**Computational and Applied Mathematics Seminar**

**Fall 2020**

**Mondays at 4:10 p.m. via ZOOM talks**

(Note the starting time maybe a few minutes later following the adjusted class times on MWF required by the University)

The CAM Seminar is organized in the ISU Mathematics Department. It brings speakers from inside and outside of ISU, raising issues and exchanging ideas on topics of current interest in the are of computational and applied mathematics.

- November 16

** Title: **Fast sparse grid simulations of high order schemes for high dimensional problems

** Yongtao Zhang **

** Department of Applied and Computational Mathematics and Statistics,** **University of Notre Dame **

**Abstract: **Mathematical models arising from biological or physical problems are often PDE problems defined on spatial domains with high dimensions. However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In this talk, I will present our recent results on designing sparse grid Krylov implicit integration factor (IIF) scheme for solving high spatial dimension convection-diffusion-reaction equations, and sparse grid weighted essentially non-oscillatory (WENO) scheme for solving hyperbolic PDEs. Our goal is to apply sparse grid techniques in high order schemes to achieve more efficient computations than that in their regular performance in solving multidimensional PDEs. A challenge is how to design the schemes on sparse grids such that comparable high order accuracy of the schemes in smooth regions of the solutions can still be achieved as that for computations on regular single grids. For sparse grid WENO scheme, additional challenge is that essentially non-oscillatory stability in non-smooth regions of the solutions needs to be preserved. We apply sparse-grid combination approach to overcome these difficulties. WENO interpolation was applied for the prolongation part in sparse-grid combination technique to deal with discontinuous solutions. Numerical examples defined on domains with up to four spatial dimensions are presented to show that significant computational times are saved, while both accuracy and stability of the original schemes are maintained for numerical simulations on sparse grids. The methods are then applied to Fokker-Planck equations, and kinetic problems modeled by high dimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids. This is joint work with Dong Lu, Shanqin Chen, and Xiaozhi Zhu for different parts of the project.

**ZOOM Link: **

Time: Nov 16, 2020 04:00 PM Central Time (US and Canada)

Join from a PC, Mac, iPad, iPhone or Android device:

Please click this URL to start or join. https://iastate.zoom.us/j/93881399673

Or, go to https://iastate.zoom.us/join and enter meeting ID: 938 8139 9673

Join from dial-in phone line:

Dial: +1 301 715 8592 or +1 312 626 6799

Meeting ID: 938 8139 9673

Participant ID: Shown after joining the meeting

International numbers available: https://iastate.zoom.us/u/acXmFrks2y

- November 23

- August 31
**(3:10-4:00pm)**

** Title: **From integrating to learning dynamics: new studies on Linear Multistep Methods

** Qiang Du **

** Applied Physics and Applied Mathematics, and Data Science Institutes,** **Columbia University**

**Abstract: **Numerical integration of given dynamic systems can be viewed as a forward problem with the learning of unknown dynamics from available state observations as an inverse problem. Solving both forward and inverse problems forms the loop of informative and intelligent

scientific computing. This lecture is concerned with the application of Linear multistep methods (LMMs) in the inverse problem setting that has

been gaining importance in data-driven modeling of complex dynamic processes via deep/machine learning. While a comprehensive mathematical theory of LMMs as popular numerical integrators of prescribed dynamics has been developed over the last century and has become textbook materials in numerical analysis. there seems to be a new story when LMMs are used in a black box machine learning formulation for learning dynamics from observed states. A natural question is concerned with whether a convergent LMM for integrating known dynamics is also suitable for discovering unknown dynamics. We show that the conventional theory of consistency, stability and convergence of LMM for time integration must be reexamined for dynamics discovery, which leads to new results on LMM that have not been studied before. We present refined concepts and algebraic criteria to assure stable and convergent discovery of dynamics in some idealized settings. We also apply the theory to some popular LMMs and make some interesting observations on their second characteristic polynomials.

** **

- September 07
**(Rescheduled to September 21 due to technical issues)**

** **

- September 14

** Title: TBD**

** Cancelled **

** Abstract: TBD**

** ZOOM Link: **

- September 21

** Title: **Complex Geometry and Optimal Transport

** Gabriel Khan **

** Department of Mathematics, Iowa State University**

**Abstract: **In this talk, we consider the Monge problem of optimal transport, which seeks to find the most cost-efficient movement of resources. In particular, we study the regularity (i.e. continuity/smoothness) of this transport. In recent work (joint with J. Zhang), we show that there is a connection between this question and the ``anti-bisectional curvature" of certain Kahler metrics. In this talk, we'll discuss several applications of these results (the second of which is joint with F. Zheng). First, we will answer a question in mathematical finance about the regularity of pseudo-arbitrages, which are investment strategies which beat the market almost surely in the long run. Second, by studying the behavior of anti-bisectional curvature along Kahler-Ricci flow, we will be able to show new results about Kahler-Einstein metrics.

**ZOOM Link: **

- September 28

** Title: **Nonlocal traffic flow models and the prevention of traffic jams

**Changhui Tan **

** Department of Mathematics, University of South Carolina**

**Abstract: **In this talk, I will discuss a family of traffic flow models. The classical Lighthill-Whitham-Richards model is known to have a finite time blowup for all generic initial data, which represents the creation of traffic jams. I will introduce a family of nonlocal traffic flow models, with look-ahead interactions. Such models can be derived from discrete cellular automata models.We show a remarkable phenomenon that the nonlocal slowdown interaction prevents traffic jams, under suitable settings. This talk is based on joint works with Thomas Hamori, Yongki Lee and Yi Sun.

