Math Colloquium - Fall 2024

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September 6

Danial Szyld, Temple University

Host: Yassine Boubendir

Randomized Householder-Cholesky QR Factorization with Multisketching

We present and analyze a new randomized algorithm called rand_cholQR for computing tall-and-skinny QR factorizations. Using one or two random sketch matrices, it is  proved that with high probability, its orthogonality error is bounded by a constant of the order of unit roundoff for any numerically full-rank matrix.An evaluation of the performance of rand_cholQR on a NVIDIA A100 GPUdemonstrates that for tall-and-skinny matrices, rand_cholQR} with multiple sketch matrices is nearly as fast as, or in some cases faster than, the state-of-the-art CholeskyQR2. Hence, compared to CholeskyQR2, rand_cholQR is more stable with almost no extra computational or memory cost, and therefore a superior algorithm both in theory and practice.

Joint work with Andrew J. Higgins, Erik G. Boman, and Ichitaro Yamazaki.

September 13

Shahriar Afkami, NJIT

Numerical methods for complex flows

I will give an overview of recent developments in numerical modeling of some complex flow problems. Examples include interfacial flows, fluid-solid interactions, viscoelastic fluids, and flows with surfactants. I will attempt to focus on methods for discretizing continuum-level differential constitutive equations for viscoelastic flows, and for interfacial flows, on an Eulerian description based on the Volume-of-Fluid method. Outstanding issues and avenues for further progress will be discussed.

September 20

Igor Aronson, Penn State University

Host: Kela Lushi

Confined Bacterial Suspensions

Previous experiments have shown [1,2] that the complex spatiotemporal vortex structures emerging in motile bacterial suspensions are susceptible to weak geometrical constraints. By a combination of continuum theory and experiments, we have shown how artificial obstacles guide the flow profile and reorganize topological defects, which enables the design of bacterial vortex lattices with tunable properties. In more recent studies, we observed the emergence of spatiotemporal chaos in a bacterial suspension confined in a cylindrical well. As the well radius increases, we observed a bifurcation sequence from a steady-state vortex to periodically reversing vortices, four pulsating vortices, and, finally, to spatiotemporal chaos (active turbulence). The results of experiments are rationalized by the analysis of the continuum model for bacterial suspensions based on the complex Swift-Hohenberg equations. Furthermore, the bifurcation sequence is explained by reduction to amplitude equations for the three lowest azimuthal modes. Equations of motion are then reconstructed from experimental data.  The results indicate that the vortex reversal precedes the onset of spatiotemporal chaos in confined active systems. 
 
[1] D Nishiguchi, IS Aranson, A Snezhko, A Sokolov, Engineering bacterial vortex lattice via direct laser lithography, Nature communications 9 (1), 4486        115       (2018)
[2] H Reinken, D Nishiguchi, S Heidenreich, A Sokolov, M Bär, S. H. L. Klapp & I. S. Aranson,  Organizing bacterial vortex lattices by periodic obstacle arrays. Commun Phys 3, 76 (2020)

September 27

Michael Siegel, NJIT

A fast mesh-free boundary integral method for two-phase flow with soluble surfactant

We present an accurate and efficient boundary integral (BI) method to simulate the deformation of drops and bubbles in Stokes flow with soluble surfactant. Surfactant which is soluble advects and diffuses in bulk fluids and adsorbs/desorbs from interfaces. Since the fluid velocity depends on the bulk surfactant concentration C, the advection-diffusion equation governing C is nonlinear. This precludes the Green’s function formulation necessary for a BI method. However, in the physically representative large Peclet number limit an analytical reduction of the surfactant dynamics does (surprisingly) admit a Green’s function formulation. Unfortunately, fast algorithms developed for similar boundary integral formulations in the case of the heat equation do not easily apply. We present a new fast algorithm for our formulation which gives a mesh-free method of solving the full moving interface problem, including soluble surfactant. The method applies to other problems involving advection-diffusion in the large Peclet number limit. Several challenging examples will be presented.

This is joint work with Michael Booty (NJIT), Samantha Evans (NJIT), and Johannes Tausch (SMU).

