## ACM Seminar

The Applied and Computational Mathematics (ACM) seminar series since 2020.

Click HERE for the schedule.

### Discontinuous Galerkin methods with local time stepping for the nonlinear shallow water equations

#### Speaker: Yulong Xing (Ohio State University)

Shallow water equations (SWEs) with a non-flat bottom topography have been widely used to model flows in rivers and coastal areas. In this presentation, we will talk about the applications of high-order well-balanced and positivity-preserving discontinuous Galerkin methods to this system. With carefully chosen numerical fluxes, we will show that the proposed methods preserve the still water steady state exactly, and at the same time maintain the non-negativity of the water height. For the temporal discretization, we propose the high order ADER-differential transform approach. Local time stepping strategy will also be studied to allow elements of different sizes to use different time steps. One- and two-dimensional numerical tests are performed to verify the well-balanced property, high-order accuracy, and good resolution for general solutions.

Time: March 19, 2021 3:30pm-4:30pm

Location: Virtually via Zoom

Host: Lili Ju

### Analysis and numerics for nematic liquid crystals

#### Speaker: Shawn Walker (Louisiana State University)

I start with an overview of nematic liquid crystals (LCs) and their applications, including how they are modeled, such as Oseen-Frank, Landau-de Gennes, and the Ericksen model. For the rest of the talk, I will focus on Landau-de Gennes (LdG) and Ericksen.

Next, I describe some of the analytical difficulties of these models. For example, the Ericksen model exhibits a non-smooth constraint (for the PDE solution), and the LdG model with uniaxiality enforced as a hard constraint is also non-smooth. I will then discuss related numerical analysis issues that arise and how we handle these difficulties with a structure-preserving finite element method (FEM) for computing energy minimizers. We prove stability and consistency of the method without regularization, and \(\Gamma\)-convergence of the discrete energies towards the continuous one as the mesh size goes to zero. Numerical simulations will be presented in two and three dimensions, some of which include non-orientable line fields, using a provably robust minimization scheme. Finally, I will conclude with some current problems and an outlook to future directions.

Time: March 5, 2021 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Qi Wang

### From signal representations to musical creation: a geometric approach

#### Speaker: Carmine Emanuele Cella (University of California, Berkeley)

Carmine Emanuele Cella, assistant professor in music and technology at CNMAT, will present work done in the last years in searching good signal representations that permit high-level manipulation of musical concepts. After the definition of a geometric approach to signal representation, the theory of sound-types and its application to music will be presented. Finally, recent research on assisted orchestration will be shown and some possible musical applications will be proposed, with connections to deep learning methods.

This is a joint seminar with the Composition Seminar in the School of Music.

Time: February 26, 2021 2:30pm-4:00pm

Location: Virtually via Zoom

Host: Qi Wang and Yi Sun

### Numerical methods for solving nonlinear differential equations from homotopy methods to machine learning

#### Speaker: Wenrui Hao (Pennsylvania State University)

Many systems of nonlinear differential equations are arising from engineering and biology and have attracted research scientists to study the multiple solution structure such as pattern formation. In this talk, I will present several methods to compute the multiple solutions of nonlinear differential equations. First, I will introduce the homotopy continuation technique to compute the multiple steady states of nonlinear differential equations and also to explore the relationship between the number of steady-states and parameters. Then I will use the machine learning techniques to solve nonlinear differential equations and learn the multiple solutions by developing a randomized Newton's method for the neural network discretization. Several benchmark problems will be used to illustrate these ideas.

Time: February 19, 2021 3:30pm-4:30pm

Location: Virtually via Zoom

Host: Qi Wang

### Aggregation with intrinsic interactions on Riemannian manifolds

#### Speaker: Razvan Fetecau (Simon Fraser University)

We consider a model for collective behaviour with intrinsic interactions on Riemannian manifolds. We establish the well-posedness of measure solutions, defined via optimal mass transport, on several specific manifolds (sphere, hypercylinder, rotation group \(SO(3)\)), and investigate the mean-field particle approximation. We study the long-time behaviour of solutions, where the primary goal is to establish sufficient conditions for a consensus state to form asymptotically. The analytical results are illustrated with numerical experiments that exhibit various asymptotic patterns.

