**Computational and Applied Mathematics Seminar**

**Fall 2023**

**Mondays at 2:15-3:05p.m (ZOOM/WebEx or in-person (Room Carver 401) talks)**

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 area of computational and applied mathematics.

- December 08
**(3-5pm, Carver 294)**

** Title: **Global Wellposedness and Traveling Waves of Chemotaxis Models

** Tong Li, The University of Iowa**

**Abstract: **We study global existence and long-time behavior of solutions for hyperbolic-parabolic PDE models of chemotaxis. We show the existence and the stability of traveling wave solutions to a system of nonlinear conservation laws derived from the Keller-Segel model. We find oscillatory traveling wave solutions to an attractive chemotaxis system which are biologically relevant. Traveling wave solutions of chemotaxis models with growth are also investigated.

** Title: **The direct and inverse scattering for the third-order equation

** Tuncay Aktosun, University of Texas at Arlington**

**Abstract: **

** Title: **Geometric viewpoints to classical regularity theory

** Lihe Wang, The University of Iowa**

**Abstract: **We will discuss some geometric viewpoints and new proofs for DeGiorgi theory and partial regularity theorems to NSE and other elliptic PDEs.

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- August 28

** Title: Organizational Meeting for CAM**

**Abstract: **We will discuss about the scope of CAM for 2023-2024.

- September 04 (
**Labor Day**)

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- September 11
**(Carver 401)**

** Title: **Simplex Interpolations and Applications in Computational Electromagnetics

** Jiming Song, Department of Electrical and Computer Engineering, Iowa State University**

**Abstract: **The interpolation is widely used in computational electromagnetics (CEM). Interpolation and anterpolation are the key parts of the multilevel fast multipole algorithm (MLFMA) to reduce the complexity of a matrix-vector multiply. One of other fields using interpolation in CEM is the efficiently evaluations of Green’s functions in layered media, periodic Green’s functions (PGF), and PGF in layered media. In general, it is very time-consuming to evaluate the PGF because PGF involves the double summations of infinite series and Sommerfeld integral for the layered media.

The bivariate interpolation errors of plane waves over right triangles, equilateral triangles and rectangles are analyzed. For the right triangular interpolations, the data points used for interpolation are chosen based on the area with minimum root-mean-square (RMS) error. For the same order interpolation, the number of data points for right triangular interpolations is up to 50% less than number of points for the rectangular interpolations. The optimal tetrahedron is proposed in the 3D improved simplex interpolation (ISI) method, which is more efficient and accurate than the existing simplex interpolation (SI). As an example, the triangular and 3D simplex interpolations are used to efficiently evaluate the periodic Green’s functions. Due to non-uniformly sampling along the θ-direction for the three-dimensional MLFMA, the isosceles triangular interpolation is proposed by selecting region with the smallest error. Numerical results prove that the isosceles triangular interpolation is more effective and accurate than the rectangular interpolation.

**Bio: **Dr. Song currently is a Professor at Iowa State University’s Department of Electrical and Computer Engineering. His research has dealt with modeling and simulations of interconnects on lossy silicon and RF components, electromagnetic wave scattering using fast algorithms, the wave propagation in metamaterials, acoustic and elastic wave propagation and non-destructive evaluation, and transient electromagnetic field. He received the NSF Career Award in 2006 and is an IEEE Fellow and ACES Fellow.** **

- September 18

** Title: **

**Abstract: **

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- September 25

** Title: **A dynamic mass-transport method for Poisson-Nernst-Planck systems

** Hailiang Liu, Iowa State University**

**Abstract: ** In this talk, I will present some recent work on developing structure preserving (i.e., positivity preserving, mass conservative, and energy dissipating) methods for numerically simulating Poisson-Nernst-Planck (PNP) systems. Motivated by Benamou-Brenier's dynamic formulation of the quadratic Wasserstein metric, we introduce a Wasserstein-type distance suitable for our problem setting, we then construct a variational scheme which falls into the Jordan--Kinderlehrer--Otto framework. The variational scheme is a first order (in time) approximation of the original PNP system. To reduce computational cost, we further approximate the constraints and the objective function in the underlying Wasserstein-type distance, such approximations won't destroy the first order accuracy. With a standard spatial discretization, we obtain a finite dimensional strictly convex minimization problem with linear constraints. The admissible set in the variational problem is a subset of the probability space and the Wasserstein-type distance is nonnegative, therefore our scheme is a positivity preserving, mass conservative, and energy dissipating.

