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August 28— "Eulerian and Lagrangian transport in wall-bounded turbulent flows"

• Presenter: Andrew Grace, CU-APPM
• Abstract:ÌýThe dynamics of fluids in Earth's waters and atmosphere profoundly affect many aspects of our lives, including weather forecasting, disaster response, and our climate. In this talk, I will present some results surrounding fluid mixing, turbulence, and Lagrangian particle dispersion, which represent several of the important underlying physical processes in these applications. I will begin by providing an overview of how we model the motion of incompressible fluids (the incompressible Navier-Stokes equations under the Boussinesq approximation) and show examples of numerical solutions to these equations in different contexts. I will then highlight recent results focused on how we can apply these equations to model turbulent transport of solid particles in buoyancy driven and mechanically driven turbulent flows. Specifically, I will discuss how multi-scale fluid structures in wall-bounded turbulent flows mediate dispersion processes near solid surfaces, and how we can model these interactions. Finally, I will finish the talk by highlighting several exciting avenues for future research.

September 25 — "Oscillation of graph eigenfunctions and its applicationsÌý"

• Presenter: Gregory Berkoilaiko, Texas A&M

• Abstract:ÌýOscillation 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. ÌýIts 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. ÌýConversely, 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. Ìý

  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. ÌýWe 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. ÌýFirst, 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. ÌýSecond, 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. ÌýFinally, 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 Joint work with Jared Bronski (UIUC ) and Mark Goresky (Princeton).

October 2 — Ìý"Applied Math for the Heart; Take a few PDEs and call me in the morning"

• Presenter: Flavio Fenton, Georgia Institute of Technology

• Abstract: 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. ÌýThen, 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).Ìý 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.Ìý 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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November 6— "Flexible and Efficient Spatial Extremes Estimation and Emulation via Variational Autoencoders"

• Presenter: Christopher Wikle, University of Missouri

• Abstract: The world is full of extreme events.Ìý 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.).Ìý Such extreme events typically occur in spatial and/or temporal clusters.Ìý 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.Ìý 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.Ìý 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.Ìý 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).Ìý

  Primary References: ,

December 4— "TBH"

• Presenter: Maria D'Orsogna, California State University Northridge,
• Abstract:
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