Department of Mathematics
Applied Mathematics Seminar, Spring 2018

Mondays 4:00 PM - 5:00 PM, LCB 219




Special Joint Statistics/Stochastics and Applied Math Seminar January 12. LCB 219 at 4pm.
Speaker: Daniel Sanz-Alonso, Department of Applied Mathematics, Brown University
Title: New Perspectives on Importance Sampling
Abstract: Importance sampling is a building block of many algorithms in computational statistics, perhaps most notably particle filters. It is the importance sampling step that often limits the accuracy of these algorithms. In this talk I will introduce a new way of understanding importance sampling based on information theory. I will argue that the fundamental problem facing algorithms based on importance sampling can be understood in terms of the distance between certain measures. The results give new understanding on the potential use of importance sampling and particle filters in high (possibly infinite) dimensional spaces.

Special Joint Statistics/Stochastics and Applied Math Seminar January 17. LCB 219 at 4pm.
Speaker: Kevin Moon, Genetics Department and Applied Math Program, Yale University
Title: Nonparametric Estimation of Distributional Functionals in Machine Learning
Abstract: Distributional functionals are integrals of functionals of probability densities and include functionals such as information divergence, mutual information, and entropy. Distributional functionals have many applications in the fields of information theory, statistics, signal processing, and machine learning. Many existing nonparametric distributional functional estimators have either unknown convergence rates or are difficult to implement. In this talk, I present multiple applications of distributional functional estimation, focusing on the problems of dimensionality reduction, extending machine learning tasks to distributional features, and estimating the optimal probability of error (the Bayes error) of a classification problem. I then present a simple, computationally tractable nonparametric estimator of a wide class of distributional functionals that achieves parametric convergence rates under certain smoothness conditions. The asymptotic distribution of the estimator is also derived.

Special Joint Statistics/Stochastics and Applied Math Seminar January 19. LCB 219 at 3pm.
Speaker: Harish Bhat, Department of Mathematics, University of California, Merced
Title: Statistical Estimation and Inference for Stochastic Differential Equations
Abstract: We consider the problem of estimating parameters in stochastic differential equation (SDE) models from discrete-time observations (time series). A key bottleneck is computation of the log likelihood. I will describe a method, density tracking by quadrature (DTQ), to compute densities and likelihoods in low-dimensional systems. DTQ consists of using quadrature to solve, at each time step, the Chapman-Kolmogorov equation associated with a time-discretization of the SDE. After motivating DTQ, I will discuss theoretical and empirical convergence results. These results include convergence in L^1 to both the exact pdf of the Markov chain (with exponential convergence rate), and to the exact pdf of the SDE (with linear convergence rate). I will then propose two methods for nonparametric estimation in the high-dimensional SDE setting. The first is an extension of equation discovery techniques used in the deterministic context, while the second is an expectation maximization approach.

February 5
Speaker: Masashi Mizuno, Department of Mathematics, Nihon University, Japan
Title: Evolution of grain boundaries with lattice orientations and triple junctions effect
Abstract: Grain boundaries and their evolution are strongly related to many properties of materials, like metals and alloys. Mathematically, for planer network case, the motion of the grain boundaries is often described by the geometric curve evolution equations. In these equations, the dynamics of the grain boundaries are assumed to depend only on the shape of the grains. However, grain lattice orientation structure (thus misorientations, which are differences between lattice orientations of neighboring grains), as well as the mobility of the triple junctions of the grains also play an important role on the evolution of the grain boundaries. In this talk, I will introduce a new model of the grain boundary migration that takes into account the dynamics of misorientations, as well as the mobility of the triple junctions. Next, mathematical analysis of the model, in particular, asymptotic behavior of the solution will be presented.

This is a joint work with Yekaterina Epshteyn (The University of Utah) and Chun Liu (Illinois institute of Technology).

