# Browsing by Author "Mattingly, Jonathan C"

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Item Open Access A Generalized Lyapunov Construction for Proving Stabilization by Noise(2012) Kolba, Tiffany NicoleNoise-induced stabilization occurs when an unstable deterministic system is stabilized by the addition of white noise. Proving that this phenomenon occurs for a particular system is often manifested through the construction of a global Lyapunov function. However, the procedure for constructing a Lyapunov function is often quite ad hoc, involving much time and tedium. In this thesis, a systematic algorithm for the construction of a global Lyapunov function for planar systems is presented. The general methodology is to construct a sequence of local Lyapunov functions in different regions of the plane, where the regions are delineated by different behaviors of the deterministic dynamics. A priming region, where the deterministic drift is directed inward, is first identified where there is an obvious choice for a local Lyapunov function. This priming Lyapunov function is then propagated to the other regions through a series of Poisson equations. The local Lyapunov functions are lastly patched together to form one smooth global Lyapunov function.

The algorithm is applied to a model problem which displays finite time blow up in the deterministic setting in order to prove that the system exhibits noise-induced stabilization. Moreover, the Lyapunov function constructed is in fact what we define to be a super Lyapunov function. We prove that the existence of a super Lyapunov function, along with a minorization condition, implies that the corresponding system converges to a unique invariant probability measure at an exponential rate that is independent of the initial condition.

Item Open Access A practical criterion for positivity of transition densities(Nonlinearity, 2015-07-10) Herzog, David P; Mattingly, Jonathan C© 2015 IOP Publishing Ltd & London Mathematical Society.We establish a simple criterion for locating points where the transition density of a degenerate diffusion is strictly positive. Throughout, we assume that the diffusion satisfies a stochastic differential equation (SDE) on Rd with additive noise and polynomial drift. In this setting, we will see that it is often the case that local information of the flow, e.g. the Lie algebra generated by the vector fields defining the SDE at a point x ∈ Rd, determines where the transition density is strictly positive. This is surprising in that positivity is a more global property of the diffusion. This work primarily builds on and combines the ideas of Arous and Lé andre (1991 Décroissance exponentielle du noyau de la chaleur sur la diagonale. II Probab. Theory Relat. Fields 90 377-402) and Jurdjevic and Kupka (1985 Polynomial control systems Math. Ann. 272 361-8).Item Open Access A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs(Electronic Journal of Probability, 2011-05-09) Hairer, Martin; Mattingly, Jonathan CWe present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operatorμt can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection on a subspace of sufficiently regular functions. Then the eigenfunctions of μt with small eigenvalues have only a very small component in the image of Π." We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced in [HM06]. One of the main novel technical tools is an almost sure bound from below on the size of "Wiener polynomials," where the coefficients are possibly non-adapted stochastic processes satisfying a Lips chitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris' lemma, which is unavailable in the present context. We conclude by showing that the two-dimensional stochastic Navier-Stokes equations and a large class of reaction-diffusion equations fit the framework of our theory.Item Open Access Absolute Continuity of Singular SPDEs and Bayesian Inference on Dynamical Systems(2023) Su, LangxuanWe explore the interplay among probability, stochastic analysis, and dynamical systems through two lenses: (1) absolute continuity of singular stochastic partial differential equations (SPDEs); (2) Bayesian inference on dynamical systems.

In the first part, we prove that up to a certain singular regime, the law of the stochastic Burgers equation at a fixed time is absolutely continuous with respect to the corresponding stochastic heat equation with the nonlinearity removed. The results follow from an extension of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time. To deal with the singularity, we introduce a novel decomposition in the spirit of Da Prato-Debussche and Gaussian chaos decomposition in singular SPDEs, by separating out the noise into different levels of regularity, along with a number of renormalization techniques. The number of levels in this decomposition diverges to infinite as we move to the stochastic Burgers equation associated with the KPZ equation. This result illustrates the fundamental probabilistic structure of a class of singular SPDEs and a notion of ``ellipticity'' in the infinite-dimensional setting.

