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

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 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

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 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

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 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

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Alexander Kiselev, and Changhui Tan

Annals of PDE, Volume 8, No. 2, Article 16, 69pp. (2022)

Abstract

The question about behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger formally derived a nonlocal nonlinear partial differential equation which models dynamics of roots of polynomials under differentiation. In this paper, we connect rigorously solutions of Steinerberger’s PDE and evolution of roots under differentiation for a class of trigonometric polynomials. Namely, we prove that the distribution of the zeros of the derivatives of a polynomial and the corresponding solutions of the PDE remain close for all times. The global in time control follows from the analysis of the propagation of errors equation, which turns out to be a nonlinear fractional heat equation with the main term similar to the modulated discretized fractional Laplacian \((-\Delta)^{1/2}\).

	doi:10.1007/s40818-022-00135-4
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264

 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

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 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

 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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