{pdf=https://arxiv.org/pdf/2007.15758v1.pdf|100%|600|google}

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

- ACM Seminar

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

Access to the video of the talk

- ACM Seminar

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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- ACM Seminar

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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- ACM Seminar

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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- ACM Seminar

- The preprint is available on arXiv:2007.15758.

{pdf=https://arxiv.org/pdf/2007.15758v1.pdf|100%|600|google}

- Euleralignment system
- EulerPoisson equation
- Radial symmetry

- This is a joint work with
*Chen Li*and*Lining Tong*. - The preprint is available on arXiv:2007.08356.

{pdf=https://arxiv.org/pdf/2007.08356v1.pdf|100%|600|google}

- Euleralignment system
- global regularity
- Singular interaction

- This is a joint work with
*Qianyun Miao*and*Liutang Xue*. - The preprint is available on arXiv:2004.03652.

{pdf=https://arxiv.org/pdf/2004.03652v1.pdf|100%|600|google}

- Euleralignment system
- misalignment
- global regularity

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

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- ACM Seminar

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

- ACM Seminar