a complex analytic function in a square domain in C. The scheme can be viewed as a generalization of the classical approach to finding roots of a function on an interval by first approximating it by a polynomial in the Chebyshev basis, followed by diagonalizing the so-called “colleague matrices.” This extension to the complex domain is based on several observations that enable the construction of polynomial bases that satisfy three-term recurrences and are reasonably well-conditioned, giving rise to “generalized colleague matrices.” We also introduce a special-purpose QR algorithm for finding eigenvalues of the resulting structured matrices stably and efficiently. I will demonstrate the effectiveness of the approach via several numerical examples.

https://seas.yale.edu/faculty-research/faculty-directory/vladimir-rokhlin

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Applied Mathematics Department Colloquium - Deanna Needell

Joshua Jeng — Thu, 05 Dec 2024 20:29:46 +0000

Applied Mathematics Department Colloquium - Deanna Needell Joshua Jeng Thu, 12/05/2024 - 13:29

Deanna Needell, Department of Mathematics, University of California - Los Angeles (UCLA)

Fairness and Foundations in Machine Learning

In this talk, we will address areas of recent work centered around the themes of fairness and foundations in machine learning as well as highlight the challenges in this area. We will discuss recent results involving linear algebraic tools for learning, such as methods in non-negative matrix factorization that include tailored approaches for fairness. We will showcase our derived theoretical guarantees as well as practical applications of those approaches. Then, we will discuss new foundational results that theoretically justify phenomena like benign overfitting in neural networks. Throughout the talk, we will include example applications from collaborations with community partners, using machine learning to help organizations with fairness and justice goals.

Bio: Deanna Needell earned her PhD from UC Davis before working as a postdoctoral fellow at Stanford University. She is currently a full professor of mathematics at UCLA, the Dunn Family Endowed Chair in Data Theory, and the Executive Director for UCLA's Institute for Digital Research and Education. She has earned many awards including the Alfred P. Sloan fellowship, an NSF CAREER and other awards, the IMA prize in Applied Mathematics, is a 2022 American Mathematical Society (AMS) Fellow and a 2024 Society for industrial and applied mathematics (SIAM) Fellow. She has been a research professor fellow at several top research institutes including the SLMath (formerly MSRI) and Simons Institute in Berkeley. She also serves as associate editor for several journals including Linear Algebra and its Applications and the SIAM Journal on Imaging Sciences, as well as on the organizing committee for SIAM sessions and the Association for Women in Mathematics.

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Applied Mathematics Department Colloquium - Flavio Fenton

Joshua Jeng — Thu, 07 Nov 2024 20:28:19 +0000

Applied Mathematics Department Colloquium - Flavio Fenton Joshua Jeng Thu, 11/07/2024 - 13:28

Applied Math for the Heart; Take a few PDEs and call me in the morning.

The heart is an electro-mechanical system in which, under normal conditions, electrical waves propagate in a coordinated manner to initiate an efficient contraction. In pathologic states, single and multiple rapidly rotating spiral and scroll waves of electrical activity can appear and generate complex spatiotemporal patterns of activation that inhibit contraction and can be lethal if untreated. Despite much study, many questions remain regarding the mechanisms that initiate, perpetuate, and terminate reentrant waves in cardiac tissue.

In this talk, we will discuss how we use a combined experimental, numerical and theoretical approach to better understand the dynamics of cardiac arrhythmias. We will show how mathematical modeling of cardiac cells simulated in tissue using large scale GPU simulations can give insights on the nonlinear behavior that emerges when the heart is paced too fast leading to tachycardia, fibrillation and sudden cardiac death. Then, how we can use state-of-the-art optical mapping methods with voltage-sensitive fluorescent dyes to actually image the electrical waves and the dynamics from simulations in live explanted animal and human hearts (donated from heart failure patients receiving a new heart). I will present numerical and experimental data for how period-doubling bifurcations in the heart can arise and lead to complex spatiotemporal patterns and multistability between single and multiple spiral waves in two and three dimensions. Then show how control algorithms tested in computer simulations can be used in experiments to continuously guide the system toward unstable periodic orbits in order to prevent and terminate complex electrical patterns characteristic of arrhythmias. We will finish by showing how these results can be applied in vitro and in vivo to develop a novel low energy control algorithm that could be used clinically that requires only 10% of the energy currently used by standard methods to defibrillate the heart.