October 4

Jeremy Hoskins, University of Chicago

Host: Travis Askham

Edge effects at insulator interfaces

In this talk we will discuss computational methods for forward and inverse problems involving interfaces and nonlocal operators. Such problems arise naturally in a number of contexts including, inter alia, quantum optics, topological insulators, acoustics, and optics. In particular, in the first part of the talk we will focus on the problem of singular waveguides separating insulating phases in two-space dimensions. The insulating domains are modeled by a massive Schrödinger equation and the singular waveguide by appropriate jump conditions along the one-dimensional interface separating the insulators. We will discuss two integral equation based methods for solving this problem, discuss guarantees on solvability, and fast, efficient algorithms for approximating the solution. In the second part of the talk, we will turn to discussing an inverse scattering problem related to models of photon propagation in quantum optics.

October 11

Roni Barak Ventura, NJIT

Host: Zoi-Heleni Michalopoulou

Quantitative research on firearm violence - a methodological push

Firearm violence is a major public health issue in the United States (US), where rates of firearm injury are among the highest in the world and steadily increasing. Even though firearms violence has a profound impact on people's health and well-being, research on the topic is lagging due to lack of funding and available data. In order to inform policymakers and stakeholders regarding effective strategies to reduce firearm violence, new and creative quantitative methodologies need to be developed. In this talk, I will present a spatiotemporal econometric model that estimates monthly firearm ownership in every US state from two cogent proxies (background checks per capita and fraction of suicides committed with a firearm). Building on this model, I will discuss my past and present research on the integration of information theory and causal inference to identify state firearm laws that effectively reduce firearm harms.

October 18

Sergey Dyachenko, University at Buffalo

Host: Xinyu Zhao

The Stokes Waves on Ideal Fluid: Modulational Instability and Wave Breaking

The long-standing problem of linear stability of surface waves on 2D fluid is solved in conformal variables for Stokes up to nearly extreme steepness. The stability spectrum of Stokes waves exhibits recurrent transitions, multiple modulation, or Benjamin-Feir instability branches. We show that all Stokes waves are, in fact, unstable, but the nature of these instabilities varies — in some cases it leads directly to wave-breaking, and, in others, to modulational disturbance of ocean swell. We discuss the profound change in the numerical approach that allowed us to consider nearly extreme Stokes waves.

October 25

Charles Epstein, Flatiron Institute

Host: Travis Askham

Analysis and Numerics for Open Wave-guides

Standard models for opto-electronic/photonic devices often involve wave-guides in dielectric media delineated by changes in the refractive index, but without a hard boundary. Such devices, called open wave-guides,  are difficult to analyze and simulate because they typically extend to infinity, with perturbations that are not compactly supported. I will describe a new approach to solving this class of problems that reduces the scattering problem to a transmission problem across an infinite artificial interface, and then to a Fredholm system of integral equations on the interface. The infinite extent of the perturbations requires new types of radiation conditions in order to uniquely specify a solution, which I will explain. Finally, the method has been numerically implemented in a range of interesting cases by Tristan Goodwill, Shidong Jiang, Manas Rachh and Jeremy Hoskins. If time permits, I will explain how this is accomplished.

November 1

Dhairya Malhotra, Flatiron Institute

Host: Travis Askham

Efficient Convergent Boundary Integral Methods for Slender Bodies

The dynamics of active and passive filaments in viscous fluids is frequently used as a model for many complex fluids in biological systems such as: microtubules which are involved in intracellular transport and cell division;flagella and cilia which aid in locomotion. The numerical simulation of such systems is generally based on slender-body theory which give asymptotic approximations of the solution. However, these methods are low-order and cannot enforce no-slip boundary conditions to high-accuracy, uniformly over the boundary. Boundary-integral equation methods which completely resolve the fiber surface have so far been impractical due to the prohibitive cost of current layer-potential quadratures for such high aspect-ratio geometries. In this talk, we will present new quadrature schemes which make such computations possible and new integral equation formulations which lead to well-conditioned linear systems upon discretization. We will present numerical results to show the efficiency of our methods.

