# Browsing by Author "Mattingly, JC"

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Item Open Access A weak trapezoidal method for a class of stochastic differential equations(Communications in Mathematical Sciences, 2011-03-01) Anderson, DF; Mattingly, JCWe present a numerical method for the approximation of solutions for the class of stochastic differential equations driven by Brownian motions which induce stochastic variation in fixed directions. This class of equations arises naturally in the study of population processes and chemical reaction kinetics. We show that the method constructs paths that are second order accurate in the weak sense. The method is simpler than many second order methods in that it neither requires the construction of iterated It̂o integrals nor the evaluation of any derivatives. The method consists of two steps. In the first an explicit Euler step is used to take a fractional step. The resulting fractional point is then combined with the initial point to obtain a higher order, trapezoidal like, approximation. The higher order of accuracy stems from the fact that both the drift and the quadratic variation of the underlying SDE are approximated to second order. © 2011 International Press.Item Open Access An adaptive Euler-Maruyama scheme for SDEs: Convergence and stability(IMA Journal of Numerical Analysis, 2007-01-01) Lamba, H; Mattingly, JC; Stuart, AMThe understanding of adaptive algorithms for stochastic differential equations (SDEs) is an open area, where many issues related to both convergence and stability (long-time behaviour) of algorithms are unresolved. This paper considers a very simple adaptive algorithm, based on controlling only the drift component of a time step. Both convergence and stability are studied. The primary issue in the convergence analysis is that the adaptive method does not necessarily drive the time steps to zero with the user-input tolerance. This possibility must be quantified and shown to have low probability. The primary issue in the stability analysis is ergodicity. It is assumed that the noise is nondegenerate, so that the diffusion process is elliptic, and the drift is assumed to satisfy a coercivity condition. The SDE is then geometrically ergodic (averages converge to statistical equilibrium exponentially quickly). If the drift is not linearly bounded, then explicit fixed time step approximations, such as the Euler-Maruyama scheme, may fail to be ergodic. In this work, it is shown that the simple adaptive time-stepping strategy cures this problem. In addition to proving ergodicity, an exponential moment bound is also proved, generalizing a result known to hold for the SDE itself. © The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.Item Open Access An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations(Communications in Contemporary Mathematics, 1999-11-01) Mattingly, JC; Sinai, Ya GItem Open Access Anomalous dissipation in a stochastically forced infinite-dimensional system of coupled oscillators(Journal of Statistical Physics, 2007-09-01) Mattingly, JC; Suidan, TM; Vanden-Eijnden, EWe study a system of stochastically forced infinite-dimensional coupled harmonic oscillators. Although this system formally conserves energy and is not explicitly dissipative, we show that it has a nontrivial invariant probability measure. This phenomenon, which has no finite dimensional equivalent, is due to the appearance of some anomalous dissipation mechanism which transports energy to infinity. This prevents the energy from building up locally and allows the system to converge to the invariant measure. The invariant measure is constructed explicitly and some of its properties are analyzed. © 2007 Springer Science+Business Media, LLC.Item Open Access Approximations of Markov Chains and High-Dimensional Bayesian Inference(2015) Mattingly, JC; Johndrow, J; Mukherjee, S; Dunson, DItem Open Access Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations(Probability Theory and Related Fields, 2011-03-01) Hairer, M; Mattingly, JC; Scheutzow, MThere are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris' theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such "asymptotic couplings" were central to (Mattingly and Sinai in Comm Math Phys 219(3):523-565, 2001; Mattingly in Comm Math Phys 230(3):461-462, 2002; Hairer in Prob Theory Relat Field 124:345-380, 2002; Bakhtin and Mattingly in Commun Contemp Math 7:553-582, 2005) on which this work builds. As in Bakhtin and Mattingly (2005) the emphasis here is on stochastic differential delay equations. Harris' celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are "small" (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a "small set" by the much weaker notion of a "d-small set," which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris' theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations. In this framework, the usual theory of Harris chains does not apply, since there are natural examples for which there exist no small sets (except for sets consisting of only one point). This gives a solution to the long-standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it. © 2009 Springer-Verlag.Item Open Access Contractivity and ergodicity of the random map x →(Theory of Probability and its Applications, 2003-06-26) Mattingly, JCThe long time behavior of the random map xn → xn+1 = |xn-θn| is studied under various assumptions on the distribution of the θn. One of the interesting features of this random dynamical system is that for a single fixed deterministic θ the map is not a contraction, while the composition is almost surely a contraction if θ is chosen randomly with only mild assumptions on the distribution of the θ's. The system is useful as an explicit model where more abstract ideas can be explored concretely. We explore various measures of convergence rates, hyperbolically from randomness, and the structure of the random attractor.Item Open Access Convergence of numerical time-averaging and stationary measures via