## AWM at SIAM 2017 Abstracts

#### Monday, July 10, 2017, 10:30 am – 12:30 pm, Room 305

### AWM Workshop:

Recent Advances in Numerical Analysis and Scientific Computing–Part I of II

### A New Convergence Analysis of Finite Element Methods for Elliptic Distributed Optimal Control Problems with Pointwise State Constraints

#### Susanne Brenner, Department of Mathematics, Louisiana State University, brenner@math.lsu.edu

We will discuss finite element methods for elliptic distributed optimal control problems with pointwise state constraints on two and three dimensional convex polyhedral domains formulated as fourth order variational inequalities.We will present a new convergence analysis that is applicable to *C*^{1} finite element methods, classical nonconforming finite element methods and discontinuous Galerkin methods.

### Super-convergence of the Asymptotic Approximation of Linear Kinetic Equation with Spectral Methods

#### Zheng Chen, Oak Ridge National Laboratory, chenz1@ornl.gov

In this work, we prove some convergence properties for a semi-discrete, moment-based approximation of a model kinetic equation in one dimension. This approximation is equivalent to a standard spectral method in the velocity variable of the kinetic distribution and, as such, is accompanied by standard algebraic estimates of the form *N*^{−q}, where *N* is the number of modes and *q* depends on the regularity of the solution. However, in the multiscale setting, we show that the error estimate can be expressed in terms of the scaling parameter ε, which measures the ratio of the mean-free-path to the domain in the system. In particular we show that the error in the spectral approximation is ?(ε*N* +1 ). More surprisingly, the coefficients of the expansion satisfy some super convergence properties. In particular, the error of the *l*th coefficient of the expansion scales like ?(ε^{2N}) when *l* = 0 and ?(ε^{2N+2−l}) for all 1 ≤ *l* ≤ *N*. This result is significant, because the low-order coefficients correspond to physically relevant quantities of the underlying system. Numerical tests will also be presented to support the theoretical results.

### Numerical Methods for the Chemotaxis Models

#### Yekaterina Epshteyn, Department of Mathematics, University of Utah, epshteyn@math.utah.edu

In this talk, we will introduce and discuss several recently developed numerical methods for the approximation and simulation of the chemotaxis and related models in Biology. Numerical experiments to demonstrate the stability and high-order accuracy of the proposed methods for chemotaxis systems will be presented. Ongoing research projects will be discussed as well.

### The Effect of the Sensitivity Parameter in Weighted Essentially Non-oscillatory Methods

#### Yulia Hristova, University of Michigan-Dearborn, yuliagh@umich.edu

Weighted essentially non-oscillatory methods (WENO) were developed to capture shocks in the solution of hyperbolic conservation laws while maintaining stability and without smearing the shock profile. WENO methods accomplish this by assigning weights to a number of candidate stencils, according to the smoothness of the solution on the stencil. These weights favor smoother stencils when there is a significant smoothness difference, while combining all of the stencils to attain higher order when the stencils are all smooth. When WENO methods were initially developed, a small parameter ε was introduced to avoid division by zero. Over time, it has become apparent that ε plays the role of the sensitivity parameter in the stencil selection. In addition, the oscillations in the numerical solution also depend on the size of ε. In this talk I will show that the value of ε must be below a certain critical threshold ε_{c}, the dependence of this threshold on the function used and on the size of the jump discontinuity captured. I will also discuss some results about the size of the oscillations when ε < ε_{c}, and their dependence on the size of ε, the function used, and the size of the jump discontinuity. This work is joined with Bo Dong, Sigal Gottlieb, Yan Jiang, and Haijin Wang.

#### Monday, July 10, 2017, 4:00 pm – 6:00 pm, Room: 305

### AWM Workshop:

Recent Advances in Numerical Analysis and Scientific Computing–Part II of II

### A Moving Mesh WENO Method Based on Exponential Polynomials for One-dimensional Conservation Laws

#### Yan Jiang, Michigan State University, jiangyan@math.msu.edu

We develop a novel WENO scheme with non-polynomial bases, in particular, the exponential bases, with a high order moving mesh method for one-dimensional (1D) hyperbolic conservation laws, which is shown to be effective in resolving shocks and other complex solution structures. A collection of numerical examples is presented to demonstrate high order accuracy and robustness of the method in capturing smooth and non-smooth solutions, including the strong δ shock arising from the weakly hyperbolic pressureless Euler equations.

