Applied Mathematics Seminar, Fall 2014

## Mondays 3:55 PM - 5:00 PM, LCB 222

• 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 2014. Grading is based on attendance and giving at least one talk 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.

September 19, Joint Stochastics/Applied Math/Math Biology Seminar, Note Time 3:00 - 4:00 pm, Room LCB 219
Speaker: Peter Bossaerts, Finance Department, University of Utah
Title: Human reaction to extreme events and its biological foundations
Abstract:The talk will discuss recent work on how humans react to extreme events. A deeper understanding of human reactions requires proper definition of an extreme event (linking it to the concept of an outlier in statistical analysis) and its nature (distinguishing between leptokurtic and platykurtic settings, and whether the outlier is transient or reflects more persistent changes). Evidently, the noradrenergic system in the human brain provides crucial support for tracking outliers, while signals in the anterior insula allow humans to distinguish between transient and fundamental outliers. Overall, human reaction to outliers that signal fundamental changes is better adapted than reaction to transient outliers. The latter type of outlier is ubiquitous in modern large-scale social institutions, however, such as financial markets, internet and air traffic.

September 22
Speaker: Tatyana Sorokina, Department of Mathematics, Towson University
Title: Dimension of splines and intrinsic supersmoothness
Abstract:The phenomenon, known assupersmoothness" was first observed for bivariate splines and attributed to the polynomial nature of splines. Using only standard tools from multivatiate calculus it is easy to show that if we continuously glue two smooth functions along a curve with a corner", the resulting continuous function must be differentiable at the corner, as if to compensate for the singularity of the curve. Moreover, locally, this property characterizes non-smooth curves. We generalize this phenomenon to higher order derivatives. Additionally, we show how supersmoothness helps to find dimension of multivariate splines.
This talk is a part of the special session of CMDS13.

September 29
Speaker: Graeme Milton, Department of Mathematics, University of Utah
Title: The searchlight effect in hyperbolic media
Abstract:Hyperbolic media in which the dielectric tensor has both positive and negative eigenvalues have been shown to defeat the diff raction limit, and allow features at very small wavelengths to be resolved as demonstrated through hyperlenses. Even in quasistatics the underlying equation resembles a wave equation. Whereas a circular hole in a dielectric media has a simple dipolar field around it, we will see that a circular hole in an almost lossless hyperbolic media, has surrounding quasistatic fields which diverge along characteristic lines tangent to the hole, and which have finite total energy absorption along these lines, even as the loss in the media tends to zero. In a hyperbolic medium a dipole with small polarizability can dramatically influence the dipole moment of a distant polarizable dipole, if it is appropriately placed. We call this the searchlight e ffect, as the enhancement depends on the orientation of the line joining the polarizable dipoles and can be varied by changing the frequency. For some particular polarizabilities the enhancement can actually increase the further the polarizable dipoles are apart, like the way quarks interact more strongly the further they are apart.
Graeme Milton, R.C.McPhedran, A. Sihvola

October 6
Speaker: Maxence Cassier, Department of Mathematics, University of Utah
Title: Analysis of two time-dependent wave propagation phenomena: 1) Space-time focusing on unknown scatterers; 2) Limiting amplitude principle in a medium composed of a dielectric and a metamaterial.
Abstract:This talk consists of two independent parts related to my Ph.D research thema. In the first one, we are motivated by this challenging question: in a propagative medium which contains several unknown scatterers, how can one generate a wave that focuses selectively on one scatterer not only in space, but also in time? In other words, we look for a wave that "hits hard at the right spot". The technique proposed here is based on DORT method (French acronym for Decomposition of the Time Reversal Operator) which leads to space focusing properties in the frequency domain. The second part is devoted to a transmission problem between a dielectric and a metamaterial. The question we consider here is the following : does the limiting amplitude principle hold in such a medium? This principle defines the stationary regime as the large time asymptotic behavior of a system subject to a periodic excitation. An answer is proposed here in the case of an infinite two-layered medium composed of a dielectric and a particular metamaterial (Drude model).

October 20
Speaker: Andrejs Treibergs, Department of Mathematics, University of Utah
Title: Which Electric Fields are Realizable in Conducting Materials?
Abstract:The usual problem is that we are given the conductivity $\sigma$ and seek the electric field in the material, $\nabla u$, by solving the conductivity equation for the potential $$\operatorname{div}(\sigma\nabla u) = 0.$$ We ask instead, given a field $\nabla u$, is it possible to find a conductivity scalar or matrix $\sigma$ to satisfy the conductivity equation? In other words, is the field is realizable? We show periodic fields are isotropically realizable by solving a first order PDE, although the conductivity may not be periodic. We can characterize fields that are isotropically realizable by a periodic conductivity. We also give conditions for periodic realizability by matrix conductivities.

This is joint work with Marc Briane and Graeme Milton.

October 27
Speaker: Elena Cherkaev, Department of Mathematics, University of Utah
Title: Spectral representation for composite materials in forward and inverse homogenization problems
Abstract:The spectral representation of the effective properties of composites is a Stieltjes analytic representation which relates the n-point correlation functions of the microstructure to the moments of the spectral measure of an operator depending on the geometry of composite. The talk discusses an inverse homogenization problem of deriving information about the microgeometry from known effective properties, the approach is based on reconstruction of the spectral measure. The spectral measure which contains all information about the microstructure, can be uniquely recovered from effective measurements known in an interval of frequency. In particular, the volume fractions of materials in the composite and an inclusion separation parameter, as well as the spectral gaps at the ends of the spectral interval, can be uniquely reconstructed. I will discuss identification of microstructural parameters from electromagnetic and viscoelastic effective measurements and show an extension to nonlinear composites.