Poisson equations(SIAM Journal on Numerical Analysis, 2010-07-07) Mattingly, JC; Stuart, AM; Tretyakov, MVNumerical approximation of the long time behavior of a stochastic di.erential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantages of this approach are its simplicity and universality. It works equally well for a range of explicit and implicit schemes, including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus, and we study only smooth test functions. However, we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.Item Open Access Convergence of Stratified MCMC Sampling of Non-Reversible DynamicsEarle, G; Mattingly, JCWe present a form of stratified MCMC algorithm built with non-reversible stochastic dynamics in mind. It can also be viewed as a generalization of the exact milestoning method, or form of NEUS. We prove convergence of the method under certain assumptions, with expressions for the convergence rate in terms of the process's behavior within each stratum and large scale behavior between strata. We show that the algorithm has a unique fixed point which corresponds to the invariant measure of the process without stratification. We will show how the speeds of two versions of the new algorithm, one with an extra eigenvalue problem step and one without, relate to the mixing rate of a discrete process on the strata, and the mixing probability of the process being sampled within each stratum. The eigenvalue problem version also relates to local and global perturbation results of discrete Markov chains, such as those given by Van Koten, Weare et. al.Item Open Access Coupling and Decoupling to bound an approximating Markov Chain(2017-07-27) Johndrow, JE; Mattingly, JCThis simple note lays out a few observations which are well known in many ways but may not have been said in quite this way before. The basic idea is that when comparing two different Markov chains it is useful to couple them is such a way that they agree as often as possible. We construct such a coupling and analyze it by a simple dominating chain which registers if the two processes agree or disagree. We find that this imagery is useful when thinking about such problems. We are particularly interested in comparing the invariant measures and long time averages of the processes. However, since the paths agree for long runs, it also provides estimates on various stopping times such as hitting or exit times. We also show that certain bounds are tight. Finally, we provide a simple application to a Markov Chain Monte Carlo algorithm and show numerically that the results of the paper show a good level of approximation at considerable speed up by using an approximating chain rather than the original sampling chain.Item Open Access Diffusion limits of the random walk metropolis algorithm in high dimensions(Annals of Applied Probability, 2012-06-01) Mattingly, JC; Pillai, NS; Stuart, AMDiffusion limits of MCMC methods in high dimensions provide a useful theoretical tool for studying computational complexity. In particular, they lead directly to precise estimates of the number of steps required to explore the target measure, in stationarity, as a function of the dimension of the state space. However, to date such results have mainly been proved for target measures with a product structure, severely limiting their applicability. The purpose of this paper is to study diffusion limits for a class of naturally occurring high-dimensional measures found from the approximation of measures on a Hilbert space which are absolutely continuous with respect to a Gaussian reference measure. The diffusion limit of a random walk Metropolis algorithm to an infinite-dimensional Hilbert space valued SDE (or SPDE) is proved, facilitating understanding of the computational complexity of the algorithm. © 2012 Institute of Mathematical Statistics.Item Open Access Ergodicity and Lyapunov functions for Langevin dynamics with singular potentials(2017-11-30) Herzog, DP; Mattingly, JCWe study Langevin dynamics of $N$ particles on $R^d$ interacting through a singular repulsive potential, e.g.~the well-known Lennard-Jones type, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof of the main result relies on an explicit construction of a Lyapunov function. In contrast to previous results for such systems, our result implies geometric convergence to equilibrium starting from an essentially optimal family of initial distributions.Item Open Access Geometric ergodicity of a bead-spring pair with stochastic Stokes forcing(Stochastic Processes and their Applications, 2012-12-01) Mattingly, JC; McKinley, SA; Pillai, NSWe consider a simple model for the fluctuating hydrodynamics of a flexible polymer in a dilute solution, demonstrating geometric ergodicity for a pair of particles that interact with each other through a nonlinear spring potential while being advected by a stochastic Stokes fluid velocity field. This is a generalization of previous models which have used linear spring forces as well as white-in-time fluid velocity fields. We follow previous work combining control theoretic arguments, Lyapunov functions, and hypo-elliptic diffusion theory to prove exponential convergence via a Harris chain argument. In addition we allow the possibility of excluding certain "bad" sets in phase space in which the assumptions are violated but from which the system leaves with a controllable probability. This allows for the treatment of singular drifts, such as those derived from the Lennard-Jones potential, which is a novel feature of this work. © 2012 Elsevier B.V. All rights reserved.Item Open Access Geometric ergodicity of Langevin dynamics with Coulomb interactionsLu, Y; Mattingly, JCThis paper is concerned with the long time behavior of Langevin dynamics of {\em Coulomb gases} in $\mathbf{R}^d$ with $d\geq 2$, that is a second order system of Brownian particles driven by an external force and a pairwise repulsive Coulomb force. We prove that the system converges exponentially to the unique Boltzmann-Gibbs invariant measure