Overall, I will present recent advancements in identifying and quantifying chaotic dynamics in the heart, beginning with mathematical models and extending to experimental validation. This work demonstrates how applied mathematics enables the development of innovative methods to control and terminate arrhythmias, with promising potential for clinical applications.

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Applied Mathematics Department Colloquium - Nick Trefethen

Anonymous — Thu, 03 Oct 2024 06:00:00 +0000

Applied Mathematics Department Colloquium - Nick Trefethen Anonymous (not verified) Thu, 10/03/2024 - 00:00

Nick Trefethen, Professor of Applied Mathematics in Residence, Harvard University

The AAA Algorithm for Rational Approximation

With the introduction of the AAA algorithm in 2018 (Nakatsukasa-Sete-T., SISC), the computation of rational approximations changed from a hard problem to an easy one. We've been exploring the implications of this transformation ever since. This talk will review the algorithm and then present about 15 demonstrations of applications in various areas including interpolation of missing data, analytic continuation, analysis of solutions of dynamical systems, Wiener-Hopf and Riemann-Hilbert problems, function extension, model order reduction, and Laplace, Stokes, and Helmholtz calculations.

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Applied Mathematics Department Colloquium - Ioannis Kevrekidis

Anonymous — Thu, 12 Sep 2024 19:54:09 +0000

Applied Mathematics Department Colloquium - Ioannis Kevrekidis Anonymous (not verified) Thu, 09/12/2024 - 13:54

Ioannis Kevrekidis, Bloomberg Distinguished Professor, Department of Applied Mathematics and Statistics, Johns Hopkins Whiting School of Engineering

No Equations, No Variables, No Space and No Time: Data and the Modeling of Complex Systems

I will give an overview of a research path in data driven modeling of complex systems over the last 30 or so years – from the early days of shallow neural networks and autoencoders for nonlinear dynamical system identification, to the more recent derivation of data driven “emergent” spaces in which to better learn generative PDE laws. In all illustrations presented, I will try to point out connections between the “traditional” numerical analysis we know and love, and the more modern data-driven tools and techniques we now have – and some mathematical questions they hopefully make possible for us to answer.

Yannis Kevrekidis studied Chemical Engineering at the National Technical University in Athens. He then followed the steps of many alumni of that department to the University of Minnesota, where he studied with Rutherford Aris and Lanny Schmidt (as well as Don Aronson and Dick McGehee in Math). He was a Director's Fellow at the Center for Nonlinear Studies in Los Alamos in 1985-86 (when Soviets still existed and research funds were plentiful). He then had the good fortune of joining the faculty at Princeton, where he taught Chemical Engineering and also Applied and Computational Mathematics for 31 years; seven years ago he became Emeritus and started fresh at Johns Hopkins (where he somehow is also Professor of Urology). His work always had to do with nonlinear dynamics (from instabilities and bifurcation algorithms to spatiotemporal patterns to data science in the 90s, nonlinear identification, multiscale modeling, and back to data science/ML); and he had the additional good fortune to work with several truly talented experimentalists, like G. Ertl's group in Berlin. Currently -on leave from Hopkins- he works with the Defense Sciences Office at DARPA. When young and promising he was a Packard Fellow, a Presidential Young Investigator and the Ulam Scholar at Los Alamos National Laboratory. He holds the Colburn, CAST Wilhelm and Walker awards of the AIChE, the Crawford and the Reid prizes of SIAM, he is a member of the NAE, the American Academy of Arts and Sciences, and the Academy of Athens.

