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ACMS Abstracts: Spring 2021

Christina Kurzthaler (Princeton)

Complex Transport Phenomena

Abstract: Self-propelled agents are intrinsically out of equilibrium and exhibit a variety of unusual transport features. In this talk, I will discuss the spatiotemporal dynamics of catalytic Janus colloids characterized in terms of the intermediate scattering function. Our findings show quantitative agreement of our analytic theory for the active Brownian particle model with experimental observations from the smallest length scales, where translational diffusion and self-propulsion dominate, up to the larges ones, which probe the rotational diffusion of the active agents. In the second part of this talk, I will address the hydrodynamic interactions between sedimenting particles and surfaces with corrugated topographies, omnipresent in biological and microfluidic environments. I will present an analytic theory for the roughness-induced mobility and discuss the sedimentation behavior of a sphere next to periodic and randomly structured surfaces.

Antoine Remond-Tiedrez (UW)

Instability of an Anisotropic Micropolar Fluid

Abstract: Many aerosols and suspensions, or more broadly fluids containing a non-trivial structure at a microscopic scale, can be described by the theory of micropolar fluids. The resulting equations couple the Navier-Stokes equations which describe the macroscopic motion of the fluid to evolution equations for the angular momentum and the moment of inertia associated with the microcopic structure. In this talk we will discuss the case of viscous incompressible three-dimensional micropolar fluids. We will discuss how, when subject to a fixed torque acting at the microscopic scale, the nonlinear stability of the unique equilibrium of this system depends on the shape of the microstructure.

Hugo Touchette (Stellenbosch University)

Large deviation theory: From physics to mathematics and back

Abstract: I will give a basic overview of the theory of large deviations, developed by Varadhan (Abel Prize 2007) in the 1970s, and of its applications in statistical physics. In the first part of the talk, I will discuss the basics of this theory and its historical sources, which can be traced back in mathematics to Cramer (1938) and Sanov (1960) and, on the physics side, to Einstein (1910) and Boltzmann (1877). In the second part, I will show how the theory can be applied to study equilibrium and nonequilibrium systems and to express many key concepts of statistical physics in a clear mathematical way.

Tijana Pfander (Ludwig-Maximilians-University of Munich)

Towards next generation data assimilation algorithms for convective scale applications

Abstract: The initial state for a geophysical numerical model is produced by combining observational data with a short-range model simulation using a data assimilation algorithm. Particularly challenging is the application of these algorithms in weather forecasting at the convective scale. For convective scale applications, high resolution nonlinear numerical models are used. In addition, intermittent convection is present in the simulations and observations, often leading to errors in locations and intensity of convective storms. In addition, the state vector has a large size, one third of which contains variables whose non-negativity needs to be preserved, and the estimation of the state vector has to be done frequently in order to catch fast changing convection. Finally, often, not only one, but rather an ensemble of predictions is needed in order to correctly specify, for example, the uncertainty of rain at a particular location, even further increasing the computational considerations. In current practice, many data assimilation methods do not preserve the non-negativity of variables and rely on Gaussian assumptions. We present an algorithm that could be used for weather forecasting at the convective scale, that is based on the ensemble Kalman filter (EnKF) and quadratic programming. This algorithm outperforms the EnKF as well as the EnKF with the lognormal change of variables for all ensemble sizes. For a model that was designed to mimic the important characteristics of convective motion, preserving non-negativity of rain and conserving mass reduce the error in all fields; they prevent the data assimilation algorithm from producing artificial mass or artificial rain. Finally, important reduction in the computational costs has been recently achieved, making it possible to apply this algorithm in high dimensional weather forecasting problems in the future.

Quanling Deng (UW)

Spectral approximation of elliptic operators by softFEM, isogeometric analysis, and the hybrid high-order method

