# Abstracts

## Invited Talks

## Nathan Albin

Kansas State University

*Fourier Spectral Methods for Nonlocal Operators*

Peridynamics is a model of continuum mechanics that incorporates nonlocal interactions in materials. I will discuss accurate and efficient numerical methods that use Fourier series to approximate a class of nonlocal operators that are frequently used in peridynamics. The methods have uncovered some interesting phenomena that can arise in peridynamics-like models. This is joint work with Bacim Alali.

## Daozhi Han

Missouri University of Science and Technology

*Modeling and Numerical Methods for Two-phase Flow in Superposed Free Flow and Porous Media*

In this talk we introduce a diffuse interface model for two-phase flows in a fluid layer overlying a porous media. The model satisfies an energy law, based on which the global existence of weak solution as well as weak-strong uniqueness is established. We then present a first-order in time, decoupled, unconditionally stable numerical method for solving the diffuse interface model.

## Xiaoming He

Missouri University of Science and Technology

*A Decoupled, Linear, and Unconditionally Energy Stable Finite Element Method for a Two-phase Ferrohydrodynamics Model*

In this talk, we present numerical approximations of a phase-field model for two-phase ferrofluids, which consists of the Navier-Stokes equations, the Cahn-Hilliard equation, the magnetostatic equations, as well as the magnetic field equation. By combining the projection method for the Navier-Stokes equations and some subtle implicit-explicit treatments for coupled nonlinear terms, we construct a decoupled, linear, fully discrete finite element scheme to solve the highly nonlinear and coupled multi-physics system efficiently. The scheme is provably unconditionally energy stable and leads to a series of decoupled linear equations to solve at each time step. Through numerous numerical examples in simulating benchmark problems such as the Rosensweig instability and droplet deformation, we demonstrate the stability and accuracy of the numerical scheme.

## Hailiang Liu

Iowa State University

*Selection Dynamics for Deep Neural Networks*

I shall present a partial differential equation framework for deep residual neural networks and for the associated learning problem. This is done by carrying out the continuum limits of neural networks with respect to width and depth. We study the wellposedness of the forward problem, and further establish several optimal conditions for the inverse deep learning problem. This presentation is based on a recent joint work with Peter Markowich (KAUST).

## Yuan Liu

Wichita State University

*Adaptive Multiresolution Ultra-Weak Discontinuous Galerkin Methods for Dispersive Wave Equations *

In this talk, we will present a class of adaptive multiresolution ultra-weak discontinuous Galerkin (UWDG) methods for solving some nonlinear dispersive wave equations including the Korteweg-de Vries equation and its two-dimensional generalization, the Zakharov-Kuznetsov equation. For the KdV equation, standard UWDG scheme is implemented with mulitiresolution basis. For the ZK equation, which contains mixed derivative terms, a new UWDG formulation is proposed. The L2 stability is established for this new scheme on regular meshes, and the optimal error estimate with a novel local projection is obtained for a simplified ZK equation. Adaptivity is achieved based on multiresolution and is particularly effective for capturing solitary wave structures. Numerical examples are presented to demonstrate the accuracy and capability of our methods. This is joint work with Juntao Huang, Yong Liu, Zhanjing Tao and Yingda Cheng.

## Tianshi Lu

Wichita State University

*Superconvergence of Discontinuous Galerkin Method for Linear and Nonlinear Advection Equations*

We studied the superconvergence property of the discontinuous Galerkin method with upwind-biased numerical fluxes for linear advection equations. The asymptotic error of the cell average and the post-processed solution were derived. The superconvergence of the post-processed solution for nonlinear equations was also analyzed. Numerical examples were given for demonstration.

