Metastable distributions of Markov chains with rare transitions and related problems that result in differential equations with nonstandard boundary conditions
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.
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.
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.
Dr. Emanuel Lazar, Department of Materials Science and Engineering, University of Pennsylvania
Local Structure Analysis in Atomic Systems
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.
Scalable methods for machine learning and sparse signal recovery
The abundance of large, distributed web-based data sets and the recent popularity of cloud computing platforms has opened many doors in machine learning and statistical modeling. However, these resources present a host of new algorithmic challenges. Practical algorithms for large-scale data analysis must scale well across many machines, have low communication requirements, and have low (nearly linear) runtime complexity to handle extremely large problems.
In this talk, we discuss alternating direction methods as a practical and general tool for solving a wide range of model-fitting problems in a distributed framework. We then focus on new "transpose reduction" strategies that allow extremely large regression problems to be solved quickly on a single node. We will study the performance of these algorithms for fitting linear classifiers and sparse regression models on tera-scale datasets using thousands of cores.
Dynamics near the subcritical transition of the 3D Couette flow
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 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.
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (Schrödinger-Ginzburg-Landau) equation.
Dr. Isabel Beichl,Mathematical and Computational Science Division, National Institute of Standards and Technology (NIST)
Sequential Importance Sampling for Counting Linear Extensions
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
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
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
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.