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- October 02

** Title: **Exploring the Gaussian Process Nature of Wide Neural Networks: Insights from Deep Equilibrium Models

** Tianxiang Gao, Iowa State University**

**Abstract: **Neural networks with wide layers have attracted significant attention due to their equivalence to Gaussian processes, enabling perfect fitting of training data while maintaining generalization performance, known as benign overfitting. However, existing results mainly focus on shallow or finite-depth networks, necessitating a comprehensive analysis of wide neural networks with infinite-depth layers, such as neural ordinary differential equations (ODEs) and deep equilibrium models (DEQs). In this paper, we specifically investigate the deep equilibrium model (DEQ), an infinite-depth neural network with shared weight matrices across layers. Our analysis reveals that as the width of DEQ layers approaches infinity, it converges to a Gaussian process. Remarkably, this convergence holds even when the limits of depth and width are interchanged, which is not observed in typical infinite-depth Multilayer Perceptron (MLP) networks. Furthermore, we demonstrate that the associated Gaussian vector remains non-degenerate for any pairwise distinct input data, ensuring a strictly positive smallest eigenvalue of the corresponding kernel matrix. These findings serve as fundamental elements for studying the training and generation of DEQs, laying the groundwork for future research in this area.

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- October 09
**(Monday, 2:15-3:05p.m, Carver 401)**

** Title: **Scientific Machine Learning for Computational Wave Imaging Problems: from Carbon Zero Emissions to Breast Cancer Detection

** Youzuo Lin, Los Alamos National Lab **

**Abstract: **AI for Science (aka “AI4Science”) is currently one of the most prominent topics in the machine learning community. In this talk, I will focus on a specific scientific problem: computational wave imaging. Computational wave imaging provides a way to infer otherwise unobservable physical properties of a medium (such as internal density and bulk modulus) from measurements of a wave signal that propagates through the medium. Scientific applications include seismic imaging of the earth, acoustic imaging in materials, and ultrasound tomography in medicine. There are currently two main approaches to solving computational wave imaging problems: those based on physics and those based on machine learning (ML). Among conventional physics-based methods, full waveform inversion (FWI) can provide high-resolution, quantitatively accurate, estimates of medium acoustic properties. However, FWI can be computationally expensive and subject to ill-posedness and “cycle skipping” (a kind of ill-posedness that is particular to wave equations). Recently, ML-based computational methods have been developed to address these issues. Some success has been attained when an abundance of simulations and labels are available. Nevertheless, when applied to a moderately different real-world dataset, ML models usually suffer from weak generalizability. In my talk, I will discuss the details of our recent research effort leveraging both data and underlying physics to address the critical issues of weak generalizability and data scarcity. Particularly, I will go through the advantages and disadvantages of our ML techniques in solving scientific problems of monitoring carbon sequestration using seismic inversion and detecting breast cancers using ultrasound tomography.

Bio:

Dr. Lin is a Senior Scientist and team leader in the Earth Physics Team at Los Alamos National Laboratory (LANL). His research interests lie in scientific machine learning methods and their applications. Particularly, he has worked on various scientific problems including inverse problems and computational imaging, subsurface clean and renewable energy exploration, ultrasound tomography for breast cancer detection, and UAV image analysis. He has published more than 90 articles in top journals and conference proceedings. He is also a co-inventor on several U.S. patents on ultrasound imaging techniques.

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- October 13 (
**Friday, 3-5pm, Carver 294)**

** Title: **Conservative, Positivity Preserving and Energy Dissipative Numerical Methods for the Poisson-Nernst-Planck Equations

** Zhongming Wang, Florida International University**

**Abstract: **We develop and analyze some numerical methods for solving the Poisson-Nernst-Planck (PNP) equations. The numerical schemes, including finite difference method and discontinuous Galerkin method, respect three desired properties that are possessed by analytical solutions: I) conservation, II) positivity of solution, and III) free-energy dissipation. Advantages of different types of methods are discussed. Numerical experiments are performed to validate the numerical analysis. Modified PNP system that incorporating size and solvation effect is also studied to demonstrate the effectiveness of our schemes in solving realistic application. This is joint work with D. Jie, H. Liu, P. Yin, H. Yu and S. Zhou.

