Applied Mathematics Seminar, Spring 2017

## Mondays 4:00 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 Spring 2016. 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.

January 9. Special Applied Math Seminar.
Speaker: Ricardo Alonso, Department of Mathematics, PUC-Rio
Title: The 1-D dissipative Boltzmann equation
Abstract: We discuss elementary properties of the 1-D dissipative Boltzmann equation. In particular, we show the optimal cooling rate of the model, give existence and uniqueness of measure solutions, and prove the existence of a non-trivial self-similar profile.

January 13 (Friday). Special Applied Math/Statistics/Stochastics Seminar
Note Time, 4:15pm - 5:15pm and Place LCB 222.

Speaker: Dane Taylor,Department of Mathematics, University of North Carolina-Chapel Hill
Title: Optimal layer aggregation and enhanced community detection in multilayer networks
Abstract: Inspired by real-world networks consisting of layers that encode different types of connections, such as a social network at different instances in time, we study community structure in multilayer networks. We study fundamental limitations on the detectability of communities by developing random matrix theory for the dominant eigenvectors of matrices that encode random networks. Specifically, we study modularity matrices that are associated an aggregation of network layers. Layer aggregation can be beneficial when the layers are correlated, and it represents a crucial step for discretizing time-varying networks (whereby time layers are binned into time windows). We explore two methods for layer aggregation: summing the layers' adjacency matrices and thresholding this summation at some value. We identify layer-aggregation strategies that minimize the detectability limit, indicating good practices (in the context of community detection) for how to aggregate layers, discretize temporal networks, and threshold pairwise-interaction data matrices.

January 18 (Wednesday). Special Applied Math/Statistics/Stochastics Seminar.
Note Time, Wednesday 4pm and Place JWB 335.

Speaker: Wenjing Liao, Department of Mathematics, Johns Hopkins University
Title: Multiscale adaptive approximations to data and functions near low-dimensional sets
Abstract: High-dimensional data are often modeled as samples from a probability measure in $R^D$, for $D$ large. We study data sets exhibiting a low-dimensional structure, for example, a $d$-dimensional manifold, with $d$ much smaller than $D$. In this setting, I will present two sets of problems: low-dimensional geometric approximation to the manifold and regression of a function on the manifold. In the first case we construct multiscale low-dimensional empirical approximations to the manifold and give finite-sample performance guarantees. In the second case we exploit these empirical geometric approximations of the manifold to construct multiscale approximations to the function. We prove finite-sample guarantees showing that we attain the same learning rates as if the function was defined on a Euclidean domain of dimension $d$. In both cases our approximations can adapt to the regularity of the manifold or the function eve when this varies at different scales or locations. All algorithms have complexity $C n\log (n)$ where $n$ is the number of samples, and the constant $C$ is linear in $D$ and exponential in $d$.

January 30. Special Applied Math Seminar.
Speaker: Michele Coti Zelati, Department of Mathematics, University of Maryland College Park
Title: Stochastic perturbations of passive scalars and small noise inviscid limits
Abstract: We consider a class of invariant measures for a passive scalar driven by an incompressible velocity field on a periodic domain. The measures are obtained as limits of stochastic viscous perturbations. We prove that the span of the H1 eigenfunctions of the transport operator contains the support of these measures, and apply the result to a number of examples in which explicit computations are possible (relaxation enhancing, shear, cellular flows). In the case of shear flows, anomalous scalings can be handled in view of a precise quantification of the enhanced dissipation effects due to the flow.

February 3 (Friday). Special Applied Math/Statistics/Stochastics Seminar. Note Time is 3pm - 4pm and Room is LCB 219.
Speaker: Dan Shen, Department of Mathematics and Statistics, University of South Florida
Title: PCA Asymptotics & Analysis of Tree Data
Abstract: A general asymptotic framework is developed for studying consistency properties of principal component analysis (PCA). Our framework includes several previously studied domains of asymptotics as special cases and allows one to investigate interesting connections and transitions among the various domains. More importantly, it enables us to investigate asymptotic scenarios that have not been considered before, and gain new insights into the consistency, subspace consistency and strong inconsistency regions of PCA and the boundaries among them. In addition, we studied the asymptotic properties of these sparse PC directions for scenarios with fixed sample size and increasing dimension (i.e. High Dimension, Low Sample Size (HDLSS)). Second we develop statistical methods for analyzing tree-structured data objects. This work is motivated by the statistical challenges of analyzing a set of blood artery trees, which is from a study of Magnetic Resonance Angiography (MRA) brain images of a set of 98 human subjects. The non-Euclidean property of tree space makes the application of conventional statistical analysis, including PCA, to tree data very challenging. We develop an entirely new approach that uses the Dyck path representation, which builds a bridge between the tree space (a nonEuclidean space) and curve space (standard Euclidean space). That bridge enables the exploitation of the power of functional data analysis to explore statistical properties of tree data sets.

February 24 (Friday). LCB 222, 4:15pm - 5:30pm.
Speaker: Gunilla Kreiss, Department of Information Technology, Uppsala University
Title: Error analysis for finite difference methods: What is the relation between the truncation error in a single point, and the global error?
Abstract: In many high order accurate finite difference methods for partial differential equations cases the local truncation error is significantly larger at a few points near boundaries than at interior points. The effect can be analyzed by Laplace transform techniques. In this talk we shall discuss how results from one space dimension can be extended to higher space dimensions. We will in particular consider the second order wave equation.

March 6
Speaker: Graeme Milton, Department of Mathematics, University of Utah
Title: On the elastic moduli of 3-d printed materials.
Abstract: 3-d printing gives us unprecedented ability to tailor microstructures to achieve desired goals. From the mechanics perspective one would like, for example, to know how to design structures that guide stress, in the same way that conducting fibers are good for guiding current. In that context the natural question is: what are the possible pairs of (average stress, average strain) that can exist in the material. A more grand question is: what are the possible effective elasticity tensors that can be achieved by structuring a material with known moduli. This is a highly non-trivial problem: in 3-dimensions elasticity tensors have 18 invariants and even an object as simple as a distorted hypercube in 18 dimensions requires about 4.7 million numbers to specify it. Here we review some of the progress that has been made on this question.
This is joint work with Marc Briane and Davit Harutyunyan.

March 20
Speaker: Saverio Spagnolie, Department of Mathematics, University of Wisconsin-Madison
Title: The sedimentation of flexible filaments
Abstract: The deformation and transport of elastic filaments in viscous fluids play central roles in many biological and technological processes. Compared with the well-studied case of sedimenting rigid rods, the introduction of filament compliance may cause a significant alteration in the long-time sedimentation orientation and filament geometry. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations and shapes which match excellently with full numerical simulations. In the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. Incorporating the dynamics of a single filament into a mean-field theory, we show how flexibility affects a well established concentration instability in a sedimenting suspension. Related aspects of boundary integral equations, boundary effects, and viscous erosion will also be touched upon.

March 22 (Wednesday).
Note Time, Wednesday 1pm - 2pm and Place JFB B-1.

Speaker: Nick Trefethen, Department of Mathematics, University of Oxford
Title: CUBATURE, APPROXIMATION, AND ISOTROPY IN THE HYPERCUBE
Abstract: The hypercube is the standard domain for computation in higher dimensions. We explore two respects in which the anisotropy of this domain has practical consequences. The first is the matter of axis-alignment in low-rank compression of multivariate functions. Rotating a function by a few degrees in two or more dimensions may change its numerical rank completely. The second concerns algorithms based on approximation by multivariate polynomials, an idea introduced by James Clerk Maxwell. Polynomials defined by the usual notion of total degree are isotropic, but in high dimensions, the hypercube is exponentially far from isotropic. Instead one should work with polynomials of a given "Euclidean degree". The talk will include numerical illustrations, a theorem based on several complex variables, and a discussion of "Padua points".

March 24 (Friday).
Note Time, Friday 2pm - 2:30pm and Place JWB 333.

Speaker: Ross McPhedran, School of Physics, University of Sydney
Title: Macdonald's Theorem for Analytic Functions
Abstract: This informal talk is about a theorem related to analytic functions, which I came across in an Obituary of Professor H.M. Macdonald, after whom the Bessel K functions are named. I find the theorem interesting, and am hoping to learn whether it has been forgotten, or is known, possibly under another name. The following is the abstract from a paper recently placed on arxiv (1702.03458). A proof is reconstructed for a useful theorem on the zeros of derivatives of analytic functions due to H. M. Macdonald, which appears to be now little known. The Theorem states that, if a function $f(z)$ is analytic inside a region bounded by a contour on which the modulus of $f(z)$ is constant, then the number of zeros of $f(z)$ and of its derivative in the region differ by unity. The proof is accompanied by Mathematica illustrations.

April 3.
Speaker: Chrysoula Tsogka, Department of Mathematics, University of Crete
Title: Multifrequency interferometric imaging with intensity-only measurements
Abstract: We consider the problem of coherent imaging using intensity-only measurements. The main challenge in intensity-only imaging is recovering phase information that is not directly available in the data, but is essential for coherent image reconstruction. Imaging without phases arises in many applications such as crystallography, ptychography and optics where images are formed from the spectral intensities.
The earliest and most widely used methods for imaging with intensity-only measurements are alternating projection algorithms. The basic idea is to project the iterates on the intensity data sequentially in both the real and the Fourier spaces. Although these algorithms are very efficient for reconstructing the missing phases in the data, and performance is often good in practice, they do not always converge to the true, missing phases. This is especially so if strong constrains or prior information about the object to be imaged, such as spatial support and non-negativity, are not reliably available.
Rather than using phase retrieval methods, we propose a different approach in which well-designed illumination strategies exploit the spatial and frequency diversity inherent in the problem. These illumination strategies allow for the recovery of interferometric data that contain relative phase information which is all that is needed to reconstruct a so-called holographic image. There is no need for phase reconstruction in this approach. Moreover, we show that this methodology leads to holographic images that suffer no loss of resolution compared with those that use full phase information. This is so when the media through which the probing signals propagate are assumed to be homogeneous.
We also consider inhomogeneous media where wavefront distortions can arise. In such media the incoherence in the recovered interferometric data can be reduced by restricting them to small spatial and frequency offsets. Using an efficient implementation of this restriction process we obtain holographic images with a somewhat reduced resolution compared to the homogeneous medium case.
The robustness of our approach will be explored with numerical simulations carried out in an optical (digital) microscopy imaging regime.

April 10.
Speaker: Mark Allen, Department of Mathematics, BYU
Title: A Free Boundary Problem on Cones
Abstract: The one phase free boundary problem shares a well-known connection to area-minimizing surfaces. In this talk we review this connection and then discuss the one-phase problem on rough surfaces, and in particular cones. After reviewing results of the author with Chang Lara for the one-phase problem on two-dimensional cones, we revisit the connection to area-minimizing surfaces to gain insight into the problem on higher dimensional cones. We then present new results on when the free boundary is allowed to pass through the vertex of a three-dimensional cone as well as results for higher dimensional cones.

May 15. Note Room is LCB 215. Time is Monday May 15th 4pm - 5pm.
Speaker: Dmitriy Leykekhman, Department of Mathematics, University of Connecticut
Title: On positivity of the discrete Green's function and discrete Harnack inequality
Abstract: In this talk I will discuss some recent results obtained for the finite element discrete Green's function and its positivity. The first result shows that on smooth two-dimensional domains the discrete Green's function with singularity in the interior of the domain must be strictly positive throughout the computational domain once the mesh is sufficiently refined. As an application of this result, we establish a discrete Harnack inequality for piecewise linear discrete harmonic functions. In contrast to the discrete maximum principle, the result is valid for general quasi-uniform shape regular meshes except for a condition on the layer near the boundary.

May 22. Note Room is LCB 323. Time is Monday May 22 4pm - 5pm.
Speaker: Roland Pulch, Institute for Mathematics and Informatics, University of Greifswald
Title: Model order reduction for dynamical systems with random parameters
Abstract: We consider dynamical systems consisting of ordinary differential equations or differential algebraic equations, which include random parameters for uncertainty quantification. The resulting random processes are expanded into a series with unknown time-dependent coefficient functions and given orthogonal basis polynomials depending on the parameters. The stochastic model can be solved numerically by either a stochastic Galerkin method or a stochastic collocation method. Both approaches yield (weakly) coupled dynamical systems, whose dimensions are much larger than the original systems. Hence we apply techniques of model order reduction to decrease the dimensionality. A reduced-order model also implies a sparse representation of a quantity of interest. The focus is on linear dynamical systems, because a detailed analysis is feasible via a transfer function defined in the frequency domain. We present results of numerical computations for test examples modeling electric circuits or mechanical systems.

July 21. Note Room is LCB 323. Time is Friday July 21 3pm - 4pm.
Speaker: Ryan Murray, Department of Mathematics, Penn State
Title: Analytical approaches to overfitting in classification problems.
Abstract: Recently a variety of techniques from analysis and PDE have been used to rigorously study problems in statistical machine learning. This talk will discuss recent work which uses modern analytical techniques (such as Young measures, gamma convergence and fractional Sobolev spaces) to rigorously study "overfitting", which is a prevalent difficulty in machine learning. The discussion will be aimed towards non-experts: no prior expertise in machine learning or analysis will be assumed.

July 31. Note Room is LCB 323. Time is Monday July 31 4pm - 5pm.
Speaker: Robert Viator, IMA, University of Minnesota
Title: Opening Band Gaps in Two-dimensional Photonic Crystals
Abstract: High-contrast photonic crystals have been studied intensely by both mathematicians and physicists for the past several decades. A desirable property of such media is the existence of band gaps, i.e. bands of frequencies at which no wave can propagate in the given medium. In this talk, we will identify an explicit lower bound on material contrast necessary to open band gaps in two-component, two-dimensional photonic crystals, along with the location and estimates on the size of these gaps. All of these properties are determined entirely by spectral quantities associated with the geometry of the crystalline structure, including Dirichlet and Neumann-Poincare spectra.

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