In the second part, we establish connections between large deviations and a class of generalized Bayesian inference procedures on dynamical systems. We show that posterior consistency can be derived using a combination of classical large deviation techniques, such as Varadhan's lemma, conditional/quenched large deviations, annealed large deviations, and exponential approximations. We identified the divergence term as the Donsker-Varadhan relative entropy rate, also related to the Kolmogorov-Sinai entropy in ergodic theory. As an application, we prove new quenched/annealed large deviation asymptotics and a new Bayesian posterior consistency result for a class of mixing stochastic processes. In the case of Markov processes, one can obtain explicit conditions for posterior consistency, when estimates for log-Sobolev constants are available, which makes our framework essentially a black box. We also recover state-of-the-art posterior consistency on classical dynamical systems with a simple proof. Our approach has the potential of proving posterior consistency for a wide range of Bayesian procedures in a unified way.

Item Open Access Analyzing the Effects of Partisan Correlation on Election Outcomes Using Order Statistics(2019-04-24) Wiebe, ClaireThe legislative representation of political parties in the United States is dependent not only on way that legislative district boundaries are drawn, but also on the way in which people are distributed across a state. That is, there exists a level of partisan correlation within the spacial distribution of an electorate that affects legislative outcomes. This work aims to study the effect of this partisan clustering on election outcomes and related metrics using analytic models and order statistics. Two models of North Carolina, one with a uniformly distributed electorate and one with a symmetrically clustered electorate, are considered both independently and in comparison. These models are used to study expected election outcomes, the proportionality of legislative representation for given state-wide vote fraction, and the sensitivity of vote share to seat share across different correlation length scales. The findings provide interesting insight into the relationship between district proportionality and legislative proportionality, the extent to which the minority party is expected to be underrepresented in seat share for given state-wide vote share and correlation length, and the extent to which the responsiveness of seat share is dependent on state wide vote share and correlation length.Item Open Access Dimensionality Reduction and Learning on Networks(2011) Balachandran, PrakashMachine learning is a powerful branch of mathematics and statistics that allows the automation of tasks that would otherwise require humans a long time to perform. Two particular fields of machine learning that have been developing in the last two decades are dimensionality reduction and semi-supervised learning.

Dimensionality reduction is a powerful tool in the analysis of high dimensional data by reducing the number of variables under consideration while approximately preserving some quantity of interest (usually pairwise distances). Methods such as Principal Component Analysis (PCA) or Isometric Feature Mapping (ISOMAP) do this do this by embedding the data, equipped with a nonnegative, symmetric, similarity kernel or adjacency matrix into Euclidean space and finding a linear subspace or low dimensional submanifold which best fits the data, respectively.

When the data takes the form of network data, how to perform such dimensionality reduction intrinsically, without resorting to an embedding, that can be extended to the case of nonnegative, non-symmetric adjacency matrices remains an important open problem. In the first part of my dissertation, using current techniques in local spectral clustering to partition the network using a Markov process induced by the adjacency matrix, we deliver an intrinsic dimensionality reduction of the network in terms of a non-Markov process on a reduced state space that approximately preserves transitions of the original Markov process between clusters. By iterating the process, one obtains a family of non-Markov processes on successively finer state spaces representing the original network ands its diffusion at different scales, which can be used to approximate the law of the original process at a particular time scale. We give applications of this theory to a variety of synthetic data sets and evaluate its performance accordingly.

Next, consider the case of detecting astronomical phenomenon solely in terms of the light intensities observed. There already exists a large database of prior recorded phenomena that has been categorized by humans as a function of the observed light intensity. Given these so-called class labels then, how can we automate the procedure of extending these class labels to the massive amount of data that is currently being observed? This is the problem of concern in semi-supervised learning.

In the second part of my thesis, we consider data sets in which relations between data points are more complex than simply pairwise. Examples include gene networks where the the data points are random variables, and similarities of a subset are measured by non-independence of the corresponding random variables. Such data sets can be illustrated as a hypergraph, and the natural question for diagnosis becomes: how does one perform transductive inference (a particular form of semi-supervised learning)? Using the simple case of pairwise and threewise similarities, we construct a reversible random walk on undirected edges induced by threewise relations (faces). By pulling the random walk back to a random walk on the vertex set and mixing it with the random walk induced by pairwise similarities, we perform diffusive transductive inference. We present applications and results of this technique, any analyze its performance on a variety of data sets.