November 8

Miao-jung Yvonne Ou, University of Delaware

Host: Thi Phong

Fractional time derivatives, Dispersive systems, and Herglotz-Nevanlinna functions

Fractional time derivatives have been applied in many applications such as anomalous diffusion and electromagnetic waves in dispersive media due to its ability to describe non-classic phenomena. However, it is nonlocal in time by definition and poses challenges in developing a numerical solver. There are many types of fractional time derivatives - Caputo derivative, Riemann-Liouville derivatives, Caputo-Fabrizio, etc. Despite of the difference in them, they are all defined as a time convolution operator with a kernel. The kernel function has to satisfy a causality condition, which means its Fourier Laplace transform has to be a Herglotz-Nevanlinna function. This link can be utilized not only to reveal the physical meaning of these fractional time derivatives but also inform about a time-domain quadrature for handling the convolution term in a numerical solver. In this talk, all the above will be explained and a numerical example will be given.

November 15

Linda Cummings, NJIT

Thin film fluid dynamics in materials science

Thin fluid films with free surfaces are ubiquitous in nature (e.g. a layer of rainwater on a rock face) and in a range of industrial applications (e.g. painting or coating processes). Flows of this kind are characterized by a separation in length scales, with the typical film height being small compared to the characteristic lateral length scale, which allows asymptotic “long-wave” methods to be used to simplify the problem and reduce it to a nonlinear parabolic partial differential equation (first-order in time, fourth-order in space) governing the evolution of the film height, h. Such flows can be especially interesting when the fluid has complex rheology, when it interacts with the underlying substrate, and/or when other external energy sources are involved in driving flow.

We will discuss these flows, focusing on two different scenarios: (i) nanoscale thin dewetting films of nematic liquid crystals (NLCs) that interact strongly with the substrate; and (ii) the evolution of laser-irradiated metal films that, on melting, break up into droplets relevant for metallic nanoparticle manufacture. The governing PDEs are strongly nonlinear and may be stiff, hence to describe experimental-scale scenarios we implement a highly efficient numerical scheme on GPUs, finding excellent qualitative agreement with experimental results (where available).

November 22

Wen Zhou, New York University

Host: Chenlu Shi

Optimal nonparametric inference on network effects with dependent edges

Testing network effects in weighted directed networks is a foundational problem in econometrics, sociology, and psychology. Yet, the prevalent edge dependency poses a significant methodological challenge. Most existing methods are model-based and come with stringent assumptions, limiting their applicability. In response, we introduce a novel, fully nonparametric framework that requires only minimal regularity assumptions. While inspired by recent developments in U-statistic literature (Chen and Kato, 2019; Zhang and Xia, 2022), our approach notably broadens their scopes. Specifically, we identified and carefully addressed the challenge of  indeterminate degeneracy in the test statistics -- a problem that aforementioned tools do not handle. We established Berry-Esseen type bounds for the accuracy of type-I error rate control. With original analysis, we also proved the minimax power optimality of our test. Simulations underscore the superiority of our method in computation speed, accuracy, and numerical robustness compared to competing methods. We also applied our method to the U.S. faculty hiring network data and discovered intriguing findings.

December 6

Nat Trask, University of Pennsylvania

Host: Victor Matveev

Structure preserving machine learning for probabilistic digital twins

Motivated by the ever-increasing success of machine learning in language and vision models, many aim to build AI-driven tools for scientific simulation and discovery. Contemporary techniques drastically lag behind their comparatively mature counterparts in modeling and simulation however, lacking rigorous notions of convergence, physical realizability, uncertainty quantification, and verification+validation that underpin prediction in high-consequence engineering settings. One reason for this is the use of "off-the-shelf" ML architectures designed for language/vision without specialization to scientific computing tasks. In this work, we establish connections between graph neural networks and the finite element exterior calculus (FEEC). FEEC forms the backbone of modern mixed finite element methods, tying the discrete topology of geometric descriptions of space (cells, faces, edges, nodes and their connectivity) to the algebraic structure of conservations laws (the div/grad/curl theorems of vector calculus). By building a differentiable learning architecture mirroring the construction of Whitney forms, we obtain a de Rham complex supporting FEEC, allowing us to learn models combining the robustness of traditional FEM with the drastic speedups and data assimilation capabilities of ML. We then introduce a novel UQ framework based on optimal recovery in reproducing Hilbert spaces, allowing the model to quantify epistemic uncertainty, providing practical notions of trust where the model may be reliably employed. Finally, we present an architecture we have recently developed which admits conditional generative modeling, allowing one to sample from the space of finite element models consistent with given observational data in near real time.

Last Updated: December 2, 2024