Jun Kitagawa : A convergent Newton algorithm for semi-discrete optimal transport
- Applied Math and Analysis ( 229 Views )The optimal transport (Monge-Kantorovich) problem is a variational problem involving transportation of mass subject to minimizing some kind of energy, and it arises in connection with many parts of math, both pure and applied. In this talk, I will discuss a numerical algorithm to approximate solutions in the semi-discrete case. We propose a damped Newton algorithm which exploits the structure of the associated dual problem, and using geometric implications of the regularity theory of Monge-Amp{\`e}re equations, we are able to rigorously prove global linear convergence and local superlinear convergence of the algorithm. This talk is based on joint work with Quentin M{\’e}rigot and Boris Thibert.
Laurent Demanet : Time upscaling of wave equations via discrete symbol calculus
- Applied Math and Analysis ( 144 Views )The complexity of solving the time-dependent wave equation via traditional methods scales faster than linearly in the complexity of the initial data. This behavior is mostly due to the necessity of timestepping at the CFL level, and is hampering the resolution of large-scale inverse scattering problems such as reflection seismology, where massive datasets need to be processed. In this talk I will report on some algorithmic progress toward time upscaling of the wave equation using discrete symbol calculus for pseudodifferential and Fourier integral operators. Joint work with Lexing Ying from UT Austin.
Linfeng Zhang : Neural network models and concurrent learning schemes for multi-scale molecular modelling
- Applied Math and Analysis ( 220 Views )We will discuss two issues in the context of applying deep learning methods to multi-scale molecular modelling: 1) how to construct symmetry-preserving neural network models for scalar and tensorial quantities; 2) how to efficiently explore the relevant configuration space and generate a minimal set of training data. We show that by properly addressing these two issues, one can systematically develop deep learning-based models for electronic properties and interatomic and coarse-grained potentials, which greatly boost the ability of ab-initio molecular dynamics; one can also develop enhanced sampling techniques that are capable of using tens or even hundreds of collective variables to drive phase transition and accelerate structure search
Roman Shvydkoy : Topological models of emergent dynamics
- Applied Math and Analysis ( 105 Views )In this talk we will introduce a new class of flocking models that feature emergence of global alignment via only local communication. Such models have been sought for since the introduction of Cucker-Smale dynamics which showed global unconditional alignment for models with substantially strong non-local interaction kernels. We introduce a set of new structural components into the communication protocol, including singular kernel, and topological adaptive diffusion, that enhance alignment mechanisms with purely local interactions. We highlight some challenges that arise in the problem of global well-posendess and stability of flocks.
Mark Iwen : Signal Recovery via Discrete Measurement Matrices
- Applied Math and Analysis ( 132 Views )We will discuss a class of binary measurement matrices having a simple discrete incoherence property. These matrices can be shown to have both useful analytic (i.e., restricted isometry and l1-approximation properties) and combinatorial (i.e., group testing and expander graph related) structure which allows them to be utilized for sparse signal approximation in the spirit of compressive sensing. In addition, their structure allows the actual signal recovery process to be carried out by highly efficient algorithms once measurements have been taken. One application of these matrices and their related recovery algorithms is their application to the development of sublinear-time Fourier methods capable of accurately approximating periodic functions using far fewer samples and run time than required by standard Fourier transform techniques.
Alun Lloyd : Drug Resistance in Acute Viral Infections
- Applied Math and Analysis ( 149 Views )A wide range of viral infections, such as HIV or influenza, can now be treated using antiviral drugs. Since viruses can evolve rapidly, the emergence and spread of drug resistant virus strains is a major concern. We shall describe within and between host models that can help indicate settings in which resistance is more or less likely to be problematic. In particular, we shall discuss the potential for the emergence of resistance in the context of human rhinovirus infection, an acute infection that is responsible for a large fraction of 'common cold' cases.
Cynthia Rudin : 1) Regulating Greed Over Time: An Important Lesson For Practical Recommender Systems and 2) Prediction Uncertainty and Optimal Experimental Design for Learning Dynamical Systems
- Applied Math and Analysis ( 101 Views )I will present work from these two papers:
1) Regulating Greed Over Time. Stefano Traca and Cynthia Rudin. 2015
Finalist for 2015 IBM Service Science Best Student Paper Award
2) Prediction Uncertainty and Optimal Experimental Design for Learning Dynamical Systems. Chaos, 2016.