## Songting Luo

Iowa State University

*A Fixed-point Iteration Method for High Frequency Helmholtz Equations*

In order to obtain globally valid solutions for high frequency Helmholtz equations efficiently without ‘pollution effect’, we transfer the problem into a fixed-point problem related to an exponential operator, and the associated functional evaluations are achieved by unconditionally stable operator-splitting based pseudospectral schemes such that large step sizes are allowed to reach the approximated fixed point efficiently for pre-scriped accuracy requirement. The Anderson acceleration is incorporated to accelerate the convergence. Both two-dimensional and three-dimensional numerical experiments are presented to demonstrate the method.

## Dinh-Liem Nguyen

Kansas State University

*Orthogonality Sampling Methods for Solving Electromagnetic Inverse Scattering Problems*

Broadly speaking, inverse scattering problems are the problems of determining information about an object (scatterer) from measurements of the field scattered from that object. Solving these inverse problems is challenging since they are in general highly nonlinear and severely ill-posed problems. In this talk, we will discuss our recent results on orthogonality sampling methods for solving electromagnetic inverse scattering problems. Compared with classical sampling methods (e.g. linear sampling method, factorization method) the orthogonality sampling methods are simpler to implement, do not require regularizations, and are more robust with respect to noise in the data.

## James Rossmanith

Iowa State University

*Strategies to Improve the Stability Properties of Explicit Discontinuous Galerkin Schemes*

In this work we consider explicit time stepping strategies for discontinuous Galerkin schemes applied to hyperbolic partial differential equations. We provide two extensions of the classical Lax-Wendroff approach in this context and provide rigorous stability proofs. This is joint work with Sam Van Fleet.

## Jue Yan

Iowa State University

*Cell-average Based Neural Network Fast Solvers for Time Dependent Partial Differential Equations*

In this talk, we present the recently developed cell-average based neural network (CANN) method for time dependent problems. CANN method is motivated by finite volume scheme and is based on the integral/weak formulation of partial differential equations. A simple shallow feed forward network is applied to learn the solution average difference between two neighboring time steps. Well trained network parameter can be interpreted as scheme coefficients of an explicit one-step finite volume like scheme. While convergence orders are observed, quite a few unusual properties are found with CANN method that are not common to conventional numerical methods. CANN method is found being relieved from explicit scheme CFL restriction on small time step size. Very large time step size can be applied which makes the method an extremely fast and efficient solver. CANN method can sharply evolve contact discontinuity with almost zero numerical diffusion. Shock and rarefaction waves are well captured for nonlinear hyperbolic conservation laws. The method has been successfully applied to high dimensional parabolic PDEs and high order PDEs like KdV equations.

## Xu Zhang

Oklahoma State University

*Immersed Finite Element Methods for Fluid Flow Interface Problems*

In this talk, we introduce a high-order immersed finite element (IFE) method for solving Navier-Stokes equation with discontinuous viscosity coefficient across fluid interface. Immersed P2-P1 Taylor-Hood finite element space are developed for spatial discretization without requiring mesh to align interfaces. The existence and uniqueness of IFE basis functions are established. In spatial discretization, we use an enhanced partially penalized IFE method with ghost penalties. In temporal discretization, theta-scheme and backward difference differentiation formulas are adopted. Extensive numerical experiments show that the proposed method is optimal-order convergent for both velocity and pressure in both stationary and moving interface cases.

## Yanzhi Zhang

Wichita State University

*Meshfree Methods for Nonlocal Problems with the Fractional Laplacian*

Recently, the fractional Laplacian has received great attention in modeling complex phenomena that involve long-range interactions. However, its nonlocality introduces considerable challenges in both mathematical analysis and numerical simulations. So far, numerical methods for the fractional Laplacian still remain limited. In this talk, I will discuss our recent meshfree methods for nonlocal problems with fractional Laplacian. The properties of these methods will be discussed, and some applications of nonlocal problems with the fractional Laplacian will also be demonstrated.

## Poster Presentations

## John Carter

Missouri University of Science and Technology

*SAV Ensemble Algorithms for MHD Equations*

We study a scalar auxilary variable ensemble method for fast computation of MHD flows. The purpose of the ensemble is to efficiently solve for multiple realizations of the problem with J different initial conditions and/or body forces. We follow the Generalized SAV method, which results in an unconditionally stable decoupled scheme and allows more flexibility in how scalar variable may be defined.