Abstract: In this talk, I will first introduce the variational formulation for the numerical spectral approximation of the second-order elliptic operators, followed by the introduction of the particular methods: softFEM, isogeometric analysis (IGA), and the hybrid high-order (HHO) method. The main idea of softFEM is to reduce the stiffness of the variational problem by subtracting to the standard stiffness bilinear form a least-squares penalty on the gradient jumps across the mesh interfaces. I will discuss briefly the motivation and why one wants to soften the stiffness of the resulting systems arising from the classical FEM. I will present a sharp upper bound on the softness parameter weighting the stabilization bilinear form so as to maintain coercivity for the softFEM bilinear form and then prove that softFEM delivers the optimal convergence rates as the standard Galerkin FEM approximation for the eigenvalues and the eigenvectors. The main idea of IGA is to apply highly-smooth basis functions within the Galerkin FEM framework. For this method, I will present dispersion analysis and develop analytical eigenpairs for the resulting generalized matrix eigenvalue problems. Lastly, the HHO method is formulated using cell and face unknowns which are polynomials of some degree. The key idea for the discrete eigenvalue problem is to introduce a discrete operator where the face unknowns have been eliminated. Using the abstract theory of spectral approximation of compact operators in Hilbert spaces, we prove that the eigenvalues converge as $h^{2t}$ and the eigenfunctions as $h^{t}$ in the $H^1$-seminorm, where $h$ is the mesh-size, $t\in [s,k+1]$ depends on the smoothness of the eigenfunctions, and $s>\frac12$ results from the elliptic regularity theory. The convergence rates for smooth eigenfunctions are thus $h^{2k+2}$ for the eigenvalues and $h^{k+1}$ for the eigenfunctions. The first two methods are continuous Galerkin (CG) methods while the third one is a discontinuous Galerkin (DG) method. I will make comparisons by showing some numerical examples.

Oscar Bruno (Caltech)

"Interpolated Factored Green Function" Method for accelerated solution of Scattering Problems

Abstract: We present a novel "Interpolated Factored Green Function" method (IFGF) for the accelerated evaluation of the integral operators in scattering theory and other areas. Like existing acceleration methods in these fields, the IFGF algorithm evaluates the action of Green function-based integral operators at a cost of O(N\log N) operations for an N-point surface mesh. Importantly, the proposed method does not utilize previously-employed acceleration elements such as the Fast Fourier transform (FFT), special-function expansions, high-dimensional linear-algebra factorizations, translation operators, equivalent sources, or parabolic scaling. Instead, the IFGF strategy, which leads to an extremely simple algorithm, capitalizes on slow variations inherent in a certain Green-function "analytic factor", which is analytic up to and including infinity, and which therefore allows for accelerated evaluation of fields produced by groups of sources on the basis of a recursive application of classical interpolation methods. In particular, the IFGF method runs on a small memory footprint, and, as it does not utilize the Fast Fourier Transforms (FFT), it is better suited than other methods for efficient parallelization in distributed-memory computer systems. Related integral equation techniques and associated device-optimization problems will be mentioned, including a novel time-domain scattering solver that accurately and effectively solves time-domain problems of arbitrary duration via Fourier transformation in time. (IFGF work in collaboration with graduate student Christoph Bauinger. Device-optimization work in collaboration with former postdoc Constantine Sideris and former students Emmanuel Garza and Agustin Fernandez-Lado. Time-domain work, in collaboration with former graduate student Thomas Anderson.)

Yulong Lu (UMass)

Theoretical guarantees of machine learning methods for statistical sampling and PDEs in high dimensions

Abstract: Neural network-based machine learning methods, including the most notably deep learning have achieved extraordinary successes in numerous fields. In spite of the rapid development of learning algorithms based on neural networks, their mathematical analysis are far from understood. In particular, it has been a big mystery that neural network-based machine learning methods work extremely well for solving high dimensional problems.

In this talk, I will demonstrate the power of neural network methods for solving two classes of high dimensional problems: statistical sampling and PDEs. In the first part of the talk, I will present a universal approximation theorem of deep neural networks for representing high dimensional probability distributions. In the second part of the talk, I will discuss the generalization error analysis of the Deep Ritz Method for solving high dimensional elliptic PDEs. For both problems, our theoretical results show that neural networks-based methods can overcome the curse of dimensionality.

Michelle DiBenedetto (University of Washington)

Small particles in surface gravity waves: Stokes drift and Stokes flow

Abstract: Plastic pollution in the ocean breaks down and persists as small particles, or microplastics. Accurately assessing microplastics sources and sinks requires a thorough understanding of the transport and dispersal of microplastics in the ocean. In this talk, I will consider microplastics transport in surface gravity waves, as waves control many transport processes at the ocean surface and along the coasts where microplastic pollution is abundant. In particular, I will focus on how the characteristics of microplastic particles, such as their size and shape, affect their horizontal and vertical transport. Vertical transport, or settling, of non-neutrally buoyant particles, is shown to increase under surface gravity waves. In addition, particle orientation is considered. Using a finite-amplitude correction, I show how non-spherical particles tend to a preferred orientation under waves. This orientational behavior is a consequence of how the particles sample the wave flow and is found to be the angular analog of Stokes drift. The mean preferred orientation is found to be solely a function of the particle’s shape. The implications of these results are then discussed with relevance to real microplastics in the ocean.