Benjamin Letham, Portia A. Letham, Cynthia Rudin, and Edward Browne.
There is an important aspect of practical recommender systems that we noticed while competing in the ICML Exploration-Exploitation 3 data mining competition. The goal of the competition was to build a better recommender system for Yahoo!'s Front Page, which provides personalized new article recommendations. The main strategy we used was to carefully control the balance between exploiting good articles and exploring new ones in the multi-armed bandit setting. This strategy was based on our observation that there were clear trends over time in the click-through-rates of the articles. At certain times, we should explore new articles more often, and at certain times, we should reduce exploration and just show the best articles available. This led to dramatic performance improvements.
As it turns out, the observation we made in the Yahoo! data is in fact pervasive in settings where recommender systems are currently used. This observation is simply that certain times are more important than others for correct recommendations to be made. This affects the way exploration and exploitation (greed) should change in our algorithms over time. We thus formalize a setting where regulating greed over time can be provably beneficial. This is captured through regret bounds and leads to principled algorithms. The end result is a framework for bandit-style recommender systems in which certain times are more important than others for making a correct decision.
If time permits I will discuss work on measuring uncertainty in parameter estimation for dynamical systems. I will present "prediction deviation," a new metric of uncertainty that determines the extent to which observed data have constrained the model's predictions. This is accomplished by solving an optimization problem that searches for a pair of models that each provide a good fit for the observed data, yet have maximally different predictions. We develop a method for estimating a priori the impact that additional experiments would have on the prediction deviation, allowing the experimenter to design a set of experiments that would most reduce uncertainty.
Matthew Simpson : The mathematics of Hirschsprungs Disease
- Applied Math and Analysis ( 139 Views )Hirschsprung's Disease is a relatively common human congenital defect where the nervous system supporting our gut (the enteric nervous system) fails to develop properly. During embryonic development, the enteric nervous system forms as a result of neural crest cell invasion. Neural crest cells migrate from the hindbrain to the anal end of the gastrointestinal tract. This is one of the longest known cell migration paths, both spatially and temporally, occurring during vertebrate embryogenesis. Neural crest cell invasion is complicated by the simultaneous expansion of underlying tissues and the influence of several growth factors. This presentation outlines a combined experimental and mathematical approach used to investigate and deduce the mechanisms responsible for successful neural crest cell colonization. This approach enables previously hypothesized mechanisms for neural crest cell colonization of the gut tissues to be refuted and refined. The current experimental and mathematical results are focused on population-scale approaches. Further experimental details of cell-scale properties thought to play an important role will be presented. Preliminary discrete modelling results aiming to realize the cell-scale phenomena will also be discussed and outlined as future work.
Peter Markowich : A PDE System Modeling Biological Network Formation
- Applied Math and Analysis ( 93 Views )Transportation networks are ubiquitous as they are possibly the most important building blocks of nature. They cover microscopic and macroscopic length scales and evolve on fast to slow times scales. Examples are networks of blood vessels in mammals, genetic regulatory networks and signaling pathways in biological cells, neural networks in mammalian brains, venation networks in plant leafs and fracture networks in rocks. We present and analyze a PDE (Continuum) framework to model transportation networks in nature, consisting of a reaction-diffusion gradient-flow system for the network conductivity constrained by an elliptic equation for the transported commodity (fluid).
Tyler Whitehouse : Consistent signal reconstruction and the geometry of some random polytopes
- Applied Math and Analysis ( 94 Views )Consistent reconstruction is a linear programming technique for reconstructing a signal $x\in\RR^d$ from a set of noisy or quantized linear measurements. In the setting of random frames combined with noisy measurements, we prove new mean squared error (MSE) bounds for consistent reconstruction. In particular, we prove that the MSE for consistent reconstruction is of the optimal order $1/N^2$ where $N$ is the number of measurements, and we prove bounds on the associated dimension dependent constant. For comparison, in the important case of unit-norm tight frames with linear reconstruction (instead of consistent reconstruction) the mean squared error only satisfies a weaker bound of order $1/N$. Our results require a mathematical analysis of random polytopes generated by affine hyperplanes and of associated coverage processes on the sphere. This is joint work with Alex Powell.
Jill Pipher : Geometric discrepancy theory: directional discrepancy in 2-D
- Applied Math and Analysis ( 90 Views )Discrepancy theory originated with some apparently simple questions about
sequences of numbers. The discrepancy of an infinite sequence is a
quantitative measure of how far it is from being uniformly distributed.
Precisely, an infinite sequence { a1,a2, ...} is said to be uniformly
distributed in [0, 1] if
lim_{n\to\infty} (1/n|{a1, a2,...an} intersect [s,t]|) = t-s.
If a sequence {ak} is uniformly distributed, then it is also the case
that for all (Riemann) integrable functions f on [0, 1],
lim_{n\to\infty} (1/n\sum_{k=1}^n f(ak))=\int_0^1 f(x)dx.
Thus, uniformly distributed sequences provide good numerical schemes
for approximating integrals. For example, if alpha is any irrational
number in [0, 1], then the fractional part {alphak}:=ak is uniformly
distributed. Classical Fourier analysis enters here, in the form of
Weyl's criterion.
The discrepancy of a sequence with respect to its first n entries is
D({ak},n) := sup_{s
See PDF.
Qin Li : Low rankness in forward and inverse kinetic theory
- Applied Math and Analysis ( 112 Views )Multi-scale kinetic equations can be compressed: in certain regimes, the Boltzmann equation is asymptotically equivalent to the Euler equations, and the radiative transfer equation is asymptotically equivalent to the diffusion equation. A lot of detailed information is lost when the system passes to the limit. In linear algebra, it is equivalent to being of low rank. I will discuss such transition and how it affects the computation: mainly, in the forward regime, inserting low-rankness could greatly advances the computation, while in the inverse regime, the system being of low rank typically makes the problems significantly harder.
Xiuqing Chen : Global weak solution for kinetic models of active swimming and passive suspensions
- Applied Math and Analysis ( 95 Views )We investigate two kinetic models for active suspensions of rod-like and ellipsoidal particles, and passive suspensions of dumbbell beads dimmers, which couple a Fokker-Planck equation to the incompressible Navier-Stokes or Stokes equation. By applying cut-off techniques in the approximate problems and using compactness argument, we prove the existence of the global weak solutions with finite (relative) entropy for the two and three dimensional models. For the second model, we establish a new compact embedding theorem of weighted spaces which is the key in the compactness argument. (Joint work with Jian-Guo Liu)
Lin Lin : Elliptic preconditioner for accelerating the self consistent field iteration of Kohn-Sham density functional theory
- Applied Math and Analysis ( 126 Views )Kohn-Sham density functional theory (KSDFT) is the most widely used electronic structure theory for molecules and condensed matter systems. Although KSDFT is often stated as a nonlinear eigenvalue problem, an alternative formulation of the problem, which is more convenient for understanding the convergence of numerical algorithms for solving this type of problem, is based on a nonlinear map known as the Kohn-Sham map. The solution to the KSDFT problem is a fixed point of this nonlinear map. The simplest way to solve the KSDFT problem is to apply a fixed point iteration to the nonlinear equation defined by the Kohn-Sham map. This is commonly known as the self-consistent field (SCF) iteration in the condensed matter physics and chemistry communities. However, this simple approach often fails to converge. The difficulty of reaching convergence can be seen from the analysis of the Jacobian matrix of the Kohn-Sham map, which we will present in this talk. The Jacobian matrix is directly related to the dielectric matrix or the linear response operator in the condense matter community. We will show the different behaviors of insulating and metallic systems in terms of the spectral property of the Jacobian matrix. A particularly difficult case for SCF iteration is systems with mixed insulating and metallic nature, such as metal padded with vacuum, or metallic slabs. We discuss how to use these properties to approximate the Jacobian matrix and to develop effective preconditioners to accelerate the convergence of the SCF iteration. In particular, we introduce a new technique called elliptic preconditioner, which unifies the treatment of large scale metallic and insulating systems at low temperature. Numerical results show that the elliptic preconditioner can effectively accelerate the SCF convergence of metallic systems, insulating systems, and systems of mixed metallic and insulating nature. (Joint work with Chao Yang)
Laura Miller : Scaling effects in heart development: Changes in bulk flow patterns and the resulting forces
- Applied Math and Analysis ( 92 Views )When the heart tube first forms, the Reynolds number describing intracardial flow is only about 0.02. During development, the Reynolds number increases to roughly 1000. The heart continues to beat and drive the fluid during its entire development, despite significant changes in fluid dynamics. Early in development, the atrium and ventricle bulge out from the heart tube, and valves begin to form through the expansion of the endocardial cushions. As a result of changes in geometry, conduction velocities, and material properties of the heart wall, the fluid dynamics and resulting spatial patterns of shear stress and transmural pressure change dramatically. Recent work suggests that these transitions are significant because fluid forces acting on the cardiac walls, as well as the activity of myocardial cells which drive the flow, are necessary for correct chamber and valve morphogenesis.
In this presentation, computational fluid dynamics was used to explore how spatial distributions of the normal forces and shear stresses acting on the heart wall change as the endocardial cushions grow, as the Reynolds number increases, and as the cardiac wall increases in stiffness. The immersed boundary method was used to simulate the fluid-structure interaction between the cardiac wall and the blood in a simplified model of a two-dimensional heart. Numerical results are validated against simplified physical models. We find that the presence of chamber vortices is highly dependent upon cardiac cushion height and Reynolds number. Increasing cushion height also drastically increases the shear stress acting on the cushions and the normal forces acting on the chamber walls.
Franca Hoffmann : Gradient Flows: From PDE to Data Analysis.
- Applied Math and Analysis ( 175 Views )Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in various areas of mathematics ranging from PDE theory to data science. In this talk, we focus on two different directions: model-driven approaches and data-driven approaches. In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs. We present recent results on properties of these equilibria and long-time asymptotics of solutions in the setting where attractive and repulsive forces are in competition. In the second part, we use and develop gradient flow theory to design novel tools for data analysis. We draw a connection between gradient flows and Ensemble Kalman methods for parameter estimation. We introduce the Ensemble Kalman Sampler - a derivative-free methodology for model calibration and uncertainty quantification in expensive black-box models. The interacting particle dynamics underlying our algorithm can be approximated by a novel gradient flow structure in a modified Wasserstein metric which reflects particle correlations. The geometry of this modified Wasserstein metric is of independent theoretical interest.
Tam Do : Vorticity Growth for Axisymmetric Flows without Swirl
- Applied Math and Analysis ( 95 Views )For 2D Euler flows, it is known that the L^\infty norm of the gradient of vorticity can grow with at most double exponential rate in time. In recent years, this bound has been proven to be sharp by Kiselev and Sverak on the unit disc. We will examine the possibility of growth in the 3D axisymmetric setting for flows without swirl component.
George Hagedorn : Some Theory and Numerics for Semiclassical Quantum Mechanics
- Applied Math and Analysis ( 146 Views )We begin with an introduction to time-dependent quantum mechanics and the role of Planck's constant. We then describe some mathematical results about solutions to the Schr\"odinger equation for small values of the Planck constant. Finally, we discuss two new numerical techniques for semiclassical quantum dynamics, including one that is a work in progress.
Gadi Fibich : Aggregate Diffusion Dynamics in Agent-Based Models with a Spatial Structure
- Applied Math and Analysis ( 115 Views )The diffusion or adoption of new products (such as fax machines, skype, facebook, Ipad, etc.) is one of the key problems in Marketing research. In recent years, this problem was often studied numerically, using agent-based models (ABMs). In this talk I will focus on analysis of the aggregate diffusion dynamics in ABMs with a spatial structure. In one-dimensional ABMs, the aggregate diffusion dynamics can be explicitly calculated, without using the mean-field approximation. In multidimensional ABMs, we introduce a clusters-dynamics approach, and use it to derive an analytic approximation of the aggregate diffusion dynamics. The clusters-dynamics approximation shows that the aggregate diffusion dynamics does not depend on the average distance between individuals, but rather on the expansion rate of clusters of adopters. Therefore, the grid dimension has a large effect on the aggregate adoption dynamics, but a small-world structure and heterogeneity among individuals have only a minor effect. Our results suggest that the one-dimensional model and the fully-connected Bass model provide a lower bound and an upper bound, respectively, for the aggregate diffusion dynamics in agent-based models with "any" spatial structure. This is joint work with Ro'i Gibori and Eitan Muller
Mark Levi : A connection between tire tracks, the stationary Schr\odingers equation and forced vibrations
- Applied Math and Analysis ( 92 Views )I will describe a recently discovered equivalence between the first two objects mentioned in the title. The stationary Schr\"odinger's equation, a.k.a. Hills equation, is ubiquitous in mathematics, physics, engineering and chemistry. Just to mention one application, the main idea of the Paul trap (for which W. Paul earned the 1989 Nobel Prize in physics) amounts to a certain property of Hill's equation. I will show that Hill's equation is equivalent to a seemingly unrelated problem of tire tracks. There is, in addition, a yet another connection between the ``tire tracks problem and the high frequency forced vibrations which I will also outline briefly.