Item Open Access Ergodic properties of highly degenerate 2D stochastic Navier-Stokes equations(Comptes Rendus Mathématique. Académie des Sciences. Paris, 2004) Hairer, Martin; Mattingly, Jonathan CThis Note presents the results from "Ergodicity of the degenerate stochastic 2D Navier-Stokes equation"; by M. Hairer and J.C. Mattingly. We study the Navier-Stokes equation on the two-dimensional torus when forced by a finite dimensional Gaussian white noise and give conditions under which the system is ergodic. In particular, our results hold for specific choices of four-dimensional Gaussian white noise. © 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Item Open Access Ergodicity for the navier-stokes equation with degenerate random forcing: Finite-dimensional approximation(Communications on Pure and Applied Mathematics, 2001-11-01) E, Weinan; Mattingly, Jonathan CWe study Galerkin truncations of the two-dimensional Navier-Stokes equation under degenerate, large-scale, stochastic forcing. We identify the minimal set of modes that has to be forced in order for the system to be ergodic. Our results rely heavily on the structure of the nonlinearity. © 2001 John Wiley & Sons, Inc.Item Open Access Error bounds for Approximations of Markov chains(2017-11-30) Johndrow, James E; Mattingly, Jonathan CThe first part of this article gives error bounds for approximations of Markov kernels under Foster-Lyapunov conditions. The basic idea is that when both the approximating kernel and the original kernel satisfy a Foster-Lyapunov condition, the long-time dynamics of the two chains -- as well as the invariant measures, when they exist -- will be close in a weighted total variation norm, provided that the approximation is sufficiently accurate. The required accuracy depends in part on the Lyapunov function, with more stable chains being more tolerant of approximation error. We are motivated by the recent growth in proposals for scaling Markov chain Monte Carlo algorithms to large datasets by defining an approximating kernel that is faster to sample from. Many of these proposals use only a small subset of the data points to construct the transition kernel, and we consider an application to this class of approximating kernel. We also consider applications to distribution approximations in Gibbs sampling. Another application in which approximating kernels are commonly used is in Metropolis algorithms for Gaussian process models common in spatial statistics and nonparametric regression. In this setting, there are typically two sources of approximation error: discretization error and approximation of Metropolis acceptance ratios. Because the approximating kernel is obtained by discretizing the state space, it is singular with respect to the exact kernel. To analyze this application, we give additional results in Wasserstein metrics in contrast to the proceeding examples which quantified the level of approximation in a total variation norm.Item Open Access Geometric Ergodicity of Two–dimensional Hamiltonian systems with a Lennard–Jones–like Repulsive Potential(arXiv preprint arXiv:1104.3842, 2011) Cooke, Ben; Mattingly, Jonathan C; McKinley, Scott A; Schmidler, Scott CItem Open Access Invariant measure selection by noise. An example(Discrete and Continuous Dynamical Systems- Series A, 2014-01-01) Mattingly, Jonathan C; Pardoux, EtienneWe consider a deterministic system with two conserved quantities and infinity many invariant measures. However the systems possess a unique invariant measure when enough stochastic forcing and balancing dissipation are added. We then show that as the forcing and dissipation are removed a unique limit of the deterministic system is selected. The exact structure of the limiting measure depends on the specifics of the stochastic forcing.Item Open Access MCMC Sampling Geospatial Partitions for Linear Models(2021) Wyse, Evan TGeospatial statistical approaches must frequently confront the problem of correctlypartitioning a group of geographical sub-units, such as counties, states, or precincts,into larger blocks which share information. Since the space of potential partitions isquite large, sophisticated approaches are required, particularly when this partitioninginteracts with other parts of a larger model, as is frequent with Bayesian inference.Authors such as Balocchi et al. (2021) provide stochastic search algorithms whichprovide certain theoretical guarantees about this partition in the context of Bayesianmodel averaging. We borrow tools from Herschlag et al. (2020) to examine a potentialapproach to sampling these clusters efficiently using a Markov Chain Monte Carlo(MCMC) approach.