under a weighted total variation distance. The proof relies on a novel construction of Lyapunov function for the Coulomb system.Item Open Access Gibbsian dynamics and the generalized Langevin equationHerzog, DP; Mattingly, JC; Nguyen, HDWe study the statistically invariant structures of the nonlinear generalized Langevin equation (GLE) with a power-law memory kernel. For a broad class of memory kernels, including those in the subdiffusive regime, we construct solutions of the GLE using a Gibbsian framework, which does not rely on existing Markovian approximations. Moreover, we provide conditions on the decay of the memory to ensure uniqueness of statistically steady states, generalizing previous known results for the GLE under particular kernels as a sum of exponentials.Item Open Access Malliavin calculus for the stochastic 2D Navier-Stokes equation(Communications on Pure and Applied Mathematics, 2006-12-01) Mattingly, JC; Pardoux, EWe consider the incompressible, two-dimensional Navier-Stokes equation with periodic boundary conditions under the effect of an additive, white-in-time, stochastic forcing. Under mild restrictions on the geometry of the scales forced, we show that any finite-dimensional projection of the solution possesses a smooth, strictly positive density with respect to Lebesgue measure. In particular, our conditions are viscosity independent. We are mainly interested in forcing that excites a very small number of modes. All of the results rely on proving the nondegeneracy of the infinite-dimensional Malliavin matrix. © 2006 Wiley Periodicals, Inc.Item Open Access Mathematically Quantifying Non-responsiveness of the 2021 Georgia Congressional Districting Plan(ACM International Conference Proceeding Series, 2022-10-06) Zhao, Z; Hettle, C; Gupta, S; Mattingly, JC; Randall, D; Herschlag, GJTo audit political district maps for partisan gerrymandering, one may determine a baseline for the expected distribution of partisan outcomes by sampling an ensemble of maps. One approach to sampling is to use redistricting policy as a guide to precisely codify preferences between maps. Such preferences give rise to a probability distribution on the space of redistricting plans, and Metropolis-Hastings methods allow one to sample ensembles of maps from the specified distribution. Although these approaches have nice theoretical properties and have successfully detected gerrymandering in legal settings, sampling from commonly-used policy-driven distributions is often computationally difficult. As of yet, there is no algorithm that can be used off-the-shelf for checking maps under generic redistricting criteria. In this work, we mitigate the computational challenges in a Metropolized-sampling technique through a parallel tempering method combined with ReCom[11] and, for the first time, validate that such techniques are effective on these problems at the scale of statewide precinct graphs for more policy informed measures. We develop these improvements through the first case study of district plans in Georgia. Our analysis projects that any election in Georgia will reliably elect 9 Republicans and 5 Democrats under the enacted plan. This result is largely fixed even as public opinion shifts toward either party and the partisan outcome of the enacted plan does not respond to the will of the people. Only 0.12% of the ∼160K plans in our ensemble were similarly non-responsive.Item Open Access METROPOLIZED FOREST RECOMBINATION FOR MONTE CARLO SAMPLING OF GRAPH PARTITIONS(SIAM Journal on Applied Mathematics, 2023-08-01) Autry, E; Carter, D; Herschlag, GJ; Hunter, Z; Mattingly, JCWe develop a new Markov chain on graph partitions that makes relatively global moves yet is computationally feasible to be used as the proposal in the Metropolis-Hastings method. Our resulting algorithm is able to sample from a specified measure on partitions or spanning forests. Being able to sample from a specified measure is a requirement of what we consider as the gold standard in quantifying the extent to which a particular map is a gerrymander. Our proposal chain modifies the recently developed method called recombination (ReCom), which draws spanning trees on joined partitions and then randomly cuts them to repartition. We improve the computational efficiency by augmenting the statespace from partitions to spanning forests. The extra information accelerates the computation of the forward and backward proposal probabilities which are required for the Metropolis-Hastings algorithm. We demonstrate this method by sampling redistricting plans on several measures of interest and find promising convergence results on several key observables of interest. We also explore some limitations in the measures that are efficient to sample from and investigate the feasibility of using parallel tempering to extend this space of measures.Item Open Access Metropolized Multiscale Forest Recombination for Redistricting(Multiscale Modeling & Simulation, 2021-01) Autry, EA; Carter, D; Herschlag, GJ; Hunter, Z; Mattingly, JCItem Open Access Noise-induced stabilization of planar flows I(Electronic Journal of Probability, 2015-10-22) Herzog, DP; Mattingly, JC© 2015 University of Washington. All rights reserved.We show that the complex-valued ODE (n ≥ 1, an+1 6≠ 0): ź = an+1zn+1 + anzn +1zn + a0; which necessarily has trajectories along which the dynamics blows up in finite time, can be stabilized by the addition of an arbitrarily small elliptic, additive Brownian stochastic term. We also show that the stochastic perturbation has a unique invariant probability measure which is heavy-tailed yet is uniformly, exponentially attracting. The methods turn on the construction of Lyapunov functions. The techniques used in the construction are general and can likely be used in other settings where a Lyapunov function is needed. This is a two-part paper. This paper, Part I, focuses on general Lyapunov methods as applied to a special, simplified version of the problem. Part II [11] extends the main results to the general setting.