February 26
Speaker: Noel Walkington, Department of Mathematics, Carnegie Mellon University
Title: Numerical Approximation of Multiphase Flows in Porous Media
Abstract: This talk will review structural properties of the equations used to model geophysical flows which involve multiple components undergoing phase transitions. Simulations of these problems only model the gross properties of these flows since a precise description of the physical system is neither available nor computationally tractable. In this context mathematics provides an essential foundation to facilitate the integration of phenomenology and physical intuition to develop robust numerical schemes that inherit essential structural and physical properties of the underlying problem.

April 2
Speaker: Graeme Milton, Department of Mathematics, The University of Utah
Title: Exact relations for Green's functions in linear PDE and boundary field equalities: a generalization of conservation laws
Abstract: Many physical problems can be cast in a form where a constitutive equation $J(x)=L(x)E(x)+h(x)$ with a source term h(x) holds for all x in $R^d$ and relates fields E and J that satisfy appropriate differential constraints, symbolized by E in $\cal{E}$ and J in $\cal{J}$ where $\cal{E}$ and $\cal{J}$ are orthogonal spaces that span the space $\cal{H}$ of square-integrable fields in which h lies. Here we show that if the moduli L(x) are constrained to take values in certain nonlinear manifolds $\cal{M}$, and satisfy suitable coercivity and boundedness conditions, then the infinite body Green's function for the problem satisfies certain exact identities. A corollary of our theory is that it also provides the framework for establishing links between the Green's functions for different physical problems, sharing some commonality in their geometry. The analysis is based on the theory of exact relations for composites, but, unlike in the theory of composites, we make no assumptions about the length scales of variations in the moduli L(x). For bodies $\Omega$ of finite extent, such that $L(x)\in \cal{M}$ for x on $\Omega$, the exact relations for the infinite body Green's function imply that the Dirichlet-to-Neumann map (DtN-map) characterizing the response of the body also satisfies exact relations. These boundary field equalities generalize the notion of conservation laws: the field inside $\Omega$ satisfies certain constraints, that leave a wide choice in these fields, but which give identities satisfied by the boundary fields, and moreover provide constraints on the fields inside the body. A consequence is the following: if a matrix valued field Q(x) with divergence-free columns takes values within $\Omega$ in a set $\cal{B}$ (independent of x) that lies on a nonlinear manifold, we find conditions on the manifold, and on $\cal{B}$, that with appropriate conditions on the boundary fluxes $q(x)=n(x)\cdot Q(x)$ (where n(x) is the outwards normal to the boundary of $\Omega$) force Q(x) within $\Omega$ to take values in a subspace $\cal{D}$. This forces q(x) to take values in $n(x)\cdot\cal{D}$. We find there are additional divergence free fields inside $\Omega$ that in turn generate additional boundary field equalities. Consequently, there exist partial Null-Lagrangians, functionals F(w,$\nabla w$) of a vector potential w and its gradient, that act as null-Lagrangians when (w,$\nabla w$) is constrained for x in $\Omega$ to take values in certain sets $\cal{A}$, of appropriate non-linear manifolds, and when w satisfies appropriate boundary conditions. The extension to certain non-linear minimization problems is also sketched.
Joint work with Daniel Onofrei.

April 9
Speaker: Rodrigo Platte, School of Mathematical and Statistical Sciences, Arizona State University
Title: Second order approximation of the MRI signal for single shot parameter assessment
Abstract: Most current methods of Magnetic Resonance Imaging (MRI) reconstruction interpret raw signal values as samples of the Fourier transform of the object. Although this is computationally convenient, it neglects relaxation and off-resonance evolution in phase, both of which can occur to significant extent during a typical MRI signal. A more accurate model, known as Parameter Assessment by Recovery from Signal Encoding (PARSE), takes the time evolution of the signal into consideration. This model uses three parameters that depend on tissue properties: transverse magnetization, signal decay rate, and frequency offset from resonance. Two difficulties in recovering an image using this model are the low SNR for long acquisition times in single-shot MRI, and the nonlinear dependence of the signal on the decay rate and frequency offset. In this talk, we address the latter issue by using a second order approximation of the original PARSE model. The linearized model can be solved using convex optimization augmented with well-stablished regularization techniques such as total variation. The sensitivity of the parameters to noise and computational challenges associated with this approximation will be discussed.