Item Open Access Multi-Scale Merge-Split Markov Chain Monte Carlo for RedistrictingAutry, Eric A; Carter, Daniel; Herschlag, Gregory; Hunter, Zach; Mattingly, Jonathan CWe develop a Multi-Scale Merge-Split Markov chain on redistricting plans. The chain is designed to be usable as the proposal in a Markov Chain Monte Carlo (MCMC) algorithm. Sampling the space of plans amounts to dividing a graph into a partition with a specified number of elements which each correspond to a different district. The districts satisfy a collection of hard constraints and the measure may be weighted with regard to a number of other criteria. The multi-scale algorithm is similar to our previously developed Merge-Split proposal, however, this algorithm provides improved scaling properties and may also be used to preserve nested communities of interest such as counties and precincts. Both works use a proposal which extends the ReCom algorithm which leveraged spanning trees merge and split districts. In this work we extend the state space so that each district is defined by a hierarchy of trees. In this sense, the proposal step in both algorithms can be seen as a "Forest ReCom." We also expand the state space to include edges that link specified districts, which further improves the computational efficiency of our algorithm. The collection of plans sampled by the MCMC algorithm can serve as a baseline against which a particular plan of interest is compared. If a given plan has different racial or partisan qualities than what is typical of the collection of plans, the given plan may have been gerrymandered and is labeled as an outlier.Item Open Access Noise-induced stabilization of planar flows ii(Electronic Journal of Probability, 2015-10-25) Herzog, David P; Mattingly, Jonathan C© 2015 University of Washington. All rights reserved.We continue the work started in Part I [6], showing how the addition of noise can stabilize an otherwise unstable system. The analysis makes use of nearly optimal Lyapunov functions. In this continuation, we remove the main limiting assumption of Part I by an inductive procedure as well as establish a lower bound which shows that our construction is radially sharp. We also prove a version of Peskir’s [7] generalized Tanaka formula adapted to patching together Lyapunov functions. This greatly simplifies the analysis used in previous works.Item Open Access Noise-induced strong stabilizationLeimbach, Matti; Mattingly, Jonathan C; Scheutzow, MichaelWe consider a 2-dimensional stochastic differential equation in polar coordinates depending on several parameters. We show that if these parameters belong to a specific regime then the deterministic system explodes in finite time, but the random dynamical system corresponding to the stochastic equation is not only strongly complete but even admits a random attractor.Item Open Access Non-local SPDE limits of spatially-correlated-noise driven spin systems derived to sample a canonical distributionGao, Yuan; Marzuola, Jeremy L; Mattingly, Jonathan C; Newhall, Katherine AWe study the macroscopic behavior of a stochastic spin ensemble driven by a discrete Markov jump process motivated by the Metropolis-Hastings algorithm where the proposal is made with spatially correlated (colored) noise, and hence fails to be symmetric. However, we demonstrate a scenario where the failure of proposal symmetry is a higher order effect. Hence, from these microscopic dynamics we derive as a limit as the proposal size goes to zero and the number of spins to infinity, a non-local stochastic version of the harmonic map heat flow (or overdamped Landau-Lipshitz equation). The equation is both mathematically well-posed and samples the canonical/Gibbs distribution related to the kinetic energy. The failure of proposal symmetry due to interaction between the confining geometry of the spin system and the colored noise is in contrast to the uncorrelated, white-noise, driven system. Specifically, the choice of projection of the noise to conserve the magnitude of the spins is crucial to maintaining the proper equilibrium distribution. Numerical simulations are included to verify convergence properties and demonstrate the dynamics.Item Open Access Non-reversible Markov chain Monte Carlo for sampling of districting mapsHerschlag, Gregory; Mattingly, Jonathan C; Sachs, Matthias; Wyse, EvanEvaluating the degree of partisan districting (Gerrymandering) in a statistical framework typically requires an ensemble of districting plans which are drawn from a prescribed probability distribution that adheres to a realistic and non-partisan criteria. In this article we introduce novel non-reversible Markov chain Monte-Carlo (MCMC) methods for the sampling of such districting plans which have improved mixing properties in comparison to previously used (reversible) MCMC algorithms. In doing so we extend the current framework for construction of non-reversible Markov chains on discrete sampling spaces by considering a generalization of skew detailed balance. We provide a detailed description of the proposed algorithms and evaluate their performance in numerical experiments.Item Open Access On Absolute Continuity for Stochastic Partial Differential Equations and an Averaging Principle for a Queueing Network(2010) Watkins, Andrea ChereseThe connection between elliptic stochastic diffusion processes and partial differential equations is rich and well understood. This connection is not very well understood when the stochastic differential equation takes values in an infinite dimensional space such as a function space. In this case, the diffusion is a stochastic partial differential equation (SPDE) and the notion of ellipticity is ambiguous. We establish a sufficient condition on the diffusion coefficient of a class of nonlinear SPDEs, which is analogous to the nondegeneracy condition in finite dimensions, that allows for the existence of a Markov transition density that is absolutely continuous with respect to an infinite dimensional Gaussian measure.

