Applied Mathematics Seminar, Spring 2018

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

• 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 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.

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.

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