### 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

### 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

### Applications of the shear-flow induced enhanced dissipation

#### Speaker: Siming He (Duke University)

In this talk, we consider the enhanced dissipation phenomena induced by shear flows. In the first part of the talk, I will introduce the idea of shear flow-induced enhanced dissipation and the recent developments on this topic. Then I will exhibit the applications of this phenomenon in various settings, ranging from suppression of chemotactic blow-ups to enhancement of chemical reactions.

Time: October 29, 2021 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Changhui Tan

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### Energetic variational approaches for active/reactive fluids and applications

#### Speaker: Chun Liu (Illinois Institute of Technology)

We present a general framework for active fluids which convert chemical energy into various types of mechanical energies. This is the extension of the classical energetic variational approaches for isothermal mechanical systems. The methods will cover a wide range of both chemical reaction kenetics, thermal and mechanical processes. This is a joint project with many collaborators, in particular, Bob Eisenberg, Yiwei Wang and Tengfei Zhang.

Time: December 3, 2021 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Qi Wang

### Structure preserving numerical methods for hyperbolic systems of conservation and balance laws

#### Speaker: Alina Chertock (North Carolina State University)

Many physical models, while quite different in nature, can be described by nonlinear hyperbolic systems of conservation and balance laws. The main source of difficulties one comes across when numerically solving these systems is lack of smoothness as solutions of hyperbolic conservation/balance laws may develop very complicated nonlinear wave structures including shocks, rarefaction waves and contact discontinuities. The level of complexity may increase even further when solutions of the hyperbolic system reveal a multiscale character and/or the system includes additional terms such as friction terms, geometrical terms, nonconservative products, etc., which are needed to be taken into account in order to achieve a proper description of the studied physical phenomena. In such cases, it is extremely important to design a numerical method that is not only consistent with the given PDEs, but also preserves certain structural and asymptotic properties of the underlying problem at the discrete level. While a variety of numerical methods for such models have been successfully developed, there are still many open problems, for which the derivation of reliable high-resolution numerical methods still remains to be an extremely challenging task.

In this talk, I will discuss recent advances in the development of two classes of structure preserving numerical methods for nonlinear hyperbolic systems of conservation and balance laws. In particular, I will present (i) well-balanced and positivity preserving numerical schemes, that is, the methods which are capable of exactly preserving some steady-state solutions as well as maintaining the positivity of the numerical quantities when it is required by the physical application, and (ii) asymptotic preserving schemes, which provide accurate and efficient numerical solutions in certain stiff and/or asymptotic regimes of physical interest.

Time: October 15, 2021 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Changhui Tan

### Neural nets and numerical PDEs

#### Speaker: Zhiqiang Cai (Purdue University)

In this talk, I will present our recent works on neural networks (NNs) and its application in numerical PDEs. The first part of the talk is to use NNs to numerically solve scalar linear and nonlinear hyperbolic conservation laws whose solutions are discontinuous. I will show that the NN-based method for this type of problems has an advantage over the mesh-based methods in terms of the number of degrees of freedom.

The second part of the talk is on our adaptive network enhancement (ANE) method. The ANE method is developed to address a fundamental, open question on how to automatically design an optimal NN architecture for approximating functions and solutions of PDEs within a prescribed accuracy. Moreover, to train the resulting non-convex optimization problem, the ANE method provides a natural process of obtaining a good initialization.

Time: October 22, 2021 2:30pm-3:30pm

Location: Virtually via Zoom

Host: Wolfgang Dahmen

### A new real-space method for the simulation of scanning transmission electron microscope images

#### Speaker: Christian Doberstein (University of South Carolina)

I will present a new method for the simulation of annular dark field (ADF) images in scanning transmission electron microscopy (STEM). While the simulation of a conventional transmission electron microscopy (TEM) image requires solving the Schrödinger equation only very few times, simulating an ADF STEM image requires solving the Schrödinger equation several times for every pixel in the output image. This makes it a computationally challenging task and it is therefore important to find algorithms that reduce the computation time to a reasonably short duration.

One of the classical approaches to simulating a STEM image is the Multislice algorithm. In this algorithm, the specimen is first divided into thin slices perpendicular to the beam direction. Afterwards, solutions to the Schrödinger equation are computed by transmitting the probe wave function (i.e. the initial condition) slice by slice through the specimen for every probe position. Recently, a new algorithm termed PRISM has been developed to speed up the Multislice computations. This algorithm makes use of the linearity of the Schrödinger equation and propagates a small set of certain elementary wave functions through the specimen instead of the probe wave functions themselves. The probe wave functions are then approximated by linear combinations of these elementary wave functions, where the number of elementary functions may be much smaller than the number of probe wave functions. Although PRISM is a mathematically elegant way to reduce the number of Multislice computations, it can introduce large errors and require prohibitive amounts of computer memory. This is due to the choice of the elementary wave functions as Dirac deltas in Fourier space and the fact that they are highly nonlocal in real space coordinates.

These problems give rise to the idea of approximating the probe wave functions by a different set of "elementary wave functions" that are localized in real space coordinates. I will present an example for such a set of elementary wave functions and show that this makes it possible to keep the speedup of PRISM while avoiding the precision and memory issues. Additionally, I will show how the Multislice computations can be performed entirely in real space coordinates using the GPU, which should further speed up the computations.

Time: September 24, 2021 3:30pm-4:30pm

Location: Virtually via Zoom