# Applied Math Colloquium - Spring 2021

Colloquia are held on Fridays at 11:30 a.m. via Webex, unless otherwise noted. Webex information will be posted at a later date.

Colloquia are held on Fridays at 11:30 a.m. via Webex, unless otherwise noted. Webex information will be posted at a later date.

To join the Applied Mathematics Colloquium mailing list visit https://groups.google.com/a/njit.edu/forum/?hl=en#!forum/math-colloquium/join (Google Profile required). To join the mailing list without a Google Profile, submit the seminar request form.

**Jeff Calder (University of Minnesota)**

Random Walks and PDEs in Graph-Based Learning

I will discuss some applications of random walks and PDEs in graph-based learning, both for theoretical analysis and algorithm development. Graph-based learning is a field within machine learning that uses similarities between datapoints to create efficient representations of highdimensional data for tasks like semi-supervised classification, clustering and dimension reduction. There has been considerable interest recently in semi-supervised learning problems with very few

labeled examples (e.g., 1 label per class). The widely used Laplacian regularization is ill-posed at low label rates and gives very poor classification results. In the first part of the talk, we will use the random walk interpretation of the graph Laplacian to precisely characterize the lowest label rate at which Laplacian regularized semi-supervised learning is well-posed. At lower

label rates, where Laplace learning performs poorly, we will show how our random walk analysis leads to a new algorithm, called Poisson learning, that is probably more stable and informative than Laplace learning. We will conclude with some applications of Poisson learning to image classification and mesh segmentation of broken bone fragments of interest in anthropology.

**Alex Gittens (RPI)**

An Algorithm for Two-Cost Budgeted Matrix Completion

Matrix completion is a ubiquitous tool in machine learning and data analysis. Most work in this area has focused on the number of observations necessary to obtain an accurate lowrank approximation. In practice, however, the cost of observations is an important limiting factor, and experimentalists may have on hand multiple modes of observation with differing noise-vs-cost trade-offs. This work considers matrix completion

subject to such constraints: a budget is imposed and the experimentalist's goal is to allocate this budget between two sampling modalities in order to recover an accurate low-rank approximation. Specifically, we consider that it is possible to obtain low noise, high cost observations of individual entries or high noise, low cost observations of entire columns. We introduce a regression-based completion algorithm for this setting and experimentally verify the performance of our approach on both synthetic and real data

sets. When the budget is low, our algorithm outperforms standard completion algorithms. When the budget is high, our algorithm has comparable error to standard nuclear norm completion algorithms and requires much less computational effort

**Denis Silantyev (Courant Institute, NYU) **

Obtaining Stokes Wave with High-Precision Using Conformal Maps and Spectral Methods on Non-Uniform Grids

Two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth has a class of solutions called Stokes waves which is fully nonlinear periodic gravity waves propagating with the constant velocity. We developed a new highly efficient method for computation of Stokes waves. The convergence rate of the numerical approximation by a Fourier series is determined by the complex singularity of the travelling wave in the complex plane above the free surface. We study this singularity and use an auxiliary conformal mapping which moves it away from the free surface thus dramatically speeding up Fourier series convergence of the solution. Three options for the auxiliary conformal map are described with their advantages and disadvantages for numerics. Their efficiency is demonstrated for computing Stokes waves near the limiting Stokes wave (the wave of the greatest height) with 100-digit precision. Drastically improved convergence rate significantly expands the family of numerically accessible solutions and allows us to study the oscillatory approach of these solutions to the limiting wave in great detail.

**Haomin Zhou (Georgia Tech)**

Optimal Transport on Graphs with Some Applications

Optimal transport theory in continuous space has been extensively studied in the past few decades. In this talk, I will present the optimal transport theory on discrete spaces. Various recent developments related to free energy, FokkerPlanck equations, as well as Wasserstein distance on graphs will be presented. some of them are surprising in the discrete case. Applications in robotics as well as Schrodinger equation on graphs will be discussed briefly.

**Nigel Mottram (Glasgow)**

Active Nematic Liquid Crystals

Active nematic liquid crystals are fluids in which continuous internal energy generation, such as in bacterial suspensions and microtubule-forming suspensions, allows for spontaneous flow generation. In such fluids, the flow-generating agent usually possesses a shape anisotropy (defined by, for instance, the long axis of the bacterium or microtubule), with this symmetry giving rise to a liquid crystalline-like phase. Internally driven flows can lead to interesting effects such as self-organisation and non-equilibrium defect configurations. In this talk I will survey the main features of dense systems of such self-propelling agents and highlight some recent results from our work in the area. For instance, the director orientation and flow of an active nematic liquid crystal confined between two parallel glass plates shows a complex interaction between orientation, self-induced flow and the flow generated through director rotation (often called backflow). In two dimensions the behaviour is even more complicated, although in certain asymptotic limits (high or low Ericksen number) we discover the importance of corner regions in driving the bulk flow. Finally, I will consider some very recent experimental results and propose an area of future work which may lead to a deeper understanding of how bacteria use self-organisation to assist spreading of thin films.

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**Chad Higdon-Topaz (Williams College)**

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**No Colloquium - Spring Recess**

**Robert Style (ETH Zurich)**

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**David Anderson (University of Wisconsin)**

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**Chris Rycroft (Harvard SEAS)**

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February 23, 2021