Nathan Glatt-Holtz : Invisicid Limits for the Stochastic Navier Stokes Equations and Related Systems
- Applied Math and Analysis ( 93 Views )One of the original motivations for the development of stochastic partial differential equations traces it's origins to the study of turbulence. In particular, invariant measures provide a canonical mathematical object connecting the basic equations of fluid dynamics to the statistical properties of turbulent flows. In this talk we discuss some recent results concerning inviscid limits in this class of measures for the stochastic Navier-Stokes equations and other related systems arising in geophysical and numerical settings. This is joint work with Peter Constantin, Vladimir Sverak and Vlad Vicol.
Jason Metcalf : Strichartz estimates on Schwarzschild black hole backgrounds
- Applied Math and Analysis ( 131 Views )In this talk, we will present some recent work on dispersive estimates for wave equations on Schwarzschild black hole backgrounds. We in particular will discuss Strichartz estimates and localized energy estimate. This is from a joint work with Jeremy Marzuola, Daniel Tataru, and Mihai Tohaneanu.
Michael Gratton : Transient and self-similar dynamics in thin film coarsening
- Applied Math and Analysis ( 144 Views )Coarsening is the phenomenon where many objects (water drops, molecular islands, particles in a freezing liquid) becoming smaller in number but larger in size in an orderly way. This talk will examine modeling one such system, nanoscopic liquid drops, through three models: a PDE for the fluid, a coarsening dynamical system for the drops, and an LSW-type ensemble model for the distribution of drops. We will find self-similar solutions for the drop population valid for intermediate times and discuss transient effects that can delay the self-similar scaling.
Mark Stern : Monotonicity and Betti Number Bounds
- Applied Math and Analysis ( 176 Views )In this talk I will discuss the application of techniques initially developed to study singularities of Yang Mill's fields and harmonic maps to obtain Betti number bounds, especially for negatively curved manifolds.
Shaoming Guo : Maximal operators and Hilbert transforms along variable curve
- Applied Math and Analysis ( 130 Views )I will present several results on the boundedness of maximal operators and Hilbert transforms along variable curves and surfaces, in dimension two or higher. Connections to the circular maximal operator, and the polynomial Carleson operator will also be discussed.
Maria Cameron : Analysis of the Lennard-Jones-38 stochastic network
- Applied Math and Analysis ( 112 Views )The problem of finding transition paths in the Lennard-Jones cluster of 38 atoms became a benchmark problem in chemical physics due to its beauty and complexity. The two deepest potential minima, the face-centered cubic truncated octahedron and an icosahedral structure with 5-fold rotational symmetry, are far away from each other in the configuration space, which makes problem of finding transition paths between them difficult. D. Wales's group created a network of minima and transition states associated with this cluster. I will present two approaches to analyze this network. The first one, a zero-temperature asymptotic approach, is based on the Large Deviation Theory and Freidlin's cycles. I will show that in the gradient case the construction of the hierarchy of cycles can be simplified dramatically and present a computational algorithm for building a hierarchy of only those Freidlin's cycles associated with the transition process between two given local equilibria. The second approach is the Discrete Transition Path Theory, a finite temperature tool. This approach allows us to establish the range of validity of the zero-temperature asymptotic and describe the transition process at still low but high enough temperatures where the zero-temperature asymptotic approach is no longer valid.