** Title: **A semi-implicit dynamical low-rank discontinuous Galerkin method for space homogeneous kinetic equations

** Peimeng Yin, The University of Texas at El Paso**

**Abstract: **Dynamical low-rank approximation (DLRA) is an emerging tool for reducing computational costs and provides memory savings when solving high-dimensional problems. In this work, we propose and analyze a semi-implicit dynamical low-rank discontinuous Galerkin (DLR-DG) method for the space homogeneous kinetic equation with a relaxation operator, modeling the emission and absorption of particles by a background medium. Both DLRA and the DG scheme can be formulated as Galerkin equations. To ensure their consistency, a weighted DLRA is introduced so that the resulting DLR-DG solution is a solution to the fully discrete DG scheme in a subspace of the classical DG solution space. Similar to the classical DG method, we show that the proposed DLR-DG method is well-posed. We also identify conditions such that the DLR-DG solution converges to the equilibrium. Numerical results are presented to demonstrate the theoretical findings. This is a joint work with Eirik Endeve, Cory Hauck, and Stefan Schnake.

- October 16

** Title: **Existence of optimal shapes in optimal control theory

** Idriss Mazari, Universite Paris-Dauphine**

** Abstract: **In this talk, we present several recent contributions (in collaboration with G. Nadin and Y. Privat) on the question of the existence of

optimal shapes in optimal control theory for bilinear models.

Motivated by applications in spatial ecology, we investigate the following problem: consider a parabolic or elliptic equation

Lu = mu + F(u) where L is a parabolic or elliptic operator, m is the control and F is a given non-linearity.

The goal is to solve the optimisation problem

Max_m \int_\Omega j(x; u)

where j is simply a cost functional, and m is an admissible control that satisfies L1 and L^\infty bounds. In other words, we assume 0\leq m\leq 1 almost everywhere, and \int_\Omega m=V_0 where V_0 is a fixed volume constraint. A basic property for such problem is to obtain the bang-bang property for maximisers. In other words, are optimal control characteristic functions of subsets of the domain on which the equation is set? Put otherwise, can optimisers be identified with subsets of the domain? What we explain in this talk is that for bilinear control problems, the answer is analogous to the Buttazzo-DalMaso theorem: if the functional we want to optimise is monotonic, then the answer to this question is positive. Our result relies on new oscillatory techniques

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- October 23
**(ZOOM)**

** Title: **Adaptive State-Dependent Diffusion for Global Optimization With and Without Gradient

** Yunan Yang, Cornell University**

**Abstract: **We develop and analyze a stochastic optimization strategy both with and without the derivative/gradient information. A key feature is the state-dependent adaptive variance. We prove global convergence in probability with the algebraic rate in both scenarios and give quantitative results in numerical examples. A striking fact is the derivative-free result, where the convergence can be achieved without explicit information about the gradient and even without comparing different objective function values as in established methods such as the simplex method and simulated annealing. This is joint work with Björn Engquist (UT Austin) and Kui Ren (Columbia University).

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- October 30

** Title: **On Equilibria for Time-Inconsistent Mean Field Games

**Zhenhua Wang, Iowa State University**

**Abstract: **We study a time-inconsistent mean field game in infinite time horizon with entropyregularization. The underlying process is a discrete time finite-state Markov Decision Process. Weintroduce the equilibrium concept which is a time-homogeneous feedback control in relaxed typethat depends on both the state and population distribution values, and provide a characterizationresult. Then we prove that an equilibrium in the mean field game is anε-equilibrium for the corre-sponding N-player game whenNis large enough. We also provide an existence result for the meanfield equilibrium when the weighted discounting structure is assumed. This is a joint work withErhan Bayraktar.