### Pseudo-time Adaptive Regularization for Non- monotone Problems

#### Sara Pollock, Wright State University, sara.pollock@wright.edu

We will discuss some of the unique challenges encountered in approximating the solutions to elliptic partial differential equations of nonmonotone type. Starting simulations on a coarse mesh, for the ultimate efficiency of the method, may trigger a sequence of divergent iterates based on linearized problems, unless some type of stabilization, or regularization is introduced. We will discuss adaptive strategies based on pseudo-time regularization to produce convergent sequences of iterates to approximate the PDE solution. The talk will be illustrated with numerical examples of the developed ideas applied to finite element discretizations of nonlinear diffusion problems.

### On the Sensitivity to the Filtering Radius in Leray Models of Incompressible Flow

#### Annalisa Quaini, Department of Mathematics, University of Houston, quaini@math.uh.edu

One critical aspect of Leray models for the Large Eddy Simulation (LES) of incompressible flows at moderately large Reynolds number (in the range of few thousands) is the selection of the filter radius. This drives the effective regularization of the filtering procedure, and its selection is a trade-off between stability (the larger, the better) and accuracy (the smaller, the better). In this paper, we consider the classical Leray-alpha and a recently introduced Leray model with a deconvolution-based indicator function, called Leray-alpha-NL. We investigate the sensitivity of the solutions to the filter radius by introducing the sensitivity systems, analyzing them at the continuous and discrete levels, and numerically testing them on two bench- mark problems.

### A BDDC Preconditioner for C^{0} Interior Penalty Methods

#### Kening Wang, University of North Florida, kening.wang@unf.edu

A balancing domain decomposition by constraints (BDDC) algorithm is constructed and analyzed for a discontinuous Galerkin method, the C^{0} interior penalty method, for biharmonic problems. The condition number of the preconditioned system is bounded by *C*(1 + ln(*H*/*h*))^{2}, where *h* is the mesh size of the triangulation, *H* is the typical diameter of subdomains, and the positive constant *C* is independent of *h* and *H*. Numerical experiments corroborate this result.

### AWM Workshop: Career Panel:

Perspectives from Women in Research

Abstracts not available

#### Tuesday, July 11, 2017, 8:00 pm – 10:00 pm, Room: West Atrium, 3rd Floor

### PP2 Minisymposterium: AWM Posters

### Tailoring Tails in Taylor Dispersion: How Boundaries Shape Chemical Deliveries in Microfluidics

#### Francesca Bernardi, University of North Carolina, USA, bernardi@live.unc.edu

Abstract. We present a study of the dispersion of a passive scalar in laminar shear flow through rectangular and elliptical channels. Through asymptotic analysis, Monte Carlo simulations and laboratory experiments, we show that the channel’s cross-sectional aspect ratio sets the longitudinal asymmetry of the tracer distribution at long time: thin channels generate distributions with sharp fronts and tapering tails, whereas thick channels produce the opposite effect. Potential applications to microfluidics will be discussed.

### A New Goal-Oriented A Posteriori Error Estimation for 2D and 3D Saddle Point Problems in Hp Adaptive Fem

#### Arezou Ghesmati, Texas A&M University, USA, aghesmati@math.tamu.edu

Abstract. We present a new approach on goal-oriented a posteriori error estimation for an automatic hp-Adaptive Finite Element Method. The method is based on the classical dual-weighted algorithm on local patches and applying the Clément type interpolation operators. The reliability and also the efficiency of the proposed a posteriori error estimator have been proved. Finally, the performance of the proposed estimator for both h- and hp-Adaptive FEM has been investigated in numerical examples for Saddle point problem.

### Sobolev Discontinuous Galerkin (dG) Methods

#### Adeline Kornelus, University of New Mexico, USA, kornelus@unm.edu

Abstract. The dG method is a popular polynomial-based method known for its spectral accuracy and geometric flexibility. Despite its nice properties, dG suffers from a restrictive time step size. In this talk, we present Sobolev dG, a novel dG method that allows for much larger time steps compared to traditional dG methods, with computational results illustrating Sobolev dGaAZs excellent time stepping properties. Sobelev dG tests with low degree polynominals while maintaining the accuracy of high-order polynominals.