April 16
Speaker: Matthew Yancey
Title: Positively Curved Graphs
Abstract: We all learned how to determine if a function is curved upward or downward in Calculus 1. Higher dimensional versions are standard fare in basic textbooks. What is more recent is the work of Lott and Villani to create an equivalent definition of curvature that depends only on a distance function, and thus generalizes to not require the space to be defined from a differentiable function. This definition is only for defining positive curvature, and does not apply to discrete spaces. Recent breakthroughs in modelling internet traffic, biological networks, and other real-world graphs have emerged by using discrete analogues of negatively curved geometric spaces, and those models have been used to understand and explain phenomenon like congestion, clustering, and bounded diameter. In this talk, we explore what a discrete analogue of Lott and Villani's work might look like and how positive curvature can be used to explain design and routing in large data centers. Since the author is a graph theorist who knew nothing about curved geometry prior to this project, the audience has no need for background.

April 17 (Tuesday). Note Time 4pm - 5pm. Room LCB 323.
Speaker: Pierre Seppecher, Universite De Toulon, France
Title: Rigorous homogenization results leading to strain gradient and generalized continua
Abstract: There exist a few rigorous result for the homogenization of elastic media which lead to generalized continua. Even if it has been proved that an homogenized energy can involve higher derivatives than the original heterogeneous one, the only examples from the litterature are limited to couple stress models. We study the homogenization of linear elastic structures which are cylinders based on thickened periodic graphs. We first reduce the problem to the homogenization of discrete lattices, i.e. periodic arrangements of nodes with elastic interactions. Taking into account the fact that flexural interactions are much weaker than the extensional ones, we are able to characterize the dependence of the homogenized energy with respect to the strain gradient and/or extra kinematical variables. Many classical exemples of generalized 1D, 2D or 3D continua can be obtained in this way.

April 23
Speaker: Katayun Barmak, Department of Applied Physics and Applied Mathematics with Materials Science and Engineering, Columbia University
Title: Grain Structure, Grain Boundary Character Distribution and Grain Growth in Thin Metallic Films
Abstract: The types and connectivity of grain boundaries strongly influence materials properties. In the macroscopic description of grain boundaries, a general grain boundary is characterized by five parameters: three parameters to specify the lattice misorientation between the adjoining crystals meeting at the boundary and two parameters to specify the inclination of the boundary plane normal. Grain boundary character distribution (GBCD) gives the relative area of boundaries with a given misorientation and a given boundary normal. Using electron back scatter diffraction in the scanning electron microscope, GBCDs of many materials with grain sizes in the micrometer range have been measured. More recently precession electron diffraction in the transmission electron microscope has allowed these measurements to be extended to nanocrystalline materials. In this talk, GBCD of three nanocrystalline metallic films, aluminum, copper and tungsten (Al, Cu and W), will be compared with their microcrystalline counterparts. Grain structure of the films in the as-deposited and annealed states will be presented and related to crystal structure and bonding. Experimental grain size distributions using image based and crystal orientation mapping based methods in Al and Cu will be compared with the distribution obtained in two-dimensional simulations of grain growth with isotropic boundary energy. The role of structure formation processes during film deposition in the observed difference between experiment and simulation will be noted.


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

Past lectures: Fall 2017, Spring 2017, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006, Spring 2006, Fall 2005, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001, Spring 2001, Fall 2000, Spring 2000, Fall 1999, Spring 1999, Fall 1998, Spring 1998, Winter 1998, Fall 1997, Spring 1997, Winter 1997, Fall 1996, Spring 1996, Winter 1996, Fall 1995.


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