CSCAMM Archives for Academic Year 2016


Metastable distributions of Markov chains with rare transitions and related problems that result in differential equations with nonstandard boundary conditions

When: Wed, August 31, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Leonid Koralov (Department of Mathematics, University of Maryland) - http://www.math.umd.edu/~koralov/
Abstract: We consider Markov chains with parameter-dependent transition rates. The asymptotic behavior of the Markov chains is established at various time scales related to value of the parameter. This result can be viewed as a generalization of the ergodic theorem to the case of parameter-dependent Markov chains. One of the interesting applications is to the study of randomly perturbed dynamical systems (i.e., diffusion processes with a small diffusion coefficient). In this case, each asymptotically stable equilibrium of the dynamical system can be associated with a state of a Markov chain. We describe the asymptotic behavior of a diffusion process with multiple trapping regions (with the vector field equal to zero outside the regions) in terms of a PDE with nonstandard boundary conditions. The talk is based on joint work with M. Freidlin and A. Wentzell.

Analysis and Simulation of Cyclic Pursuit

When: Wed, September 14, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Kevin Galloway (United States Naval Academy, Electrical and Computer Engineering Department) - https://www.usna.edu/Users/ee/galloway/index.php
Abstract: In this talk we will demonstrate that dyadic pursuit interactions employed between self-steering agents within a collective (e.g. a flock of birds or a group of mobile robots) can be used as building blocks for generating coordinated motion of the collective. Inspired by biology and motivated by potential technological applications such as team-based autonomous search-and-rescue, we will develop a mathematical framework to model the motion of self-steering particles under a variety of feedback control strategies. We focus on one particular realization based on the constant bearing (CB) pursuit strategy with the cycle graph (i.e. “cyclic pursuit”), and demonstrate how certain invariance properties result in a reduced system and interesting behaviors. This work relies on a combination of mathematical analysis tools and computational simulation to uncover the structure and wide range of behaviors exhibited by these cyclic pursuit systems. In the course of the discussion a GUI-based MATLAB program is described and demonstrated, for integrating the underlying collective dynamics (with nonholonomic constraints), and for experimenting with various combinations of pursuit laws and pursuit graph topologies.

Mathematical Modeling of Incommensurate Materials

When: Wed, September 21, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Mitchell Luskin (Department of Mathematics, University of Minnesota) - http://www.math.umn.edu/~luskin/
Abstract:
Incommensurate materials are found in crystals, liquid crystals, and quasi-crystals. Stacking a few layers of 2D materials such as graphene and molybdenum disulfide, for example, opens the possibility to tune the elastic, electronic, and optical properties of these materials. One of the main issues encountered in the mathematical modeling of layered 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system. This leads to complex commensurate-incommensurate transitions and pattern formation. Even basic concepts like the Cauchy-Born strain energy density, the electronic density of states, and the Kubo-Greenwood formulas for transport properties have not been given a rigorous analysis in the incommensurate setting. New approximate approaches will be discussed and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.

Local Structure Analysis in Atomic Systems

When: Wed, September 28, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Dr. Emanuel Lazar (Department of Materials Science and Engineering, University of Pennsylvania) - http://www.seas.upenn.edu/~mlazar/aboutme.html
Abstract: Many physical systems are modeled as large sets of atom-like particles. Understanding how such particles are arranged is thus a very natural problem, though describing this ``structure'' in an insightful yet tractable manner can be tricky. We consider several conventional methods for describing local structure and their limitations, theoretical and practical. We then introduce a topological approach more naturally suited for structure analysis and highlight its versatility and robustness. In particular, the proposed method can aid in analyzing high-temperature materials without uncontrolled modification of raw data. Several short applications to materials science are considered.

TBA

When: Wed, October 5, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Tom Goldstein (Department of Computer Science, University of Maryland) - http://www.cs.umd.edu/~tomg/
Abstract: TBA

Dynamics near the subcritical transition of the 3D Couette flow

When: Wed, October 12, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Jacob Bedrossian (Department of Mathematics, University of Maryland) - http://www2.cscamm.umd.edu/~jacob/
Abstract: Since the 1800s it has been observed that 3D stationary states of the incompressible Navier-Stokes equations display nonlinear instabilities at lower Reynolds than what can be predicted by linear theory alone. This phenomenon is now referred to as subcritical transition. We make a detailed study of this behavior near the plane, periodic Couette flow. For sufficiently regular perturbations, we determine the nonlinear stability threshold at high Reynolds number and characterize the long time dynamics of solutions near this threshold. For rougher data, we obtain an estimate of the stability threshold which agrees well with numerical experiments. The primary stabilizing mechanism is an anisotropic enhanced dissipation resulting from the mixing caused by the large mean shear; the main linear instability is a non-normal instability known as the lift-up effect. Understanding the variety of nonlinear resonances and devising the correct norms to estimate them form the core of the analysis we undertake. Joint work with Pierre Germain and Nader Masmoudi. Connections with related results on Landau damping in kinetic theory and inviscid damping in 2D fluid mechanics may also be discussed if time permits.