** ZOOM Link: **

- October 05

** Title: **CRITICAL THRESHOLDS IN 1D PRESSURELESS EULER-POISSON SYSTEMS WITH VARYING BACKGROUND

** Manas Bhatnagar**

** Department of Mathematics, Iowa State University**

**Abstract: **The Euler Poisson equations describe important physical phenomena in many applications such as semiconductor modeling and plasma physics. In this talk, we will advance our understanding of critical threshold phenomena in such systems in the presence of different forces. We will identify critical thresholds in two damped Euler Poisson systems, with and without alignment, both with attractive potential and spatially varying background state. For both systems, we give respective bounds for subcritical and supercritical regions in the space of initial configuration, thereby proving the existence of a critical threshold for each scenario. Key tools include comparison with auxiliary systems, phase space analysis of the transformed system.

** ZOOM Link: **

- October 12

**Title: **Nano-Optical Studies of Two-Dimensional Materials

** Zhe Fei **

** ****Department of Physics & Astronomy, Iowa State University**

**Abstract: **Nano-optics is a frontier of research that studies light-matter interactions in the nanometer length scale. One of the core research topics in the field of nano-optics is about polaritons, which are quasiparticles generated due to the collective oscillations of photons and varieties of polarization charges in materials. These hybrid light-matter modes are closely related to the fundamental optical & electronic properties of materials and provide practical approaches toward nanoscale light trapping and manipulation. In recent years, varieties of polaritons were discovered in layered two-dimensional (2D) materials. Due to the reduced dimensionality and exceptional sensitivity, this class of materials have shown many unique polaritonic properties and physics. In this talk, we present nano-optical studies of various types of polaritons in layered 2D materials using the state-of-the-art scanning near-field optical microscopy – a powerful technique enabling ultra-small and ultra-fast imaging and spectroscopy in a wide spectral range from terahertz to visible. With this powerful tool, we performed systematic investigations of plasmon polaritons in graphene, graphene nanostructures, and few-layer graphene, phonon polaritons in hexagonal boron nitride, and exciton polaritons in group VI transition metal dichalcogenides. Novel physics and potential applications associated with these polaritonic modes will be discussed.

**ZOOM Link: **

- October 19
**(3:10-4:00pm)**

** Title: **Accuracy and Architecture Studies of Residual Neural Network solving ODEs

** Changxin Qiu **

** Department of Mathematics, Iowa State University**

**Abstract: **We consider applying residual network (ResNet) to solve ordinary differential equations. Stochastic gradient descent method is applied to train the network to obtain the optimal parameter sets of weight matrices and biases. We apply three finite difference methods of forward Euler, Runge-Kutta2 and Runge-Kutta4 with mesh size △ to generate three sets of targets training the ResNet and carry out the target study. The well trained ResNet behave just as its counterpart of the corresponding one-step finite difference method. In particular, we carry out three studies of (1) architecture study in terms of the number of layers and neurons per layer to find the optimal ResNet structure; (2) target study to verify the ResNet can be as accurate as its counterpart; (3) solution trajectory simulation. A sequence of numerical examples are presented to demonstrate the performance of the ResNet solver.

**ZOOM Link: **

- October 26

**Title: **Graphon mean field systems: large population and long time limits

** Ruoyu Wu **

** Department of Mathematics, Iowa State University**

**Abstract: **We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with random weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. A law of large numbers result is established as the system size increases and the underlying graphons converge. Under suitable additional assumptions, we show the exponential ergodicity for the system, establish the uniform in time law of large numbers, and introduce the uniform in time Euler approximation. The precise rate of convergence of the Euler approximation is provided.

** ZOOM Link: **

- November 02

**Title: **Positive and energy stable schemes for Poisson-Nernst-Planck systems

** Wumaier Maimaitiyiming **

** Department of Mathematics, UCLA**

**Abstract: **In this talk, we present positive and energy-dissipating schemes for solving Poisson-Nernst-Planck (PNP) equations. PNP equations are nonlinear/nonlocal gradient flows in density space, and their explicit solutions are rarely available; however, solutions to such problems feature intrinsic properties such as (i) solution positivity, (ii) mass conservation, and (iii) energy dissipation. These physically relevant properties are highly desirable to be preserved at the discrete level with the least time-step restrictions. We first construct our schemes for a reduced PNP model, then extend to multi-dimensional PNP equations. The strategies in the construction of the baseline schemes include (i) reformulation of each underlying model so that the resulting system is more suitable for constructing positive schemes and (ii) semi-implicit time discretization. We show that the schemes preserve solution positivity and mass conservation for arbitrary time steps. Moreover, the discrete free energy dissipates along time with an O(1) bound on time steps. The schemes also preserve steady states. Both the first and second-order schemes are linear and can be efficiently implemented. We provide numerical tests to validate our theoretical results and illustrate the accuracy, efficiency, and capacity to preserve the solution properties of our schemes.

**ZOOM Link: **

- November 09