### An AMG Approach in Solving Graph Laplacians of Protein Networks Based on Diffusion State Distance Metrics

#### Junyuan Lin, Tufts University, USA, junyuan.lin@tufts.edu

Abstract. In this presentation, protein networks from 2016 Disease Module Identification DREAM Challenge are analyzed. We redefined the Protein-Protein Interaction networks on a new distance metric, “Diffusion State Distance’ metric, and applied a modified Algebraic Multi-grid Method to calculate the distance between each pair of nodes. Finally, we applied spectral clustering to partition the protein network into functional modules. Consequently, we ranked No.1 out of over 50 teams over the world.

### Hyperspectral Image Classification Using Parallellized Graph Clustering Methods

#### Zhaoyi Meng, University of California, Los Angeles, USA, mzhy@ucla.edu

Abstract. We introduce two data classification algorithms and investigate many-core node parallelization schemes. The new algorithms are derived from PDE solution techniques and they provide a significant performance and accuracy advantage over traditional data classification algorithms. We use OpenMP as the parallelization language to parallelize the most time-consuming parts of the algorithms and then optimize the OpenMP implementations. We show performance improvement and strong scaling behavior.

### A Reaction-Diffusion Model for Cell Polarization in Yeast

#### Marissa Renardy, Ohio State University, USA, renardy.1@osu.edu

Abstract. Cell polarization is fundamental to cellular processes such as differentiation, migration, and development. We consider cell polarization driven by a pheromone gradient in mating yeast. This is modeled by a large reaction-diffusion system. Many parameters in the system can be only crudely estimated, which leads to inaccuracy in the model. The aim of our work is to determine the sensitivity of the system to the parameters and perform data-driven parameter estimation.

### An Invariant-Region-Preserving Limiter for DG Method to Compressible Euler Equations

*Yi Jiang, Iowa State University, USA, yjiang1@iastate.edu*;

Hailiang Liu, Iowa State University, USA, hliu@iastate.edu

Abstract. We introduce an explicit invariant-region-preserving limiter for compressible Euler equations. The invariant region considered consists of positivity of density and pressure and a maximum principle of a specific entropy. The reconstructed polynomial preserves the cell average, lies entirely within the invariant region and does not destroy the high order of accuracy for smooth solutions. Numerical tests are presented to illustrate the properties of the limiter. In particular, the tests on Riemann problems show that the limiter helps to damp the oscillations near discontinuities.

### Band-Edge Solitons in the Nls Equation with Periodic Pt-Symmetric Potentials

#### Jessica Taylor, University of California, Merced, USA, jtaylor5@ucmerced.edu

Abstract. The bifurcation of nonlinear bound states from the spectral edges of the NLS equation with periodic parity-time (PT)-symmetric potentials is studied asymptotically and computationally. These modes undergo a transition near the breakdown point of the PT symmetry. The effective mass tensor and nonlinear coupling constants, which determine the structure of these modes, are analyzed in detail. The implication to collapse dynamics is discussed.

### Polynomial Preconditioned Arnoldi for Eigenvalues

*Jennifer A. Loe, Baylor University, U.S., jennifer_loe@baylor.edu*;

Ron Morgan, Baylor University, U.S., Ronald_Morgan@baylor.edu;

Mark Embree, Virginia Tech, USA, embree@vt.edu

Abstract. Polynomial preconditioning has been explored for Krylov methods for large eigenvalue problems but has not become standard, possibly due to difficulty in obtaining a helpful polynomial. We give a simple choice for a polynomial preconditioner and a stable way to compute with it. When applied to the Arnoldi method for eigenvalues, this approach can significantly reduce computational costs for difficult problems.

### Almost Sure Convergence of Particle Swarm Optimization Using Pure Adaptive Search Method

#### Ganesha Weerasinghe, Auburn University, U.S., ksw0013@auburn.edu

Abstract. Particle swarm optimization (PSO) is a population based stochastic optimization method, which has been used across a wide range of applications. Though there have been several studies about the convergence of PSO, the convergence proofs in these studies are either imprecise or under unrealistic assumptions. Here, we present a almost surely convergence analysis of PSO based on the fact that particles personal and global best values follow pure adaptive search method.

### Computational Approaches for Linear Goal-Oriented Bayesian Inverse Problems

#### Karina Koval, Courant Institute of Mathematical Sciences, New York University, USA, koval@cims.nyu.edu

Abstract. We present a comparison of different approaches for computing low rank approximations to posterior covariances arising in goal-oriented inverse problems. In particular, we study problems where both the forward model and the quantity of interest depend linearly on the uncertain parameters. We present and compare algorithms for the computation of optimal approximations, referring to recent work where the approximations are obtained as low rank updates to the priors.