Colloquium

Applied Mathematics Department Colloquium - Maria D'Orsogna

Joshua Jeng — Mon, 17 Nov 2025 23:13:00 +0000

Applied Mathematics Department Colloquium - Maria D'Orsogna Joshua Jeng Mon, 11/17/2025 - 16:13

Maria-Rita D'Orsogna; Department of Mathematics; California State University Northridge

First Passage Properties of Velocity Jump Processes

First passage problems study the time it takes for a stochastic process, such as a random walk, to reach a target for the first time. These problems arise in many applications across physics, biology, and finance, in which reaching a target can trigger irreversible downstream events such as domain exit, biochemical reactions, or financial selloffs. Classical formulations typically assume diffusive, continuous dynamics, leading to analytical expressions for the survival probability and the mean first passage time (MFPT) to the target. Many real-world stochastic phenomena, however, are more accurately described by velocity jump processes (VJPs), characterized by persistent, directed motion interrupted by random velocity changes. Despite their ubiquity, the first passage properties of VJPs remain understudied. In this talk, we will present a general framework for estimating first passage properties of VJPs with fixed speed and random reorientations that follow a given angular distribution, such as the von Mises-Fisher, wrapped Cauchy, or elliptical distribution. Asymptotic expressions are derived for the MFPT to a target in the low Knudsen number regime, where the mean free path is small compared to the distance to the target. Explicit solutions are obtained for VJPs in two- and three-dimensional circular domains under radial symmetry. Remarkably, the MFPT scaling in the narrow capture problem can differ substantially from the classical diffusive prediction.

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Applied Mathematics Department Colloquium - Christopher Wikle

Joshua Jeng — Thu, 06 Nov 2025 23:09:54 +0000

Applied Mathematics Department Colloquium - Christopher Wikle Joshua Jeng Thu, 11/06/2025 - 16:09

Christopher Wikle, Department of Statistics, University of Missouri

Flexible and Efficient Spatial Extremes Estimation and Emulation via Variational Autoencoders

The world is full of extreme events. For example, a central question in public health planning might be to assess the likelihood of extreme exposures (meteorological conditions, air pollution, social stress, etc.). Such extreme events typically occur in spatial and/or temporal clusters. Yet, the principal methodologies that statisticians deal with spatially dependent processes (Gaussian processes and Markov random fields) are not suitable for complex tail dependence structures. This is particularly true of simulation model emulation. More flexible spatial extremes models exhibit appealing extremal dependence properties but are often exceedingly prohibitive to fit and simulate from in high dimensions. Here I present recent work where we develop a new spatial extremes model that has flexible and non-stationary dependence properties, and we integrate it in the encoding-decoding structure of a variational autoencoder (XVAE), whose parameters are estimated via variational Bayes combined with deep learning. The XVAE can be used to analyze high-dimensional data or as a spatio-temporal emulator that characterizes the distribution of potential mechanistic model output states and produces outputs that have the same statistical properties as the inputs, especially in the tail. Through extensive simulation studies, we show that our XVAE is substantially more time-efficient than traditional Bayesian inference while also outperforming many spatial extremes models with a stationary dependence structure. We demonstrate our method applied to a high-resolution satellite-derived dataset of sea surface temperature in the Red Sea and to a high-resolution simulation model of a turbulent plume, such as one would find in a wildfire. We note, however, that these methods can be applied to any data set or simulation model that exhibits extremes.

This is joint work with Likun Zhang and Xiaoyu Ma (University of Missouri), Raphael Huser (KAUST), and Kiran Bhaganagar (University of Texas-San Antonio).

Primary References: https://arxiv.org/abs/2307.08079 , https://arxiv.org/abs/2502.04685

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Applied Mathematics Department Colloquium - Flavio Fenton

Joshua Jeng — Thu, 02 Oct 2025 22:09:14 +0000

Applied Mathematics Department Colloquium - Flavio Fenton Joshua Jeng Thu, 10/02/2025 - 16:09

Flavio Fenton; School of Physics; Georgia Tech

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 - Gregory Berkolaiko

Joshua Jeng — Thu, 25 Sep 2025 22:13:58 +0000

Applied Mathematics Department Colloquium - Gregory Berkolaiko Joshua Jeng Thu, 09/25/2025 - 16:13

Gregory Berkolaiko, Department of Mathematics, Texas A&M University

Oscillation of graph eigenfunctions and its applications

Oscillation theory, originally due to Sturm, seeks to connect the number of sign changes of an eigenfunction of a self-adjoint operator to the label k of the corresponding eigenvalue. Its applications run in both directions: knowing k, one may wish to estimate the zero set, or the topology of its complement, useful in clustering and partitioning problems. Conversely, knowing an eigenvector (and thus the number of its sign changes), one may want to determine if it is the ground state, useful in the linear stability analysis of solutions to nonlinear equations.

Within the setting of generalized graph Laplacians, Fiedler's theorem says that the k-th eigenvector of a tree (a graph without cycles) changes sign across exactly k-1 edges. We present a formula for the number of sign changes on a general graph, which attributes the excess sign changes to the cycles in the graph and their intersections.

This result has many interesting connections. First, it allows one to derive a simple formula for the effective mass tensor of a particular class of crystals (periodic lattices), namely the maximal abelian covers of finite graphs. Second, it can be used to efficiently determine stability of a stationary solution on a coupled oscillator network, such as the non-uniform Kuramoto model for the synchronization of a network of electrical oscillators. Finally, the determinant of the matrix which determines the excess sign changes is closely related to the graph's Kirchhoff polynomial (which counts the weighted spanning trees), hinting at connections to both Feynman amplitudes and matroids.

Based on a joint work with Jared Bronski (UI Urbana-Champaign) and Mark Goresky (IAS Princeton).

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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.

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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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