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Applied Mathematics Colloquium - Dane Taylor

Anonymous — Fri, 24 May 2024 06:00:00 +0000

Applied Mathematics Colloquium - Dane Taylor Anonymous (not verified) Fri, 05/24/2024 - 00:00

Dane Taylor, Department of Mathematics and Statistics, University of Wyoming

Consensus processes over networks: Past, present, and future

Models for consensus---that is, the reaching of agreement---have been developed, e.g., to study how group decisions are collectively made within social networks, how groups of animals collectively move, and how decentralized machine-learning (ML) algorithms train on dispersed data. In this talk, I will review applications and theories for consensus processes over networks and then describe ongoing work to extend these models. First, I will discuss the important role that network structure plays in shaping consensus dynamics, including results for how the presence of community structure may or may not impact the convergence rate for decentralized ML. Motivated by emerging scenarios in which collective decisions are made by human-AI teams, I will study consensus over a system comprised of 2 asymmetrically coupled networks, which are used to model a social network supported by a network of AI agents. I will present theory for when collective decisions are obtained optimally fast (i.e., with a maximal convergence rate) and when the resulting decisions are cooperative (i.e., the final state reflects the initial states of both networks). Time permitting, I will present another generalization in which consensus is formulated for “higher-order” networks with multiway interactions encoded by a simplicial complex (i.e., as opposed to a graph that encodes pairwise interactions). This discussion will touch on a few related mathematical areas including differential equations, random matrix theory, spectral perturbation/optimization, and algebraic topology/homology.

More information about this speaker may be found at https://www.uwyo.edu/mathstats/people/faculty/taylor.html

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Applied Mathematics Colloquium - Cecilia Diniz Behn

Anonymous — Fri, 12 Apr 2024 06:00:00 +0000

Applied Mathematics Colloquium - Cecilia Diniz Behn Anonymous (not verified) Fri, 04/12/2024 - 00:00

Cecilia Diniz Behn, Department of Applied Mathematics and Statistics, Colorado School of Mines

Dynamics of sleep homeostasis

The “homeostatic sleep drive” describes sleep need that increases during wake and decreases during sleep. Using a physiologically-based mathematical model of sleep-wake regulatory neurophysiology, we simulate sleep-wake behavior under conditions representing individuals with different sleep needs. Combining this model with experimental data, we demonstrate that 1) changes in the sensitivity to homeostatic sleep pressure predict differences in the sleep of adults with long and short typical sleep durations and their responses to sleep deprivation; and 2) changes in the dynamics of homeostatic sleep pressure predict the consolidation of sleep that occurs in most children between 2- and 5-years old and the interindividual differences associated with this transition. We analyze the dynamics of the homeostatic sleep drive and its effects on sleep patterning with mathematical tools including 1-dimensional circle maps and bifurcation analysis. This framework can be used to predict effective therapeutic interventions for individuals experiencing challenges related to the patterns, duration, and/or timing of sleep.

More information about this speaker may be found at https://people.mines.edu/cdinizbe/

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Applied Mathematics Colloquium - Mark Ward

Anonymous — Fri, 08 Mar 2024 07:00:00 +0000

Applied Mathematics Colloquium - Mark Ward Anonymous (not verified) Fri, 03/08/2024 - 00:00

Mark Ward, Department of Mathematics, Purdue University

The Data Mine Model for Partnerships

The Data Mine at Purdue University is a pioneering experiential learning community for undergraduate and graduate students of any background to learn data science. The first data-intensive experience embedded in a large learning community, The Data Mine had nearly 1300 students in academic year (AY) 2022–2023 and nearly 1700 students for AY 2023–2024. For the upcoming AY 2024-2025, The Data Mine plans to have at least 2000 undergraduate and graduate students, working on more than 100 data-intensive projects from over 50+ different Corporate Partners. Our projects are interdisciplinary, data-driven projects that are open to students from any program of study. For instance, we have many opportunities for students in Aeronautical and Astronautical Engineering, Pharmaceutical Sciences, Biomedical Engineering, Business Analytics, Manufacturing, etc. Students come together with different life experiences, different levels of technical skill, but also varying ways they navigate paths to solutions because of the variety of majors represented, resulting in a more creative and robust solution than a traditional data science program.

More information about this speaker may be found at https://www.stat.purdue.edu/people/faculty/mdw.html

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