## ACM Seminar

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

Click HERE for the schedule.

### 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

### 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

### Efficient, positive, and energy stable schemes for Poisson-Nernst-Planck systems

#### Speaker: Hailiang Liu (Iowa State University)

We are concerned with positive and energy-dissipating schemes for solving the time-dependent system of Poisson-Nernst-Planck (PNP) equations, which has found much use in the modeling of biological membrane channels and semiconductor devices. As a gradient flow in density space, this strongly coupled system of nonlinear equations can take long time evolution to reach steady states. Hence, designing efficient and stable methods is highly desirable. In this talk we shall present a class of methods with structure-preserving properties for the PNP system, and review advances around related models such as the quantum diffusion equation.

Time: February 21, 2020 2:30pm-3:30pm

Location: LC317R

Host: Changhui Tan

### Mathematical and numerical analysis to variable-order mobile-immobile time-fractional diffusion equations

#### Speaker: Xiangcheng Zheng (University of South Carolina)

We proved the wellposedness of variable-order mobile-immobile time-fractional diffusion equations and the regularity of their solutions. Optimal-order finite element approximation was presented and analyzed. Numerical experiments were carried out for demonstration.

Time: January 31, 2020 2:30pm-3:30pm

Location: LC317R

### Lagrangian Front Tracking and Applications to Conservation Law, Fluid Mixing, and Phase Transition Problems

#### Speaker: Xiaolin Li (Stony Brook University)

In this talk, I will review the history of the Lagrangian front tracking method and the computational platform built on this methodology. I will review the front tracking in the study of fluid interface instabilities, including Rayleigh-Taylor instability, Richtmyer Meshkov instability and fluid mixing induced by these instabilities. I will also introduce a fully conservative front tracking scheme and its application in the phase transition problem.

Time: March 20, 2020 2:30pm-3:30pm

Location: LC317R

Host: Xinfeng Liu

### Traveling Wave of Gray-Scott System: Results and Perspective

#### Speaker: Yuanwei Qi (University of Central Florida)

In this talk, I shall report some recent progress on the existence and multiplicity of traveling waves to one of the most important models in Turing Pattern Formation. In addition, I shall pose some questions which are wide open which demand new ideas and fresh approaches. This is a joint-work with Xinfu Chen et al.

Time: January 17, 2020 2:30pm-3:30pm

Location: LC317R

Host: Changhui Tan