Speaker: Amir Sagiv (Columbia University)

In many scientific areas, deterministic models (e.g., differential equations) use numerical parameters. Often, such parameters might be uncertain or noisy. A more honest model should therefore provide a statistical description of the quantity of interest. Underlying this numerical analysis problem is a fundamental question - if two "similar" functions push-forward the same measure, would the new resulting measures be close, and if so, in what sense? We will first show how the probability density function (PDF) of the quantity of interest can be approximated. We will then discuss how, through the lense of the Wasserstein-distance, our problem yields a simpler and more robust theoretical framework.

Finally, we will take a steep turn to a seemingly unrelated topic: the computational sampling problem. In particular, we will discuss the emerging class of sampling-by-transport algorithms, which to-date lacks rigorous theoretical guarantees. As it turns out, the mathematical machinery developed in the first half of the talk provides a clear avenue to understand this latter class of algorithms. 
 

Time: November 11, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Wolfgang Dahmen

 Speaker: Jianliang Qian (Michigan State University)

We propose a novel Hadamard-Babich ansatz consisting of an infinite series of dyadic coefficients (three-by-three matrices) and spherical Hankel functions for solving point-source Maxwell's equations in an inhomogeneous medium so as to produce the so-called dyadic Green's function. Using properties of spherical Hankel functions, we derive governing equations for the unknown asymptotics of the ansatz including the travel time function and dyadic coefficients. By proposing matching conditions at the point source, we rigorously derive asymptotic behaviors of these geometrical-optics ingredients near the source so that their initial data at the source point are well-defined. To verify the feasibility of the proposed ansatz, we truncate the ansatz to keep only the first two terms, and we further develop partial-differential-equation based Eulerian approaches to compute the resulting asymptotic solutions. Numerical examples demonstrate that our new ansatz yields a uniform asymptotic solution in the region of space containing a point source but no other caustics. 
 

Time: October 21, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Lili Ju

 Speaker: Zhilin Li (North Carolina State University)

Considering the different backgrounds of the audience, I would like to present an overview of an augmented strategy for solving PDEs hoping to find more applications of the approach. The purpose of the augmented strategy is to decouple some complex systems, rescale or preconditioning PDEs. The augmented strategy makes it possible to obtain accurate and stable discretization. The idea of the augmented strategy for a complicated problem is to introduce some augmented variable(s) along a codimension on a manifold, like a boundary integral method of the source and/or dipole strengths except that no Green's function is needed, more flexible in terms of PDEs (linear or nonlinear), boundary conditions and source terms.

Some important applications will be discussed including the treatment of pressure boundary conditions (not free variables) in Stokes and Navier-Stokes equations; rescaling and fast algorithms for interface problems with large jump ratios, a fluid flow and Darcy's coupling in which the governing equations are different in different regions; and ADI methods for parabolic interface problems, and scattering problems modeled by Maxwell equations, and solver PDEs on irregular domains.  
 

Time: September 30, 2022 2:30pm-3:30pm October 7, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Qi Wang

Xiang Bai, Qianyun Miao, Changhui Tan and Liutang Xue

Nonlinearity, Volume 37, 025007, 46pp. (2024).

Abstract

In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behavior and optimal decay estimates of the solutions as \(t\to\infty\).

	doi:10.1088/1361-6544/ad140b
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264

Yi Sun and Changhui Tan

Physica D, Volume 446, 133657 (2023)

Abstract

This paper presents a class of one-dimensional cellular automata (CA) models on traffic flows, featuring nonlocal look-ahead interactions. We develop kinetic Monte Carlo (KMC) algorithms to simulate the dynamics. The standard KMC method can be inefficient for models with global interactions. We design an accelerated KMC method to reduce the computational complexity in the evaluation of the nonlocal transition rates. We investigate several numerical experiments to demonstrate the efficiency of the accelerated algorithm, and obtain the fundamental diagrams of the dynamics under various parameter settings.

	doi:10.1016/j.physd.2023.133657
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264
	This work is supported by a UofSC VPR ASPIRE I grant

 Speaker: Ziad Musslimani (Florida State University)

In this talk we shall outline a new method to solve initial and boundary value problems of physical relevance. The idea is to use the underlying physics (such as conservation laws or dissipation rate equations) combined with a dynamic renormalization process to numerically compute ground and excited states as well as time-dependent solutions. We will apply the method on a prototypical problems that arise in physics such as Gross-Pitaevski equation and Hartree-Fock.
 