Imaging with intensity-only measurements

When: Wed, October 19, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Alexei Novikov (Department of Mathematics, Penn State University) - https://www.math.psu.edu/~anovikov/
Abstract: Imaging requires the solution of complicated inverse problems where we aim to determine the medium parameters from the measurements of the reflections of probing signals. In optics and X-ray imaging it is often difficult, or impossible, to measure the phases received at the detectors, only the intensities are available for imaging. I will introduce this problem mathematically, and explain some approaches that arise in attempting to image with intensities. I will then show results from extensive numerical simulations.

Sequential Importance Sampling for Counting Linear Extensions

When: Wed, November 9, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Dr. Isabel M. Beichl (Mathematical and Computational Science Division, National Institute of Standards and Technology (NIST)) - http://math.nist.gov/~IBeichl/
Abstract: A linear extension of a partially ordered set is a linear ordering of the vertices that respects the poset ordering. Any directed acyclic graph (DAG) defines a poset where vertex v precedes vertex w in the poset if w is reachable from v via a directed path. A linear ordering for the vertices of a DAG , called a topological sort, is equivalent to a linear extension for the associated DAG. Counting the number of topological sorts of a DAG is a well-known NP hard problem, important in in scheduling, and computational linear algebra. Because the problem is hard, approximations must be used. Monte Carlo methods for approximating using MCMC are known but they are not practical for real-world computation as they have complexity O(n^6). We will describe an alternate practical method based on sequential importance sampling. Success using SIS depends on designing an importance function that "knows" the search tree. We describe a robust importance function related to Moebius inversion. One interesting property is that our approximation is exact in case the DAG is a forest.

Adaptive Estimation in Two-way Sparse Reduced-rank Regression

When: Wed, November 16, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Tingni Sun (Department of Mathematics, University of Maryland) - http://www2.math.umd.edu/~tingni/
Abstract: This talk considers the problem of estimating a large coefficient matrix in a multiple response linear regression model in the high-dimensional setting, where the numbers of predictors and response variables can be much larger than the number of observations. The coefficient matrix is assumed to be not only of low rank, but also has a small number of nonzero rows and nonzero columns. We propose a new estimation scheme and provide its nearly optimal non-asymptotic minimax rates of estimation error under a collection of squared Schatten norm losses simultaneously. Some numerical studies will also be discussed.

Geometric Methods for the Approximation of High-dimensional Data sets and High-dimensional Dynamical Systems

When: Wed, November 30, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Mauro Maggioni (Department of Mathematics, Johns Hopkins University) - http://www.math.jhu.edu/~mauro/
Abstract:
We discuss a geometry-based statistical learning framework for performing model reduction and modeling of data sets as well as of certain classes of stochastic high-dimensional dynamical systems.

We start by discussing the problem of dictionary learning for data, and introduce a new setting for the problem and a solution based on hierarchical low-rank representation of the data, together with the corresponding statistical guarantees. We then discuss how to perform other statistical learning tasks, such as regression and estimation of distributions of the data, using the learned dictionaries.

We will then discuss the approximation of certain classes of stochastic dynamical systems: we assume only have access to a (large number of expensive) simulators that can return short simulations of high-dimensional stochastic system, and introduce a novel statistical learning framework for learning automatically a family of local approximations to the system, that can be (automatically) pieced together to form a fast global reduced model for the system, called ATLAS. ATLAS is guaranteed to be accurate (in the sense of producing stochastic paths whose distribution is close to that of paths generated by the original system) not only at small time scales, but also at large time scales, under suitable assumptions on the dynamics. We discuss applications to homogenization of rough diffusions in low and high dimensions, as well as relatively simple systems with separations of time scales, and deterministic chaotic systems in high-dimensions, that are well-approximated by stochastic differential equations.

Irreversibility, information and the second law of thermodynamics at the nanoscale

When: Wed, December 7, 2016 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Chris Jarzynski (Institute for Physical Science and Technology, University of Maryland) - http://www.ipst.umd.edu/researchandfaculty/cjarzyns.php
Abstract: What do the laws of thermodynamics look like, when applied to microscopic systems such as optically trapped colloidal particles, single molecules manipulated with laser tweezers, and biomolecular machines? In recent years it has become apparent that the fluctuations of small systems far from thermal equilibrium satisfy strong and unexpected laws, which allow us to rewrite familiar inequalities of macroscopic thermodynamics as equalities. These results in turn have spurred a renewed interest in the feedback control of small systems and the closely related Maxwell’s demon paradox. I will describe some of this progress, and will argue that it has refined our understanding of irreversibility, the second law, and the thermodynamic arrow of time.

Fast advection asymptotics for two-dimensional stochastic incompressible viscous fluids and SPDEs on graphs

When: Wed, February 1, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Sandra Cerrai (Department of Mathematics, University of Maryland) - http://www.math.umd.edu/~cerrai/
Abstract: Fast advection asymptotics for a stochastic reaction-diffusion-advection equation in a two-dimensional bounded domain will be discussed. To describe the asymptotics, one should consider a suitable class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow.

Phase Field and Free Boundary Models of Cell Motility

When: Wed, February 8, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Leonid Berlyand (Dept. of Mathematics & Materials Research Institute, Pennsylvania State University) - http://www.personal.psu.edu/lvb2/
Abstract: We study two types of models describing the motility of eukaryotic cells on substrates. The first, a phase-field model, consists of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion modified by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters and prove existence of traveling waves in the supercritical regime.

The second model type is a non-linear free boundary problem for a Keller-Segel type system of PDEs in 2D with area preservation and curvature entering the boundary conditions. We find an analytic one-parameter family of radially symmetric standing wave solutions (corresponding to a resting cell) as solutions to a Liouville type equation. Using topological tools, traveling wave solutions (describing steady motion) with non-circular shape are shown to bifurcate from the standing waves at a critical value of the parameter. Our bifurcation analysis explains, how varying a single (physical) parameter allows the cell to switch from rest to motion.

A new approach to bounds on mixing

When: Wed, February 15, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Dr. Flavien Leger (Courant Institute of Mathematical Sciences, New York University) - http://math.nyu.edu/~leger/
Abstract: We consider mixing by incompressible flows. In 2003, Bressan stated a conjecture concerning a bound on the mixing achieved by the flow in terms of an L1 norm of the velocity field. Existing results in the literature use an Lp norm with p>1. In this paper we introduce a new approach to prove such results. It recovers most of the existing results and offers new perspective on the problem. Our approach makes use of a recent harmonic analysis estimate from Seeger, Smart and Street.

Boundary layers: the Good, the Bad, and the Ugly

When: Wed, February 22, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Toan Nguyen (Department of Mathematics, Penn State University) - http://www.toannguyen.org/
Abstract: I will review recent advances concerning the Prandtl's conjecture: slightly viscous flows can be decomposed into inviscid flows, plus a Prandtl's layer near solid boundary, in the inviscid limit.

Kinetic theory of coupled oscillators

When: Wed, March 8, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Dr. Carson Chow (Mathematical Biology Section, National Institutes of Health) - https://www.niddk.nih.gov/about-niddk/staff-directory/intramural/carson-chow/Pages/research-summary.aspx
Abstract: Coupled oscillators arise in contexts as diverse as the brain, synchronized flashing of fireflies, coupled Josephson junctions, or unstable modes of the Millennium bridge in London. Generally, such systems are either studied for a small number of oscillators or in the infinite oscillator, mean field limit. The dynamics of large but finite networks of oscillators is largely unknown. Kinetic theory was developed by Boltzmann and Maxwell to show how microscopic Hamiltonian dynamics of particles could account for the thermodynamic properties of gases. Here, I will show how concepts of kinetic theory and statistical field theory can be applied to deterministic coupled oscillator and neural systems to compute dynamical finite system size effects.

Microdroplet instability for a least-action principle for incompressible droplets

When: Wed, March 15, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Robert Pego (Department of Mathematical Sciences, Carnegie Mellon University) - http://www.math.cmu.edu/~bobpego/
Abstract: The least-action problem for geodesic distance on the `manifold' of fluid-blob shapes exhibits instability due to microdroplet formation. This reflects a striking connection between Arnold's least-action principle for incompressible Euler flows and geodesic paths for Wasserstein distance. A connection with fluid mixture models via a variant of Brenier's relaxed least-action principle for generalized Euler flows will be outlined also. This is joint work with Jian-Guo Liu and Dejan Slepcev.

