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

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 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

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 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

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 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

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Changhui Tan

SIAM Journal on Mathematical Analysis, Volume 53, No 3, pp. 3040–3071 (2021).

Abstract

We study the global wellposedness of pressureless Eulerian dynamics in multidimensions, with radially symmetric data. Compared with the one-dimensional system, a major difference in multidimensional Eulerian dynamics is the presence of the spectral gap, which is difficult to control in general. We propose a new pair of scalar quantities that provides significantly better control of the spectral gap. Two applications are presented: (i) the Euler-Poisson equations: we show a sharp threshold condition on initial data that distinguish global regularity and finite time blowup; (ii) the Euler-alignment equations: we show a large subcritical region of initial data that leads to global smooth solutions.

	doi:10.1137/20M1358682
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	This work is supported by NSF grant DMS #1853001

Li Chen, Changhui Tan, and Lining Tong

Methods and Applications of Analysis, Volume 28, No.2, pp. 155-174 (2021).

Dedicated to Professor Ling Hsiao's 80th birthday.

Abstract

We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the asymptotic flocking behavior, where solutions converge to a constant steady state exponentially in time.

	doi:10.4310/MAA.2021.v28.n2.a3
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	This work is supported by NSF grant DMS #1853001