In the second part of this work, we consider a two-station queueing network that processes K job types. The first station in this network is a polling station, and we assume that the second station is operating under any nonidling service discipline. We consider diffusion-scaled versions of many of the processes governing this system, and we show that the scaled two-dimensional total workload process converges to Brownian motion in a wedge. We also show that the scaled immediate workload process for station 2 does not converge, but admits an averaging principle.

Item Open Access On SDEs with Partial Damping Inspired by the Navier-Stokes Equations(2019) Williamson, BrendanThe solution to the Navier Stokes equations on the 2D torus with stochastic forcing that is white noise in time, coloured in space has a Fourier series representation whose coordinates satisfy a countable system of Stochastic Differential Equations. Inspired by the structure of these equations, we construct a finite system of stochastic differential equations with a similar structure and explore the conditions under which the system has an invariant distribution. \\

Our main tool to prove existence of invariant distributions are Lyapunov functions, or more generally Lyapunov pairs. In particular, we construct the Lyapunov pairs piecewise over different regions and then use mollifiers to unify these disparate characterisations. We also apply some results from Algebraic Geometry and Matrix Perturbation Theory to study and exploit the geometry of the problem in high dimensions. \\

The combination of these methods allowed us to prove that a large class of the equations we constructed have an invariant distributions. Furthermore we have explicit tail estimates for these invariant distributions.

Item Open Access Optical Precursor Behavior(2007-05-07T19:07:14Z) LeFew, William R.Controlling and understanding the propagation of optical pulses through dispersive media forms the basis for optical communication, medical imaging, and other modern technological advances. Integral to this control and understanding is the ability to describe the transients which occur immediately after the onset of a signal. This thesis examines the transients of such a system when a unit step function is applied. The electromagnetic field is described by an integral resulting from Maxwell’s Equations. It was previously believed that optical precursors, a specific transient effect, existed only for only a few optical cycles and contributed only small magnitudes to the field. The main results of this thesis show that the transients arising from this integral are entirely precursors and that they may exist on longer time scales and contribute larger magnitudes to the field. The experimental detection of precursors has previously been recognized only through success comparison to the transient field resulting from an application of the method of steepest descent to that field integral. For any parameter regime where steepest descents may be applied, this work gives iterative methods to determine saddle points which are both more accurate than the accepted results and to extend into regimes where the current theory has failed. Furthermore, asymptotic formulae have been derived for regions where previous attempts at steepest descent have failed. Theory is also presented which evaluates the applicability of steepest descents in the represention of precursor behavior for any set of parameters. Lastly, the existence of other theoretical models for precursor behavior who may operate beyond the reach of steepest descent is validated through successful comparisons of the transient prediction of those methods to the steepest descent based results of this work.