Applied Mathematics Seminar, Fall 2018

## Mondays 4:00 PM - 5:00 PM, LCB 215

• This seminar can be taken for credit: Students can get 1-3 credits by registering to the Applied Math Seminar class Math 7875 Section 010 for Fall 2018. Students should talk to the seminar organizer before taking it for a credit. Grading is based on attendance and giving a talk by presenting an applied-mathematics paper (not necessarily your own). Student talks will be appropriately labeled to distinguish them from visitor talks. The seminar organizer is available to review your slides, for dry-runs etc.
• Please direct questions or comments about the seminar (or its class) to Yekaterina Epshteyn (epshteyn (at) math.utah.edu)
• Talks are announced through the applied-math mailing list. Please ask the seminar organizer for information about how to subscribe to this list.

August 27
Speaker: Eugene Mishchenko, Department of Physics and Astronomy, The University of Utah
Title: The puzzling phenomenon of minimal conductivity of graphene
Abstract: graphene is a two-dimensional crystal of carbon atoms arranged in a honeycomb lattice. Graphene happens to be a semi-metal that can, in some instances, display insulating properties yet reveal metallic behavior in others. For example, conductivity of graphene is predicted by the band theory to be metallic. In contrast, screening of Coulomb interaction between electrons is expected to be rather weak, very much like in typical insulators. Accordingly, one would expect interactions to be strong and result in significant corrections to the conductivity. Surprisingly, experiments show little such corrections, if any. I will discuss theoretical efforts expended over the last decade to understand this phenomenon.

September 10
Speaker: Braxton Osting, Department of Mathematics, The University of Utah
Title: Diffusion generated methods for target-valued maps
Abstract: A variety of tasks in inverse problems and data analysis can be formulated as the variational problem of minimizing the Dirichlet energy of a function that takes values in a certain target set and possibly satisfies additional constraints. These additional constraints may be used to enforce fidelity to data or other structural constraints arising in the particular problem considered. I'll present diffusion generated methods for solving this problem for a wide class of target sets and prove some stability and convergence results. I'll give examples of how these methods can be used for the geometry processing task of generating quadrilateral meshes, finding Dirichlet partitions, constructing smooth orthogonal matrix valued functions, and solving inverse problems for target-valued maps. This is joint work with Dong Wang and Ryan Viertel.

September 24
Speaker: Lajos Horvath, Department of Mathematics, The University of Utah
Title: Change Point Detection in the Conditional Correlation Structure of Multivariate Volatility Models
Abstract: We propose semi-parametric CUSUM tests to detect a change point in the correlation structures of non--linear multivariate models with dynamically evolving volatilities. The asymptotic distributions of the proposed statistics are derived under mild conditions. We discuss the applicability of our method to the most often used models, including constant conditional correlation (CCC), dynamic conditional correlation (DCC), BEKK, corrected DCC and factor models. Our simulations show that, our tests have good size and power properties. Also, even though the near--unit root property distorts the size and power of tests, de--volatizing the data by means of appropriate multivariate volatility models can correct such distortions. We apply the semi--parametric CUSUM tests in the attempt to date the occurrence of financial contagion from the U.S. to emerging markets worldwide during the great recession.

Joint work with Marco Barassi, Department of Economics, University of Birmingham UK and Yuqian Zhao, Department of Statistics and Actuarial Science, University of Waterloo, Canada.

October 1
Speaker: Jia Zhao, Department of Mathematics, Utah State University
Title: Thermodynamically Consistent Models and Their Structure-Preserving Numerical Approximations
Abstract: Generally speaking, all non-equilibrium dynamical models should be consistent with thermodynamical principles, saying the first and second laws of thermodynamics. Thermodynamically consistent models possess several advantages. First of all, they are physically consistent (not breaking physical laws); secondly, they usually possess good mathematical structures that warrant well-posedness of the governing partial differential equations; thirdly, they provide guidance to developing stable numerical approximations. In this talk, I will first present a general approach for deriving thermodynamically consistent models using the generalized Onsager principle and the variational principle. It turns out many existing models in literature are special cases of the generalized model formulation.Then, I will introduce the newly proposed energy quadratization (EQ) strategy to develop efficient, high-order accurate and energy-stable numerical approximations for a class of thermodynamically consistent models. Applications of this modeling and numerical paradigm will be discussed.

October 15
Speaker: Akil Narayan, Department of Mathematics and SCI, The University of Utah
Title: Low-rank multifidelity methods for parameterized scientific models
Abstract: We present an algorithm for coupling inexpensive low-fidelity model simulations with high-fidelity simulation data of parameterized differential equations. The goal is to grab a "free lunch": simulation accuracy of the high-fidelity model with algorithmic complexity of only a simplified low-fidelity model. The procedure forms an approximation with sparsely available high-fidelity simluations, and is a simple linear algebraic construction with connections to kernel learning and column skeletonization of matrices. We discuss theoretical results that establish accuracy guarantees, and introduce recent analysis providing bounds on the error committed by the algorithm.