Forecast Sensitivity to Observations in Scientific Prediction Problems

When: Wed, March 29, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Kayo Ide (University of Maryland)
Abstract: Data assimilation is a method to estimate and predict evolution of physical system by integrating computational models and observations sampled from the evolving system. Prediction can be extremely challenging when the underlying physical system is highly nonlinear, corresponding model is high-dimensional and complex, and observations are not only nonlinear but also heterogeneous and inhomogeneous. In the context of data assimilation, impact of selected sets of observations on the optimal estimation can be quantified by information theory. The concept can be extended to evaluate the impact on forecast skill. In other words, forecast skill can be traced back to the observations used in the state estimate using an adjoint technique that can be either explicit or ensemble based. Forecast Sensitivity to Observations (FSO) is a diagnostic tool that complements traditional data denial of the observing system experiments. In this talk, we will present the FSO in the operational numerical weather prediction. We will also discuss the effect of observation and model biases on FSO.

Stochastic homogenization for reaction-diffusion equations

When: Wed, April 5, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Andrej Zlatos (Department of Mathematics, University of California San Diego) - http://www.math.ucsd.edu/~zlatos/
Abstract: We study spreading of reactions in random media and prove that homogenization takes place under suitable hypotheses. That is, the medium becomes effectively homogeneous in the large-scale limit of the dynamics of solutions to the PDE. Hypotheses that guarantee this include fairly general stationary ergodic KPP reactions, as well as homogeneous ignition reactions in up to three dimensions perturbed by radially symmetric impurities distributed according to a Poisson point process.

In contrast to the original (second-order) reaction-diffusion equations, the limiting "homogenized" PDE for this model are (first-order) Hamilton-Jacobi equations, and the limiting solutions are discontinuous functions that solve these in a weak sense. A key ingredient is a novel relationship between spreading speeds and front speeds for these models (as well as a proof of existence of these speeds), which can be thought of as the inverse of a well-known formula in the case of periodic media, but we are able to establish it even for more general stationary ergodic media.

From probability to PDEs: modeling the physics of polymerization

When: Wed, April 12, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Dr. Paul Patrone (Mathematical Analysis and Modeling Group, National Institute of Standards and Technology) - https://www.nist.gov/people/paul-patrone
Abstract: In a variety of applied settings, understanding the mechanisms of polymerization remains an open and important problem. In the composites industry, for example, different processing conditions change the microstructure of polymeric materials, which can ultimately impact the final material properties. Historically, however, a detailed understanding of this microstructure has been difficult to achieve because its inherent randomness is modeled in terms of high-dimensional atomistic models.

To address this problem, I discuss a coarse-graining approach that transforms a discrete representation of polymerization models into a lower-dimensional and continuous framework. The main idea is to reformulate the polymerization process in terms of a discrete master equation whose generic structure can represent the different physical processes that dominate crosslinking. I show that appropriate limits of this master equation yield a system of generic, first-order PDEs for the probability of different sub-structures within the polymerized network. Using the method of characteristics, I then discuss how this model can be transformed into a system of nonlinear ODEs, which in some cases can be solved exactly. In a related vein, I discuss how this approach provides a deeper understanding of analytical models first posed by Flory and show numerical results that confirm the accuracy of the approach in limiting cases.

Subriemannian Geometry and Finite Time Thermodynamics

When: Wed, May 3, 2017 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Perinkulam Krishnaprasad (Institute For Systems Research & Electrical and Computer Engineering, University of Maryland) - http://www.isr.umd.edu/~krishna/
Abstract: Subriemannian geometry has its roots in optimal control problems. The Caratheodory-Chow-Rashevskii theorem on accessibility also places the subject in contact with an axiomatic approach to macroscopic thermodynamics. Explicit integrability of optimal control problems in this context is of interest. As in the case for integrability questions in mechanics, here too symmetries and conservation laws have a key role. In this talk we discuss model problems and results pertaining to such questions in isolated systems and ensembles of interacting systems. Of special interest is the stochastic control problem of determining thermodynamic cycles that draw useful work from fluctuations. This work is in collaboration with PhD student Yunlong Huang, and Dr. Eric Justh of the Naval Research Laboratory.