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

 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

 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

 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

Trevor M. Leslie, and Changhui Tan

Communications in Partial Differential Equations, Volume 48, No. 5, pp. 753-791 (2023)

Abstract

We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density and bounded velocity. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker-Smale dynamics. Our approach is largely inspired by the work of Brenier and Grenier [SIAM J. Numer. Anal, 35(6):2317-2328, 1998] on the pressureless Euler equations.

	doi:10.1080/03605302.2023.2202720
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264

Eitan Tadmor, and Changhui Tan

SIAM Journal on Mathematical Analysis, Volume 54, No. 4, pp. 4277-4296 (2022)

Abstract

We study the global wellposedness of the Euler-Monge-Ampère (EMA) system. We obtain a sharp, explicit critical threshold in the space of initial configurations which guarantees the global regularity of EMA system with radially symmetric initial data. The result is obtained using two independent approaches -- one using spectral dynamics of Liu & Tadmor [Comm. Math. Physics 228(3):435-466, 2002] and another based on the geometric approach of Brenier & Loeper [Geom. Funct. Analysis 14(6):1182--1218, 2004]. The results are extended to 2D radial EMA with swirl.

	doi:10.1137/21M1437767
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264

I have been awarded a grant from the Office of the Vice President for Research at the University of South Carolina, on a one-year project: Multiscale nonlocal models in traffic flows.

Project Summary

Mathematical models on traffic flows have been studied extensively in the past century. Many celebrated models lie in a beautiful multiscale framework. The investigations on these models play an important role in designing traffic networks and preventing traffic jams.

Recently, the fast development of self-driving vehicles and new communication technologies allow long-range interactions in traffic networks. It attracts a lot of interest in nonlocal traffic flow models.

The aim of the proposed project is to develop the mathematical theory on non-local traffic flows and understand how nonlocal interactions can help to optimize the traffic networks and avoid the creation of traffic congestions.

The PI has been actively working on a variety of multiscale nonlocal models in physical, biological, and sociological contexts. These experiences can greatly help the understanding of the nonlocal phenomena in traffic models. Preliminary investigations show intriguing behaviors and promising outcomes. Results generated from this project will be capitalized to prepare proposals for external grants from NSF and DOT.

UofSC Office of Research Awards Announcement Page

Alexander Kiselev, and Changhui Tan

SIAM Journal on Mathematical Analysis, Volume 54, No. 3, pp. 3161-3191 (2022).

Abstract

In this paper, we analyze a nonlocal nonlinear partial differential equation formally derived by Stefan Steinerberger to model dynamics of roots of polynomials under differentiation. This partial differential equation is critical and bears striking resemblance to hydrodynamic models used to described collective behavior of agents (such as birds, fish or robots) in mathematical biology. We consider periodic setting and show global regularity and exponential in time convergence to uniform density for solutions corresponding to strictly positive smooth initial data.

	doi:10.1137/21M1422859
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264

I have been awarded an NSF grant (DMS #1815667) on a three-year project: Regularity and Singularity Formation in Swarming and Related Fluid Models. It is translated to DMS #1853001 when I move to University of South Carolina.

Abstract

Swarming is a commonly observed complex biological and sociological phenomenon. The internal interaction mechanism attracts a lot of attention in physics, engineering, biology, and social sciences. This project is devoted to developing a unified mathematical theory towards the understanding of the swarming dynamics, as well as other nonlocal models that share similar structures. These models are widely considered in fluid mechanics, meteorology, astrophysics, biology, and ecology. The study of the regularity and singularity formations of these equations will provide a firm theoretical foundation for these applications, and also help consolidate the validity of these models in describing the natural phenomena.

The research will focus on understanding the nonlinear and nonlocal phenomena in swarming dynamics, and models having related structures in fluid mechanic. Three different but related models will be investigated. The first model is the Euler-Alignment system, which describes the flocking behavior in animal swarms. The goal is to develop a robust toolbox to analyze the nonlocal alignment operator and its balance with the drift nonlinearity. Similar behaviors are also observed in other fluid equations including porous medium flow, and surface quasi-geostrophic equations, which will be investigated using the same analytical techniques. The second model is the 2D inviscid Boussinesq equations. The global regularity is one of the outstanding problems in fluid dynamics. The idea is to construct solutions to capture the possible singularity formation, starting from some modified versions of the equations. The third model is the kinetic swarming system. The aim is to investigate the important relation between the kinetic equation and a variety of hydrodynamic limits. In particular, different alignment operators will be considered at the kinetic level. They are expected to lead to different macroscopic limits. All these three sub-projects will advance the mathematical understanding of nonlocal PDEs and related applications. They will also provide education and training to graduate and undergraduate students in this active field.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

NSF award page on the grant DMS #1853001

I have been awarded an NSF grant (DMS #2108264) on a three-year project: Nonlocal Transport Equations in Fluids, Swarming, and Traffic Flows. 

Abstract

Nonlocal models are relevant to many real-world phenomena and have been an area of active and growing research in recent decades. The development of a mathematical theory of nonlocal interactions plays a significant role in the understanding of complex structures, with rich applications in physics, biology, and social sciences. One example of the effects of nonlocal behavior found in nature is the collective dynamics in animal swarms, where small-scale interactions emerge into intriguing global phenomena. This project develops novel and robust analytical techniques for models that share similar nonlocality. These tools help to advance the understanding of the hidden structures of the models, and ultimately have an impact in applications, such as in traffic flow, where they can be used to study how to integrate nonlocal communications into a smart traffic network to improve efficiency and avoid traffic congestions. The training and professional development of graduate students is an integral part of the project.

The project studies three families of nonlocal transport equations. The first family includes the Euler-alignment system describing the flocking phenomenon for animal swarms. The goal is to establish a global well-posedness theory for the system in multi-dimensions, starting from imposing radial symmetry, and to apply the methodology to other models, such as the Euler-Poisson equations and more. The second includes a nonlocal transport equation which describes the evolution of the distribution of polynomial roots under repeated differentiation, the aim is to find a rigorous connection between this equation and the differentiation process. The last is a family of nonlocal traffic flow models, which have received extensive attention in the last decade, and are analyzed to understand the impact of the nonlocal interactions and how the nonlocal phenomenon can help to prevent traffic congestions.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

NSF award page on the grant DMS #2108264