November 3
Speaker: Braxton Osting, Department of Mathematics, University of Utah
Title: Geometric methods for graph partitioning
Abstract: Several geometric methods for graph partitioning have been introduced in the past few years, with wide applications in clustering, community detection, and image analysis. These methods, which I'll review, are built on graph-based analogues of total variation, motion by mean curvature, the Ginzburg-Landau functional, and the Merriman-Bence-Osher threshold dynamics. In this talk, I'll discuss a new graph partitioning method where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. The resulting eigenvalue optimization problem can be solved by a rearrangement algorithm, which we show to converge in a finite number of iterations to a local minimum of a relaxed objective function. The method compares well to state-of-the-art approaches when applied to clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations. The model has a semi-supervised extension and provides natural representatives for the clusters as well.

November 10
Speaker: Varun Shankar, Department of Mathematics, University of Utah
Title: A Radial Basis Function (RBF)-based Leray Projection Method for the Incompressible Stokes and Navier-Stokes equations
Abstract: Traditionally, the unsteady incompressible Stokes and Navier-Stokes equations have been solved by fractional-step projection methods. These methods require a specification of numerical boundary conditions for the pressure, in addition to boundary conditions for the velocity. Selecting the wrong pressure boundary conditions leads to a large time-splitting error. I will present a new method based on a discrete Leray projection that avoids any time-splitting and poisson solves. The discrete Leray projector is built with a divergence-free RBF interpolant. I will demonstrate that the method shows high orders of convergence in both space and time (6th and 4th orders) on the unsteady incompressible Stokes equations, and show preliminary results for the Navier-Stokes equations as well. This is a research-in-progress talk.

November 17
Speaker: Benjamin Webb, Department of Mathematics, Brigham Young University
Title: A New Type of Stability for Dynamical Networks
Abstract: In this talk the stability of a general class of dynamical networks is considered. We begin by giving a brief introduction into the study of networks including their graph structure and the variety of dynamics such systems exhibit. Following this, we present a criteria for the global stability of a general dynamical network and show that this type of stability is invariant with respect to the addition and removal of time delays. Since time delays can have a destabilizing effect on a system's dynamics this is a stronger form of stability, which we call intrinsic stability. By using this notion of intrinsic stability we show that it is possible to reduce a network by removing its "implicit delays". The resulting smaller network can be used to obtain improved stability estimates of the original unreduced network.

November 24
Speaker: Dongbin Xiu, Department of Mathematics and Scientific Computing and Imaging (SCI) Institute, University of Utah
Title: Multi-dimensional polynomial interpolation on arbitrary nodes
Abstract: Polynomial interpolation is well understood on the real line. In multi-dimensional spaces, one often adopts a well established one-dimensional method and fills up the space using certain tensor product rule. Examples like this include full tensor construction and sparse grids construction. This approach typically results in fast growth of the total number of interpolation nodes and certain fixed geometrical structure of the nodal sets. This imposes difficulties for practical applications, where obtaining function values at a large number of nodes is infeasible. Also, one often has function data from nodal locations that are not by "mathematical design" and are unstructured''. In this talk, we present a mathematical framework for conducting polynomial interpolation in multiple dimensions using arbitrary set of unstructured nodes. The resulting method, least orthogonal interpolation, is rigorous and has a straightforward numerical implementation. It can faithfully interpolate any function data on any nodal sets, even on those that are considered singular by the traditional methods. We also present a strategy to choose optimal'' nodes that result in robust interpolation. The strategy is based on optimization of Lebesgue function and has certain highly desirable mathematical properties.

December 1
Speaker: Andrej Cherkaev, Department of Mathematics, University of Utah
Title: Constraints in multiwell variational problems and recovering the functional from the optimal solution

Abstract: The talk deals with two problems:

1. The solution of quasiconvex envelopes for multiwell Lagrangians is an infinitely fast oscillating sequence that takes the values (Young's measures) in several wells. This values - supporting points of the envelope - are subject to certain constraints. There constraints are discussed and examples are provided.

2. In a number of applications, we could postulate that a design is optimal (for instance, it is perfected by evolution), but the goal functional is unknown. I will discuss the technique of recovering these functionals and prove as an example that "All Beams are Created Optimal"

December 8
Speaker: Ben Adcock, Department of Mathematics, Simon Fraser University
Title: Function approximation via infinite-dimensional convex optimization
Abstract: In a number of applications, including uncertainty quantification and, more generally, scattered data approximation, one is required to approximate a smooth multivariate function from a small number of pointwise samples. Classically, this task is carried out by methods such as interpolation or least-squares fitting. Yet with the advent of compressed sensing there has been an increasing focus on the use alternative techniques based on convex optimization. In this talk I will describe an infinite-dimensional framework for function approximation from pointwise samples via weighted l^1 minimization. The framework I will introduce is general, and applies to arbitrary point sets and expansion systems. I will explain why working in infinite dimensions is both theoretically and practically important, and describe the critical role that weights play in the minimization. In the second half of the talk I will address the following question: does weighted l^1 minimization always perform at least as well as classical least squares regression? An affirmative answer to this question is of practical relevance, since it means that weighted l^1 techniques should always be used in over classical techniques in applications where the primary limitation is the amount of data available. I will present a mathematical framework for examining this question, and answer it in the affirmative for the case of polynomials approximations from structured and unstructured data. Finally, I will discuss the role that sparsity plays this framework, and present some open problems and challenges.

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