## ACM Seminar

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

Click HERE for the schedule.

### Quantitative steepness, semi-FKPP reactions, and pushmi-pullyu fronts

#### Speaker: Jing An (Duke University)

We will discuss the algebraic structure of a large class of reaction-diffusion equations and use it to study the long-time behavior of the solutions and their convergence to traveling waves in the pulled and pushed regimes, as well as at the pushmi-pullyu boundary. A new quantity named as the shape defect function is introduced to measure the difference between the profiles of the solution and the traveling waves. In particular, the positivity of the shape defect function, combined with a new weighted Hopf-Cole transform and a relative entropy approach, plays a key role in the stability arguments. The shape defect function also gives a new connection between reaction-diffusion equations and reaction conservation laws at the pulled-pushed transition. This is joint work with Chris Henderson and Lenya Ryzhik.

Time: November 18, 2022 2:30pm-3:30pm

Location: LeConte 205

Host: Siming He

### On the construction of 3D incompressible Euler equilibria by magnetic relaxation

#### Speaker: Federico Pasqualotto (Duke University)

Magnetic relaxation is a conjectured general procedure to obtain steady solutions to the incompressible Euler equations by means of a long-time limit of an MHD system. In some regimes, the magnetic field is conjectured to “relax” to a steady state of the 3D Euler equations as time goes to infinity.

In this talk, I will first review the classical problem of magnetic relaxation, connecting it to questions arising in topological hydrodynamics. I will then present a general construction of steady states of the incompressible 3D Euler equations by a long-time limit of a regularized MHD system. We consider the so-called Voigt regularization, and our procedure yields non-trivial equilibria on the flat 3D torus and on general bounded domains.

This is joint work with Peter Constantin.

Time: October 28, 2022 2:30pm-3:30pm

Location: LeConte 205

Host: Siming He

### A new approach to the mean-field limit of Vlasov-Fokker-Planck equations

#### Speaker: Pierre-Emmanuel Jabin (Pennsylvania State University)

We introduce a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov-Poisson-Fokker-Planck system for plasmas in dimension 2 together with a partial result in dimension 3. The method is broadly compatible with second order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted L p norms of the marginals or observables of the system uniformly in the number of particles. This allows to treat very singular interaction kernels between the particles, including repulsive Poisson interactions. This is a joint work with D. Bresch and J. Soler.

Time: December 2, 2022 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Changhui Tan

### A measure perspective on uncertainty quantification

#### 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

### Hadamard-Babich ansatz for point-source Maxwell's equations

#### 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

### An overview of augmented strategy and applications

#### 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

### Spectral renormalizations methods in physics

#### 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

### Global solutions of quasi-geostrophic shallow water front problems

#### 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

### How Math and AI are revolutionizing biosciences

#### 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

### Orbital stability for internal waves

#### 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

#### More...

### Stabilizing phenomenon for incompressible fluids

#### Speaker: Jiahong Wu (Oklahoma State University)

The background magnetic field stabilizes and damps electrically conducting fluids, and the temperature tames and stabilizes buoyancy driven fluids. These are just two examples of a seemingly universal stabilizing phenomenon that has been experimentally and numerically observed for different types of incompressible fluids. This talk presents recent work that establishes this phenomenon as mathematically rigorous stability results. In particular, we describe the global existence and stability results for the 3D incompressible anisotropic magnetohydrodynamic system near a background magnetic field, for the Boussinesq system near the hydrostatic equilibrium, and for the Oldroyd-B model near the trivial solution.

Time: February 18, 2022 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Changhui Tan

### Numerical methods for nonlocal models: asymptotically compatible schemes and multiscale modeling

#### Speaker: Xiaochuan Tian (University of California, San Diego)

Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, for example, the peridynamics model of fracture mechanics, they also come with increased difficulty in computation with nonlocality involved. In this talk, we will give a review of the asymptotically compatible schemes for nonlocal models with a parameter dependence. Such numerical schemes are robust under the change of the nonlocal length parameter and are suitable for multiscale simulations where nonlocal and local models are coupled. We will discuss finite difference, finite element and collocation methods for nonlocal models as well as the related open questions for each type of the numerical methods.

Time: November 12, 2021 3:30pm-4:30pm

Location: Virtually via Zoom

Host: Changhui Tan

### De Giorgi method for kinetic equations

#### Speaker: Weiran Sun (Simon Fraser University)

In this talk we explain how to generalize the De Giorgi level-set method for diffusion equations to a framework for kinetic equations with singular kernels. In particular, we use the non-cutoff Boltzmann and the Landau equations as examples to show how the De Giorgi method can be used to prove the existence of \(L^2\cap L^\infty\) solutions in the near-equilibrium regime. The key idea is to make use of the strong averaging lemma to establish a nonlinear iteration for level-set energies which will give a local existence theory. We then extend the time interval to infinity by exploring the spectral structures of the linearized kinetic operators. This talk is based on recent works with Ricardo Alonso, Yoshinori Morimoto, and Tong Yang.

Time: November 5, 2021 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Changhui Tan

### Global-in-time domain decomposition methods for the coupled Stokes and Darcy flows

#### Speaker: Thi-Thao-Phuong Hoang (Auburn University)

In many engineering and biological applications (e.g., groundwater flow problems, flows in vuggy porous media, industrial filtrations, biofluid-organ interaction and cardiovascular flows), the Stokes-Darcy system is used to model the interaction of fluid flow with porous media flow, where the Stokes equations represent an incompressible fluid, and the Darcy equations represent a flow through a porous medium. The time scales in the Stokes and Darcy regions could be largely different, thus it is inefficient to use the same time step throughout the entire spatial domain.

In this talk, we present decoupling iterative algorithms based on domain decomposition for the time-dependent Stokes-Darcy model, in which different time step sizes can be used in the flow region and in the porous medium. The coupled system is formulated as a space-time interface problem based on either physical interface conditions or equivalent Robin-Robin interface conditions. Such an interface problem is solved iteratively by a Krylov subspace method (e.g., GMRES) which involves at each iteration parallel solution of time-dependent Stokes and Darcy problems. Consequently, local discretizations in both space and time can be used to efficiently handle multiphysics systems with discontinuous parameters. Numerical experiments with nonconforming time grids are considered to illustrate the performance of the proposed methods.

Time: November 19, 2021 2:30pm-3:30pm

Location: COL 2014 and Virtually via Zoom

Host: Lili Ju