October 22
Speaker: Dong Wang, Department of Mathematics, The University of Utah
Title: The threshold dynamics method and its applications
Abstract: In this talk, I will review some recent work on the threshold dynamics method for diffusion generated motion of the interface on a solid surface. We also analyze the contact line behavior of the method from asymptotic expansion and the contact line dynamic is derived for the wetting problem. Applications to image segmentation, topology optimization for fluids, and configurations of heterogeneous foams will also be presented.

October 29
Speaker: Ian Tobasco, Department of Mathematics, University of Michigan, Ann Arbor
Title: The cost of crushing: curvature-driven wrinkling of thin elastic shells
Abstract: How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on wrinkling patterns formed by thin floating shells, we develop a rigorous method via Gamma-convergence for evaluating the cost of crushing to leading order in the shell's thickness and other small parameters. The experimentally observed patterns involve regions of well-defined wrinkling alongside totally disordered regions in which no single direction of wrinkling is preferred. Our goal is to explain the appearance of such "wrinkling domains". The basic mathematical objects that emerge in the limit are (linearly) short maps from the mid-shell into the plane, and defect measures to describe the wrinkling patterns. To solve for the limiting shape, one must maximize the total area covered in the plane subject to a shortness constraint; to solve for the optimal patterns, one must minimize the total defect subject to a curvature constraint. We analyze these limiting problems using convex duality, and obtain a boundary value-like problem that completely characterizes optimal defect measures. Optimal defect measures are not in general unique. Nevertheless, in some cases their restrictions to certain sub-domains are uniquely determined, and explicit formulas exist.

November 5
Speaker: Ellis Scharfenaker, Department of Economics, The University of Utah
Title: Statistical Equilibrium Economics
Abstract: In this talk I will explore some of the reasons why the problems in the articulation of theory and measurement in economics is rooted in the standard equilibrium concept. I will discuss how methods from statistical mechanics and information theory offer a general approach to modeling social and economic phenomena.

November 19
Speaker: Brian Knaeble, Department of Mathematics, UVU
Title: Sensitivity Analysis for Causal Inference
Abstract: It is not always ethical to conduct a randomized controlled trial. Within the field of environmental epidemiology exposure scientists have amassed large amounts of observational data. When analyzing such data classic sensitivity analysis may be used to support causal inference. This talk will provide an overview of recently developed mathematical methods for improved sensitivity analysis. We will discuss how best to apply these methods for better interpretation of data arising from studies relating to local population health here along the Wasatch Front.

December 3
Speaker: Qing Xia, Department of Mathematics, The University of Utah
Title: A Domain Decomposition Approach based on Difference Potentials Method for Chemotaxis Models in 3D
Abstract: In this talk, I will present a domain decomposition approach based on Difference Potentials Method (DPM) for approximating the solution to the classical Patlak-Keller-Segel chemotaxis models in 3D. We employ DPM and uniform Cartesian meshes to handle sub-domains of complex geometric shapes, without loss of accuracy near the irregular boundaries of the sub-domains. As a result of using uniform meshes, fast Poisson solver based on FFT is employed for better efficiency of our numerical algorithms. In addition, our domain decomposition approach is capable of mesh adaptivity and is suitable for parallel computing, which further boosts the efficiency. Numerical results from 3D simulations will be given to demonstrate the significantly improved efficiency and similar accuracy of the domain decomposition approach, in comparison to the single domain approach. (Joint work with Yekaterina Epshteyn).

December 10 (Special Seminar), Room LCB 219
Speaker: Natalie Sheils, School of Mathematics, The University of Minnesota
Title: Revivals and Fractalization in the Linear Free Space Schrodinger Equation with Pseudoperiodic Boundary Conditions
Abstract: We consider the one-dimensional linear free space Schrodinger equation on a bounded interval subject to homogeneous linear boundary conditions. We prove that, in the case of pseudoperiodic boundary conditions, the solution of the initial-boundary value problem exhibits the phenomenon of revival at specific (rational'') times, meaning that it is a linear combination of a certain number of copies of the initial datum. Equivalently, the fundamental solution at these times is a finite linear combination of delta functions. At other (irrational'') times, for suitably rough initial data, e.g., a step or more general piecewise constant function, the solution exhibits a continuous but fractal-like profile. Further, we express the solution for general homogeneous linear boundary conditions in terms of numerically computable eigenfunctions. (Joint work with Peter Olver and David Smith)

December 18, Room TBA
Speaker: Mikyoung Lim, Department of Mathematical Sciences, KAIST
Title: Series solution to interface problems using the geometric function theory
Abstract: In this talk I present a new series solution method to solve the interface problems, the conductivity problem and the linear elasticity problem in the presence of inclusions. For arbitrary simply connected domain in two dimensions there exists a conformal mapping from the unit disk to the domain. By reflecting the conformal mapping via a circle, we have an exterior conformal mapping as well. The coefficients of the conformal mapping fully contain the geometry information of the domain. I will introduce a new boundary integral method to compute the conformal mapping coefficients and a series solution for the interface problems using the properties of the conformal mapping.

Seminar organizer: Yekaterina Epshteyn (epshteyn (at) math.utah.edu).

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148