Time: February 12, 2021 3:30pm-4:30pm

Location: Virtually via Zoom

Host: Changhui Tan

### Transport information dynamics with applications

#### Speaker: Wuchen Li (University of South Carolina)

In this talk, I briefly reviewed several dynamical equations, raised in optimal transport, information geometry and mean field game modeling. I will discuss the applications of these dynamics in AI and Bayesian sampling problems.

Time: January 22, 2021 2:30pm-3:30pm

Location: Virtually via Zoom

### Linear interpolation convexity/concavity in the minimization of attractive-repulsive energy

#### Speaker: Ruiwen Shu (University of Maryland)

Energy minimization problems of attractive-repulsive pairwise interactions are very important in the study of pattern formation in biological and social sciences. In this talk, I will discuss some recent progress (joint work with Jose Carrillo) on the study of Wasserstein-\(\infty\) local energy minimizers by using the method of linear interpolation convexity/concavity. In the first part, we prove the radial symmetry and uniqueness of local minimizers for interaction potentials satisfying the 'linear interpolation convexity', which generalizes the result of O. Lopes 17' for global minimizers. In the second part, we show that the failure of linear interpolation convexity could lead to the formation of small scales in the support of local minimizers, and construct interaction potentials whose local minimizers are supported on fractal sets. To our best knowledge, this is the first time people observe fractal sets as the support of local minimizers.

Time: Febrary 5, 2021 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Changhui Tan

### Self-similar Solutions of Active Scalars with Critical Dissipation

#### Speaker: Dallas Albritton (New York University)

In PDE analyses of fluid models, often we may identify a so-called critical space that lives precisely at the borderline between well-posedness and ill-posedness. What happens at this borderline? We explore this question in two active scalar equations with critical dissipation. In the critical surface quasi-geostrophic equations, we investigate the connection between non-uniqueness and large self-similar solutions that was established by Jia, Sverak, and Guillod in the Navier-Stokes equations. This is joint work with Zachary Bradshaw. In the critical Burgers equation, and more generally in critical scalar conservation laws, the analogous self-similar solutions are unique, and we show that all front-like solutions converge to a self-similar solution at the diffusive rates. This is joint work with Raj Beekie.

Time: November 13, 2020 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Changhui Tan

### Some Analytic Properties of a Singular Potential in the Laudau-de Gennes Theory for Liquid Crystals

#### Speaker: Xiang Xu (Old Dominion University)

The Landau-de Gennes theory is a type of continuum theory that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. In the free energy, there is a singular bulk potential which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. In this talk we shall discuss some analytic properties related to this singular potential.

Time: October 30, 2020 3:30pm-4:30pm

Location: Virtually via Zoom

Host: Changhui Tan

### A Proximal-gradient Algorithm for Crystal Surface Evolution

#### Speaker: Li Wang (University of Minnesota)

We develop a novel semi-implicit scheme for the crystal surface evolution equation, which suffers from significant stiffness that prevents simulation on fine spatial grids. Our method leverages the formal structure of the equation as the gradient flow of the total variation energy, with respect to a weighted \(H^{-1}\) norm. Inspired by the classical minimizing movement scheme, we reformulate the semi-implicit time discretization into an optimization problem, and then use a primal-dual hybrid gradient (PDHG) method to compute the minimizer. In one dimension, we prove the convergence of PDHG to the semi-implicit scheme, and show that, at the discrete level, our PDHG method converges at a rate independent of the grid size. This is joint work with Katy Craig, Jian-Guo Liu, Jianfeng Lu and Jeremy Marzuola.