Time: April 22, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Qi Wang

 Speaker: Qingtian Zhang (West Virginia University)

In this talk, I will introduce the vortex front problem for quasi-geostrophic shallow water equation, which is also known as Hasegawa-Mima equation in plasma science. The contour dynamic equation of the vortex front will be derived, which is a nonlocal, nonlinear dispersive equation. The existence of global solutions will be proved when the initial data is small.
 

Time: March 25, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan

Thomas Hamori and Changhui Tan

Nonlinear Analysis: Real World Applications, Volume 73, 103899, (2023).

Abstract

We study a class of traffic flow models with nonlocal look-ahead interactions. The global regularity of solutions depend on the initial data. We obtain sharp critical threshold conditions that distinguish the initial data into a trichotomy: subcritical initial conditions lead to global smooth solutions, while two types of supercritical initial conditions lead to two kinds of finite time shock formations. The existence of non-trivial subcritical initial data indicates that the nonlocal look-ahead interactions can help avoid shock formations, and hence prevent the creation of traffic jams.

	doi:10.1016/j.nonrwa.2023.103899
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264
	This work is supported by a UofSC VPR ASPIRE I grant

 Speaker: Guowei Wei (Michigan State University)

Mathematics underpins fundamental theories in physics such as quantum mechanics, general relativity, and quantum field theory. Nonetheless, its success in modern biology, namely cellular biology, molecular biology, biochemistry, genomics, and genetics, has been quite limited. Artificial intelligence (AI) has fundamentally changed the landscape of science, technology, industry, and social media in the past few years and holds a great future for discovering the rules of life. However, AI-based biological discovery encounters challenges arising from the structural complexity of macromolecules, the high dimensionality of biological variability, the multiscale entanglement of molecules, cells, tissues, organs, and organisms, the nonlinearity of genotype, phenotype, and environment coupling, and the excessiveness of genomic, transcriptomic, proteomic, and metabolomic data. We tackle these challenges mathematically. Our work focuses on reducing the complexity, dimensionality, entanglement, and nonlinearity of biological data in AI. We have introduced evolutionary de Rham-Hodge, persistent cohomology, persistent Laplacian, and persistent sheaf theories to model complex, heterogeneous, multiscale biological systems and thus significantly enhance AI's ability to handle biological datasets. Using our mathematical AI approaches, my team has been the top winner in D3R Grand Challenges, a worldwide annual competition series in computer-aided drug design and discovery for years. Using over two million genomes isolates from patients, we discovered the mechanisms of SARS-CoV-2 evolution and transmission and accurately forecast emerging SARS-CoV-2 variants.
 

Time: April 15, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Qi Wang

 Speaker: Ming Chen (University of Pittsburgh)

I will discuss the nonlinear stability of capillary-gravity waves propagating along the interface dividing two immiscible fluid layers of finite depth. The motion in both regions is governed by the incompressible and irrotational Euler equations, with the density of each fluid being constant but distinct. We prove that for supercritical surface tension, all known small-amplitude localized waves are (conditionally) orbitally stable in the natural energy space. Moreover, the trivial solution is shown to be conditionally stable when the Bond and Froude numbers lie in a certain unbounded parameter region. For the near critical surface tension regime, we show that one can infer conditional orbital stability or orbital instability of small-amplitude traveling waves solutions to the full Euler system from considerations of a dispersive PDE similar to the steady Kawahara equation. This is joint work with S. Walsh.
 

Time: March 4, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan