Applied Mathematics Seminar, Spring 2016

## 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 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 22. Special joint Applied Math and Stochastics Seminar. Note Time 3pm and Place LCB 219.
Speaker: Wenjia Jing, Department of Mathematics, The University of Chicago
Title: Imaging in random media: modeling and sifting noisy signals
Abstract: In imaging problems, waves propagating through a medium are recorded and used to locate reflectors, sources, etc. in the medium. The recorded signals are often incoherent due to multiple scattering by inhomogeneities in the medium. In optics, additional incoherencies appear due to the loss of phase information in the measurements. Imaging with such incoherent signals is challenging. In this talk, through the examples of passive array imaging in random waveguides and so-called "transient opto-elastography," I will discuss how to model such noisy signals and how to "sift" them, namely using correlation-based techniques, to extract useful information for imaging purposes.
This talk is based mainly on joint works with Habib Ammari and Josselin Garnier.

January 27 (Wednesday). Special joint Statistics/Stochastics and Applied Math Seminar.
Note Time 3pm and Place LCB 219.

Speaker: Giovanni Motta, Department of Statistics, Columbia University
Title: Local polynomials estimation of time-varying matrices for multivariate locally stationary processes
Abstract: In this paper we propose two novel non-parametric approaches to estimate, respectively, the time-varying covariance and the time-varying correlation matrix of a multivariate locally stationary process. They both improve our previous estimators of the time-varying covariance and the time-varying correlation matrices. Our previous approach to estimate a time-varying covariance matrix is based on a Kernel-type smoother with the same bandwidth for all the entries of the matrix. This estimator is positive definite by construction. However, the use of one common bandwidth for all the entries of the covariance matrix appears to be restrictive, as these curves are in general characterized by different degrees of smoothness. A possible approach is to use a kernel-type estimator based on different smoothing parameters. However, this estimator is not in general positive definite, unless some severe restrictions are imposed on the bandwidths. Our novel estimator adapts to the different degrees of smoothness of the entries of the covariance matrix and, at the same time, is positive definite by construction. Our previous approach to estimate a time-varying correlation uses the same bandwidth for both numerator (covariance) and denominator (variances). This approach guarantees that the resulting estimator is a well defined correlation matrix. However, the use of one common bandwidth for both numerator and denominator does not allow the covariance and the variances to have different degrees of smoothness. On the other hand, a kernel-type estimator based on different smoothing parameters for numerator and denominator might deliver unbounded correlations and the resulting estimated correlation matrix is not necessarily positive semi-definite. Moreover, the estimated bandwidths that are optimal for estimating the covariance and the variances are not necessarily optimal for estimating the ratio. The estimator we propose in this paper is a local average of the signs of the cross-products: it does not require distinguishing between numerator and denominator and, at the same time, is positive definite by construction.

January 29. Commutative Algebra Seminar, 3:10pm - 4:00pm, LCB 222
Speaker: Graeme Milton, Department of Mathematics, The University of Utah
Title: Superfunctions and the algebra of subspace collections
Abstract: A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of subspace collections. Operations on subspace collections are found to correspond to various operations on rational functions, such as addition, multiplication and substitution. It is established that every rational matrix valued function which is homogeneous of degree 1 can be generated from an appropriate, but not necessarily unique, subspace collection: the mapping from subspace collections to rational functions is onto, but not one to one. For some applications superfunctions may be more important than functions, as they incorporate more information about the physical problem, yet can be manipulated in much the same way as functions. Orthogonal subspace collections occur in many physical problems, but we'll show an example of the use of non-orthogonal ones, which when substituted in orthogonal ones greatly accelerate the convergence of Fast Fourier transform methods.

February 5. Special Applied Math Seminar. Note Time 3:55pm and Place LCB 225.
Speaker: Francois Monard, Department of Mathematics, University of Michigan
Title: Geodesic X-ray transforms on surfaces and tensor tomography
Abstract: In this talk, we will study what can be reconstructed about a function (or a tensor) on a surface, from knowledge of its integrals along a given family of geodesic curves, that is, its X-ray transform. The "straight-line" version of this question was first answered by J. Radon in 1917 and its solution forms the theoretical backbone of Computerized Tomography since the 1960's. In practice, variations of the refractive index do occur and bend photon paths in optics-based imaging, and this requires that the problem be studied for general curves.
In a geometric setting beyond that of the Radon transform, examples of situations impacting the qualitative invertibility and stability of these transforms are (i) the case of "simple" surfaces, (ii) the presence of conjugate points/caustics, and (iii) the presence of trapped geodesics. We will discuss positive and negative theoretical results occurring when one considers each of the scenarios above, with numerical illustrations throughout.

February 8
Speaker: Gunilla Kreiss, Department of Information Technology, Uppsala University
Title: Super convergence of finite difference solutions of PDEs
Abstract: A stable finite difference method applied to a problem with a smooth solution allows for a straightforward estimate of convergence rate based on the convergence rate of the local truncation error. In many high order cases the local truncation error at a few points near boundaries is significantly larger than at interior points. The straight forward estimat will predict a rate determined by the slowest converging local truncation error. Convergence in numerical computations is often faster than this prediction. We explore ways to improve our understanding of such super-convergence. We will in particular consider the second order wave equation.

February 10 (Wednesday). Special joint Statistics/Stochastics and Applied Math Seminar.
Note Time 4pm and Place LCB 219.

Speaker: Lizhen Lin, Department of Statistics and Data Sciences, University of Texas at Austin
Title: Nonparametric Statistical Inference of non-Euclidean data
Abstract: Over the last few decades data represented in various non-conventional forms have become increasingly more prevalent. Typical examples include diffusion matrices in diffusion tensor imaging (DTI) of neuroimaging, and various digital image data arising in biology, medicine, machine vision and other fields of science and engineering. One may also encounter data that are stored in the forms of subspaces, orthonormal frames, surfaces, and networks. Statistical analysis of such data requires rigorous formulation and characterization of the underlying space, and inference is heavily dependent on the geometry of the space. For a majority of the cases considered, the underlying spaces fall into the general category of manifolds. This talk focuses on nonparametric inference on manifolds and other non-Euclidean spaces. Appropriate notion of means (e.g., Frechet means) and variations are defined, and inference is based on the asymptotic distributions of their sample counterparts. In particular, we present omnibus central limit theorems for Frechet means for inference, from which many of the existing CLTs follows immediately. Applications are provided to some stratified spaces of recent interest, and to the space of symmetric positive definite matrices arising in diffusion tensor imaging. In addition to inferring i.i.d data, we also consider nonparametric regression problems where predictors or responses lying on manifolds.

February 12. Note Time is 3:55pm and Place LCB 225.
Speaker: Mikyoung Lim, Department of Mathematical Sciences, KAIST
Title: Spectrum of the Neumann-Poincaré operator and plasmon resonance
Abstract: In this talk we consider spectral properties of the Neumann-Poincaré(NP) operator on planar domains with corners. Recently there is rapidly growing interest in the spectral properties of the NP operator due to its relation to plasmonics and cloaking by anomalous localized resonance: Plasmon resonance occurs at eigenvalues of the NP operator and anomalous localized resonance occurs at the accumulation point of eigenvalues, respectively. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We analyse the spectrum of intersecting disks which has two corners and show computational experiments which provide the spectral properties of domains with corners. For the computations we use a modification of the Nyström method which makes it possible to construct high-order convergent discretizations of the NP operator on domains with corners.

February 19. Special joint Statistics/Stochastics and Applied Math Seminar.
Note Time 3pm and Place LCB 219.

Speaker: Shuyang Bai, Department of Mathematics and Statistics, Boston University
Title: Self-normalized resampling of long-memory time series
Abstract: Statistical inference of long-memory time series faces two challenges due to the special behavior of the sample sum 1) a non-standard fluctuation rate which is typically unknown 2) a family of non-Gaussian scaling limits arise and it is difficult to statistically determine which one. We introduce a procedure which combines two strategies: self-normalization and resampling. Such combination successfully bypasses the aforementioned challenges. To establish the validity of the procedure, a key result involving bounding the maximal correlation between two blocks of a long-memory sequence is derived. Furthermore, the same procedure also works under short memory or heavy tails. It thus provides a unified treatment for various different situations.

February 22. Special joint Statistics/Stochastics and Applied Math Seminar. Note Room is LCB 219.
Speaker: Veniamin Morgenshtern, Department of Statistics, Stanford University
Title: Super-Resolution of Positive Sources
Abstract: The resolution of all microscopes is limited by diffraction. The observed data is a convolution of the emitted signal with a low-pass kernel, the point-spread function (PSF) of the microscope. The frequency cut-off of the PSF is inversely proportional to the wavelength of light. Hence, the features of the object that are smaller than the wavelength of light are difficult to observe. In single-molecule microscopy the emitted signal is a collection of point sources, produced by blinking molecules. The goal is to recover the location of these sources with precision that is much higher than the wavelength of light. This leads to the problem of super-resolution of positive sources in the presence of noise. I will show that the problem can be solved by using convex optimization in a stable fashion. The stability of reconstruction depends on Rayleigh-regularity of the signal support, i.e., on how many point sources can occur within an interval of one wavelength. The stability estimate is complemented by a converse result: the performance of the convex algorithm is nearly optimal. I will also give a brief summary on the ongoing project, developed in collaboration with the group of Prof. W.E. Moerner, where we use these theoretical ideas to improve data processing in modern microscopes.

February 26. Joint Stochastics and Applied Math Seminar. Note Time is 3-4 pm. Room LCB 219
Speaker: Harish Bhat, Department of Applied Mathematics, University of California, Merced
Title: Density tracking by quadrature for stochastic differential equations
Abstract: We consider the problem of computing the probability density function (pdf) for a class of stochastic differential equations. In a nutshell, our method consists of using quadrature to solve, at each time step, the Chapman-Kolmogorov equation associated with a time-discretization of the stochastic differential equation. After motivating the method and comparing it with existing techniques, we will discuss convergence results. Our main result is that our pdf converges in L^1 to both the exact pdf of the Markov chain (with exponential convergence rate), and to the exact pdf of the stochastic differential equation (with linear convergence rate). We carry out numerical tests to show that the empirical performance of the method complies with theoretical convergence results. Finally, we discuss how the method can be used to construct Metropolis algorithms for posterior inference of parameters in stochastic differential equation models.

February 29.
Speaker: Davit Harutyunyan, Department of Mathematics, University of Utah
Title: Recent progress in the shell buckling theory
Abstract: In has been known that the rigidity of a shell, also under compression, is closely related to the optimal Korn's constant in the nonlinear first Korn's inequality (geometric rigidity estimate) for $W^{1,2}$ fields under the appropriate Dirichlet type boundary conditions arising from the nature of the compression. In their recent work Frisecke, James and Mueller (2002, 2006) derived a geometric rigidity estimate for plates, which gave rise to a derivation of a hierarchy of plate theories for different scaling regimes of the elastic energy depending on the thickness of the plate. FJM type theories have been derived by Gamma-convergence and rely on compactness arguments and of course the underlying nonlinear Korn's inequality. While the rigidity of plates has been understood almost completely, the rigidity, in particular the buckling of shells is almost completely open. This is due to the luck of rigidity estimates and compactness as understood by Grabovsky and Harutyunyan (2014) for cylindrical shells. In the case of shells, when there is enough rigidity, is has been understood that actually the linear first Korn's inequality can replace the nonlinear one, Grabovsky, Truskinovsky (2007). The important mathematical question is: What makes the shells more rigid than plates and how can one compare the rigidity of two different shells? In this talk we give the answer to that question by classifying shells according to the Gaussian curvature. We derive sharp first Korn's inequalities for shells of zero, positive and negative Gaussian curvature. It turns out, that for zero Gaussian curvature the amount of rigidity is $h^{1.5}$, for negative curvature it is $h^{4/3}$ and for positive curvature it is h, i.e. the positive Gaussian curvature shell is the must rigid one. Here h is the shell thickness. All three exponents are completely new in Korn's inequalities.
This is partially joint work with Yury Grabovsky.

March 21
Speaker: Dima Pesin, Department of Physics, The University of Utah
Title: Berry phase effects in electronic transport
Abstract: I will briefly review basics of Berry phase effects in electronic transport, focusing on one- and two-dimensional systems. I will then generalize to the three-dimensional case, and introduce what has become known as the Weyl semimetal: the 3D analog of graphene. I will review their transport and optical properties, focusing on non-local voltages generated by the so-called chiral anomaly, as well as effects of non-locality in their electrodynamics.

March 28
Speaker: Lajos Horvath, Department of Mathematics, The University of Utah
Title: Statistical inference based on curves
Abstract: Functional data analysis is concerned with observations that are random functions defined on a set ${\mathcal T}$. For example, $X(t)$ could denote the observation of temperature (pollution level) at a given location at time $t$. Stock prices, exchange rates are also modeled as continues curves in economics and finance. Many of such continuous time phenomena are studied, although they are measured only at discrete time points. We provide examples when the observations can be modeled as curves. We discuss how inference on the mean of random curves is performed. There are two popular techniques for this task: principal component analysis and fully functional approach. The principal component analysis transforms the data into a finite dimensional vector (dimension reduction) while the fully functional approach uses the whole sample paths directly. We compare the advantages and drawbacks of both methods.

March 30. Note Time is 4-5 pm on Wednesday. Room LCB 219
Speaker: Marc Briane, Institut de Recherche Mathematique de Rennes, France
Title: Loss of ellipticity by homogenization in 2D elasticity
Abstract: This work in collaboration with G. Francfort deals with the loss of ellipticity of two-dimensional quadratic elastic functionals by homogenization. It was shown by Geymonat, Muller, Triantafyllidis (1993) that, in the setting of linearized elasticity, a $\Gamma$-convergence result holds for highly oscillating sequences of elastic energies whose functional coercivity constant in $R^N$ is zero, while the corresponding coercivity constant on the torus remains positive. We illustrate the range of applicability of that result by finding sufficient conditions for such a situation to occur. We thereby justify the degenerate laminate construction of Gutierrez (1999). We also demonstrate that the predicted loss of strict strong ellipticity resulting from the Gutierrez construction is unique within a "laminate-like" class of microstructures.

April 11 (Student Talk)
Speaker: Todd Harry Reeb, Department of Mathematics, The University of Utah
Title: The consistency of Dirichlet graph partitions
Abstract: The Dirichlet partition problem is to partition a domain $\Omega$ into $k$ connected subdomains so that the sum of their first Dirichlet-Laplacian eigenvalues is minimized; a minimizer of this objective is called a Dirichlet partition of $\Omega$. While the original problem has its origins in shape optimization and the theory of Bose-Einstein condensates, it has more recently inspired a graph partitioning algorithm for use in image processing and data analysis. In this talk, we'll discuss both the continuum and discrete problems, and we'll give a consistency result for the discrete problem on random geometric graphs approximating $\Omega$, which states the convergence of graph partitions to an appropriate continuum partition.
This is joint work with Braxton Osting.

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