Time: October 23, 2020 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Changhui Tan

#### More...

### Approximating the Riemannian Metric from Discrete Samples

#### Speaker: Barak Sober (Duke University)

The approximation of both geodesic distances and shortest paths on point cloud sampled from an embedded submanifold \(M\) of Euclidean space has been a long-standing challenge in computational geometry. Given a sampling resolution parameter \(h\), state-of-the-art discrete methods yield \(O(h)\) provable approximation rates. In this work, we prove that the Riemannian metric of the Moving Least-Squares Manifold (Manifold-MLS) introduced by (Sober & Levin 19) has approximation rates of \(O(h^{k-1})\). In other words, the Manifold-MLS is nearly an isometry with high approximation order. We use this fact to devise an algorithm that computes geodesic distances between points on \(M\) with the same rates of convergence. Finally, we show the potential and the robustness to noise of the proposed method in some numerical simulations.

Time: October 16, 2020 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Wolfgang Dahmen

### A New Finite Element Approach for Nonlinear Eigenvalue Problems

#### Speaker: Jiguang Sun (Michigan Technological University)

We propose a new finite element approach, which is different from the classic Babuska-Osborn theory, for some nonlinear eigenvalue problems. The eigenvalue problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. Finite element methods are used for discretization. The convergence of eigenvalues/eigenvectors is proved using the abstract approximation theory for holomorphic operator functions. Then the spectral indicator method is extended to compute the eigenvalues/eigenvectors. Two nonlinear eigenvalue problems are treated using the proposed approach.

Time: October 9, 2020 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Qi Wang

### Geometric Structure of Mass Concentration Sets for Pressureless Euler Alignment Systems

#### Speaker: Trevor Leslie (University of Wisconsin, Madison)

We consider the Euler Alignment Model with smooth, slowly decaying interaction protocol. It has been known since the work of Carrillo, Choi, Tadmor and Tan in 2016 that a certain conserved quantity '\(e\)' governs the global-in-time existence or finite-time blowup of sufficiently regular solutions. We give an interpretation of the quantity e and use it to analyze the structure of the limiting density profile. We draw two striking conclusions: First, the singular support of the limiting density measure (where 'aggregation' occurs) is precisely the image of the initial zero set of \(e\), under the limiting flow map. This allows us to reverse-engineer mass concentration sets of a specified topological genus, for example. Second, the smoothness of \(e\) at time zero controls the size of mass concentration set: If \(e_0\) is \(C^k\), then the mass concentration set has Hausdorff dimension at most \(1/(k+1)\). We show that this bound is sharp by means of an explicit example. This is joint work with Lear, Shvydkoy, and Tadmor. If time allows, we will also discuss the role of e in the limiting dynamics for the case of strongly singular interaction protocols.

Time: March 25, 2020 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Changhui Tan

### The Monolithic Arbitrary Lagrangian-Eulerian (ALE) Finite Element Analysis of Moving Interface Problems

#### Speaker: Rihui Lan (University of South Carolina)

In this talk, two different moving interface problems are studied by different arbitrary Lagrangian-Eulerian (ALE) finite element methods. For the transient Stokes/parabolic coupling with jump coefficients—a linearized fluid-structure interaction (FSI) problem, a novel \(H^1\)-projection is defined to account for the mesh motion. The well-posedness and optimal convergence properties in both the energy norm and \(L^2\) norm are analyzed for this mixed-type \(H^1\)-projection, with which the stability and optimal error estimate in the energy norm are derived for both semi- and fully discrete mixed finite element approximations to the Stokes/parabolic interface problem. For the parabolic/mixed parabolic moving interface problem with jump coefficients, a stable Stokes-pair mixed FEM within a specific stabilization technique and a novel ALE time-difference scheme are developed to discretize this interface problem in both semi- and fully discrete fashion, for which the stability and optimal error estimate analyses are conducted.

Time: March 27, 2020 2:30pm-3:30pm October 2, 2020 2:30pm-3:30pm

Location: LC317R Virtually via Zoom

Host: Lili Ju