Melvin Leok (University of California, San Diego)

The Connections Between Discrete Geometric Mechanics, Information Geometry, Accelerated Optimization and Machine Learning

Abstract: Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-$h$ flow map of a Hamiltonian system on the space of probability distributions. We will also discuss how time-adaptive Hamiltonian variational integrators can be used to discretize the Bregman Hamiltonian, whose flow generalizes the differential equation that describes the dynamics of the Nesterov accelerated gradient descent method.

Cecilia Mondaini (Drexel University)

Rates of convergence to statistical equilibrium: a general approach and applications

Randomness is an intrinsic part of many physical systems. For example, it might appear due to uncertainty in the initial data, or in the derivation of the mathematical model, or also in observational measurements. In this talk, we focus on the study of convergence/mixing rates for stochastic/random dynamical systems towards statistical equilibrium. Our approach uses the weak Harris theorem combined with a generalized coupling technique to obtain such rates for infinite-dimensional stochastic systems in a suitable Wasserstein distance. We emphasize the importance of obtaining these results via algorithms that are well-defined in infinite dimensions. This allows to obtain convergence rates that are robust with respect to finite-dimensional approximations, thus beating the curse of dimensionality. In particular, we show two scenarios where this approach is applied in the context of stochastic fluid flows. First, to show that Markov kernels constructed from a suitable numerical discretization of the 2D stochastic Navier-Stokes equations converge towards the invariant measure of the continuous system. Second, to approximate the posterior measure obtained via a Bayesian approach to inverse PDE problems, particularly when applied to advection-diffusion type PDEs. Here we present an alternative proof of mixing rates for the exact preconditioned Hamiltonian Monte Carlo algorithm in infinite dimensions, a result that was still an open problem until quite recently. This talk is based on joint works with Nathan Glatt-Holtz (Tulane U).

Shukai Du (University of Wisconsin-Madison)

Generalized projection-based error analysis of HDG methods

Hybridizable Discontinuous Galerkin (HDG) methods belong to a subclass of Discontinuous Galerkin (DG) methods which can be efficiently solved by static condensation. On the other hand, many conforming and mixed Finite Element (FE) methods can be related to HDG methods with certain stabilization functions. HDG can be therefore considered as a class of methods that lies in between the traditional conforming FE and DG methods, enabling them to inherit advantages from both sides. However, this fact also suggests that the class of HDG essentially contains very different species of methods and they are usually analyzed by very different approaches.

In the talk, I will introduce the so-called generalized projection-based error analysis framework of HDG methods, enabling a unified analysis. I will mention three applications of this framework - (1) transient elastic waves, (2) static Maxwell's equations, and (3) Stokes's equations.

Matthew Junge (City University of New York)

Modeling COVID-19 Spread in Universities

Abstract: University policy surrounding COVID-19 often involves big decisions informed by minimal data. Models are a tool to bridge this divide. I will describe some of the work that came out during Summer of 2020 to inform college reopening for Fall 2020. This includes a stochastic, agent-based model on a network for infection spread in residential colleges that I developed alongside a biologist, computer scientist, and group of students [1]. Time-permitting, I will describe a new project that aims to predict the impact of vaccination on infection spread in urban universities during the Fall 2021 semester. Disclaimer: I self-identify as a "pure" probabilist who typically proves theorems about particle systems [2]. These projects arose from my feeling compelled to help out to the best of my abilities during the height of the pandemic.

Skylar Burton (University of Wisconsin-Madison)

A mathematical model of contact tracing during the 2014-2016 West African Ebola outbreak

Abstract: The 2014-2016 West African outbreak of Ebola Virus Disease (EVD) was the largest and most deadly to date. Contact tracing plays a vital role in controlling such outbreaks. Our work mechanistically represents the contact tracing process to illustrate potential areas of improvement in managing contact tracing efforts. We also explore the role contact tracing played in eventually ending the outbreak. Our ODE model of contact tracing in Sierra Leonne includes the novel features of counting the total number of people being traced and tying this directly to the number of tracers doing the work. Our work highlights the importance of incorporating changing behavior into one’s model as needed when indicated by the data and reported trends. If time permits, I will also talk about optimal control of some discrete-time models of harvesting which contradict the prevailing wisdom that harvesting earlier is always better and exhibit surprising dynamics.