## Alex Fulk

University of Kansas

*Exploring the Effects of Prescribed Fire on the Spread of Ticks*

Lyme disease is a tick-borne disease and its prevalence has consistently increased over the past several decades. Here we consider a method of control that has garnered recent interest from researchers, prescribed fire. Conflicting results on the effectiveness of this method have made it difficult to determine if it should be used. Thus, the current work focuses on the effects of prescribed burns on the abundance of ticks via an impulsive partial differential equation system. Findings indicate that ticks recover relatively quickly following a burn, but frequent, long-term prescribed burns can reduce the prevalence of ticks in and around the area being burned. Finally, we also explored the effectiveness of prescribed burns in preventing establishment of ticks into new areas. These findings indicated that frequent burning can limit, but not prevent establishment.

## Thu Le

Kansas State University

*Imaging of 3D Objects with Real Data Using Orthogonality Sampling Methods*

The electromagnetic inverse scattering problem aims to reconstruct the location and shape of an unknown object from the scattered wave data. It has wide applications in radar and nondestructive testing. We investigate the Orthogonality Sampling Method for Maxwell's equations. It is a fast, robust and stable method. Numerical results testing against 3D experimental data from the Fresnel institute will be presented.

## Xuejian Li

Missouri University of Science and Technology

*Incremental Proper Orthogonal Decomposition (POD) and Its Applications*

In this work, we present an incremental technique to address the challenge of computing POD modes and data compression for large-scale data. The algorithm is provided for both standard R^n space and weighted R^n space. We also provide error estimations to easily monitor the information loss along the incremental process. The applications are then given to producing dominant POD modes for reduced basis method and data compression for PDE-constrained optimization. Finally, the numerical examples are provided to validate the proposed method.

## Xuping Tian

Iowa State University

*AEGD: Adaptive Gradient Descent with Energy*

We propose AEGD, a new algorithm for first-order gradient-based optimization of non-convex objective functions, based on a dynamically updated ‚Äòenergy‚Äô variable. The method is shown to be unconditionally energy stable, irrespective of the base step size. We prove energy-dependent convergence rates of AEGD for both non-convex and convex objectives, which for a suitably small step size recovers desired convergence rates for the batch gradient descent. We also provide an energy-dependent bound on the stationary convergence of AEGD in the stochastic non-convex setting. The method is straightforward to implement and requires little tuning of hyper-parameters. Experimental results demonstrate that AEGD works well for a large variety of optimization problems: it is robust with respect to initial data, capable of making rapid initial progress. The stochastic AEGD shows comparable and often better generalization performance than SGD with momentum for deep neural networks.

## Trung Truong

Kansas State University

*A Stable Sampling Method for Inverse Scattering from Periodic Structures*

Inverse scattering problems for periodic structures arise from many real-life applications, especially non-destructive testing for diffraction gratings and photonic crystals. Different refractive index reconstruction and shape reconstruction techniques for these problems have been studied, however, they often perform poorly when the data is perturbed by high levels of noise due to the problems being severely ill-posed. In this work, we present a new stable shape reconstruction method based on an Orthogonality Sampling approach that is capable of inverting highly noisy data, which makes it a perfect candidate for testing against experimental and real data.

## Shiping Zhou

Missouri University of Science and Technology

*Numerical Study on High-order Fractional Laplacian*

Nonlocal problems arise naturally in different fields, such as molecular dynamics in biology, stable Levy process, anomalous diffusion phenomena in physics, and conformal geometry. The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a nonlocal pseudo-differential operator for $\alpha\in(0,2)$. In this work, we consider the extension of this operator to a high-order version with $\alpha\in(2,4)$. We propose several numerical schemes and corresponding simulations of the high-order fractional Laplacian. The performances of those methods are studied and compared in solving different problems.