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- November 06

** Title: **Generating HPC Code via Numerically Aware Program Graphs and LLMs.

** Ali Janessari, Department of Computer Science, Iowa State University**

**Abstract: **A major challenge in adopting the latest machine learning models into program analysis and code generation is source code representation. The absence of numerical awareness, composite data structure information, and improper way of presenting variables in previous representation works have limited the accuracy of these models. This talk proposes a new graph-based program representation called PERFOGRAPH that can capture numerical information and composite data structures, thus enabling it to capture programs’ intricate dependencies and semantics. Experimental results demonstrate that PERFOGRAPH outperforms existing representations and sets new state-of-the-art results by reducing the error rate by 7.4% (AMD dataset) and 10% (NVIDIA dataset) in the Device Mapping task. Additionally, we apply PERFOGRAPH in parallelizing sequential programs, a challenging task as even experienced developers need to spend considerable time finding parallelism opportunities and writing parallel versions of sequentially written programs. Our method improves the average runtime of the parallel code generated by the state-of-the-art LLMs by as high as 3.4% and 2.9% for the NAS and Rodinia Benchmarks.

** Bio:**

Ali Jannesari is an Assistant Professor in the Computer Science Department at Iowa State University (ISU). He is the Director of the Software Analytics and Pervasive Parallelism Lab. His research focuses primarily on the intersection of high-performance computing (HPC) and data science (AI). Prior to joining the faculty at ISU, Dr. Jannesari served as a Senior Research Fellow at the University of California, Berkeley. During his time in Germany, he led the Multicore Programming Group at the Technical University of Darmstadt and was a junior research group leader at RWTH Aachen University. He also worked as a PostDoc fellow at the Karlsruhe Institute of Technology (KIT) and the Bosch Research Center in Munich. Dr. Jannesari's research includes over ninety refereed articles, several of which have received awards. His research has been recognized and funded by multiple European and US funding agencies. Jannesari has initiated and contributed to the open-source parallelization framework DiscoPoP. He holds a Habilitation degree from TU Darmstadt and received his Ph.D. in Computer Science from Karlsruhe Institute of Technology (KIT).

- November 13

** Title: **A grid-overlay finite difference method for the fractional Laplacian on arbitrary bounded domains

** Weizhang Huang, University of Kansas**

**Abstract: **In this talk I will represent a grid-overlay finite difference method for the numerical approximation of the fractional Laplacian on arbitrary bounded domains. Using an unstructured simplicial mesh for the underlying domain, the method solves boundary-value problems of the fractional Laplacian based on a uniform-grid finite difference approximation and a data transfer from the unstructured mesh to the uniform grid. The method takes full advantages of both uniform-grid finite difference approximation in efficient matrix-vector multiplication via the fast Fourier transform and unstructured meshes for complex geometries and mesh adaptation. It is shown that its stiffness matrix is similar to a symmetric and positive definite matrix and thus invertible if the data transfer has full column rank and positive column sums. Piecewise linear interpolation is studied as a special example for the data transfer. It is proved that the full column rank and positive column sums of linear interpolation is guaranteed if the spacing of the uniform grid is smaller than or equal to a positive bound proportional to the minimum element height of the unstructured mesh. Moreover, a sparse preconditioner is proposed for the iterative solution of the resulting linear system for the homogeneous Dirichlet problem of the fractional Laplacian. I will present numerical examples to demonstrate that the new method has similar convergence behavior as existing finite difference and finite element methods and that the sparse preconditioning is effective. Furthermore, the new method can readily be incorporated with existing mesh adaptation strategies. Numerical results obtained by combining with the so-called MMPDE moving mesh method are also presented.

- November 27

** Title: **Towards a Discontinuous Galerkin Method for the Relativistic Vlasov-Maxwell System

** Yifan Hu, Iowa State University**

**Abstract: **We present an overview of a discontinuous Galerkin (DG) method for the relativistic Vlasov-Maxwell system (RVMS). RVMS is a 6-dimensional nonlinear system of integral-differential equations describing multi-species, collisionless, relativistic plasma. Many conventional high order DG methods suffer from diminishing timestep size due to high-dimensional nonlinearity, leading to computationally inefficient algorithms. To improve the timestep size and overall performance, we build on previous work of the Rossmanith Research Lab, and demonstrate preliminary results of a fully-discrete, one-step, regionally-implicit, high order DG method for RVMS. This method has successfully solved model problems such as linear advection, density stratified fluid, two-stream instability, weak Landau damping, and strong Landau damping.

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