arXiv subjects

Elizabeth Carlson

Publications and source records attributed to Elizabeth Carlson.

12 recordsLinked to original sources

arXiv2026

Comparing Deterministic and Stochastic Parameter Recovery Algorithms Applied to Chaotic Systems

This paper explores the effectiveness of various novel deterministic and traditional stochastic data assimilation (DA) and parameter recovery (PR) algorithms given noisy data from chaotic systems. We use semi-analytic methods to numerically construct synthetic data from the Lorenz '63 and multiscale Lorenz '96 chaotic dynamical systems, adding white noise. Our findings show that, for different noise levels, deterministic PR algorithms paired with deterministic DA algorithms are shown computationally to be overall more accurate and stable than stochastic PR algorithms. Additionally, deterministic PR methods have demonstrated greater speed and efficiency, requiring less computational power than stochastic PR methods. This suggests that future work should consider exploring the full potential of deterministic PR algorithms in the presence of noise.

nlin.CD↗

arXiv2026

Quartic Lyapunov functions for global fluid stability

A fluid system is 'globally stable' if all initial conditions eventually converge to the same state. Since Reynolds (1895) and Orr (1907), the standard way to show global stability has been the energy method, which uses the fluctuation energy as a Lyapunov function. However, the energy method fails whenever transient energy growth is possible, so it often yields overly strict stability criteria. The first broadly applicable alternative has recently been introduced (Goulart & Chernyshenko 2012; Fuentes et al. 2022), using polynomial optimization to construct non-quadratic Lyapunov functions. Unlike the energy method, however, this approach is highly technical, computationally expensive, and hard to interpret physically. Moreover, it treats only one set of parameters at a time; in particular, if it verifies global stability at a certain Reynolds number, it does not imply the same for smaller values. The present work makes progress by connecting this numerical program with new analytical and physical insights. We show how to exploit symmetries of shear flows via convenient complex variable representations, greatly reducing the problem size. We then refine key inequalities, replace several expensive computational steps with simpler analytical alternatives, and show how to prove global stability over a range of Reynolds numbers. Our analysis identifies the simplest class of non-quadratic Lyapunov functions for two-dimensional parallel shear flows: a three-parameter family of quartic polynomials. Using these Lyapunov functions, we verify global stability of 2-D plane Couette flow and plane Poiseuille flow up to higher Reynolds numbers than possible with the energy method. Our work takes a step towards an analytical theory of global fluid stability beyond the energy method, and offers structural insights that should significantly improve future numerical investigations of global stability.

physics.flu-dyn↗

arXiv2026

Operator Learning for Smoothing and Forecasting

Machine learning has opened new frontiers in purely data-driven algorithms for data assimilation in, and for forecasting of, dynamical systems; the resulting methods are showing some promise. However, in contrast to model-driven algorithms, analysis of these data-driven methods is poorly developed. In this paper we address this issue, developing a theory to underpin data-driven methods to solve smoothing problems arising in data assimilation and forecasting problems. The theoretical framework relies on two key components: (i) establishing the existence of the mapping to be learned; (ii) the properties of the operator learning architecture used to approximate this mapping. By studying these two components in conjunction, we establish novel universal approximation theorems for purely data driven algorithms for both smoothing and forecasting of dynamical systems. We work in the continuous time setting, hence deploying neural operator architectures. The theoretical results are illustrated with experiments studying the Lorenz `63, Lorenz `96 and Kuramoto-Sivashinsky dynamical systems.

stat.ML↗

arXiv2025

Determining Modes, State Reconstruction, and Intertwinement: Existence of Self-Synchronizing Intertwinements

In the companion paper of the authors, a general synchronization framework was developed in the paradigmatic context of the 2D Navier-Stokes equations that allows one to precisely study the relation between the determining modes property of the corresponding dynamical system and the ability of certain continuous data assimilation algorithms to reconstruct unobserved state variables from sufficiently many observed state variables in this system, i.e., the reconstruction property. In this framework, the determining modes property and the reconstruction property can be viewed in a unified way as the ability of certain couplings of the Navier-Stokes equations to self-synchronize; due to the bi-directionality of coupling, the coupled system is referred to as an "intertwinement." A central achievement of this framework is to deduce a conceptual equivalence between the determining modes property and the reconstruction property of continuous data assimilation algorithms. In this paper, we prove that there are at least two non-trivial classes of intertwinements to which this framework applies. Moreover, these intertwinements encompass the continuous data assimilation algorithms studied in (Olson, Titi 2003) and (Azouani, Olson, Titi 2014) as special cases. Specifically, we show that there exist two types of intertwinements that are globally well-posed and we identify conditions under which these intertwinements self-synchronize. We emphasize that the intertwinements studied here can be induced by nonlinear perturbations of the underlying system, which are subsequently coupled bi-directionally to a copy of itself. Thus, establishing global well-posedness and suitable global-in-time uniform bounds, which are crucial to proving the claimed synchronization phenomenon, requires careful consideration.

math.AP↗

arXiv2024

Model discovery on the fly using continuous data assimilation

We review an algorithm developed for parameter estimation within the Continuous Data Assimilation (CDA) approach. We present an alternative derivation for the algorithm presented in a paper by Carlson, Hudson, and Larios (CHL, 2021). This derivation relies on the same assumptions as the previous derivation but frames the problem as a finite dimensional root-finding problem. Within the approach we develop, the algorithm developed in (CHL, 2021) is simply a realization of Newton's method. We then consider implementing other derivative based optimization algorithms; we show that the Levenberg Maqrquardt algorithm has similar performance to the CHL algorithm in the single parameter estimation case and generalizes much better to fitting multiple parameters. We then implement these methods in three example systems: the Lorenz '63 model, the two-layer Lorenz '96 model, and the Kuramoto-Sivashinsky equation.

math.NA↗

arXiv2024

On the nudging approach to continuous data assimilation in the limit of infinite error feedback gain

This article studies the intimate relationship between two filtering algorithms for continuous data assimilation, the synchronization filter and the nudging filter, in the paradigmatic context of the two-dimensional (2D) Navier-Stokes equations (NSE) for incompressible fluids. In this setting, the nudging filter can formally be viewed as an affine perturbation of the 2D NSE. Thus, in the degenerate limit of zero nudging parameter, the nudging filter converges to the solution of the 2D NSE. However, when the nudging parameter of the nudging filter is large, the perturbation becomes singular. It is shown that in the singular limit of infinite nudging parameter, the nudging filter converges to the synchronization filter. In establishing this result, the article fills a notable gap in the literature surrounding these algorithms. Numerical experiments are then presented that confirm the theoretical results and probes the issue of selecting a nudging strategy in the presence of observational noise. In this direction, an adaptive nudging strategy is proposed that leverages the insight gained from the relationship between the synchronization filter and the nudging filter that produces measurable improvement over the constant nudging strategy.

math.AP↗

arXiv2024

Determining Modes, State Reconstruction, and Intertwinement: A Synchronization Framework

This article studies the interrelation between the determining modes property in the two-dimensional (2D) Navier-Stokes equations (NSE) of incompressible fluids and the reconstruction property of two filtering algorithms for continuous data assimilation applied to the 2D NSE. These two properties are realized as manifestations of a more general phenomenon of "self-synchronous intertwinement." It is shown that this concept is a logically stronger form of asymptotic enslavement, as characterized by the existence of finitely many determining modes in the 2D NSE. In particular, this stronger form is shown to imply convergence of the direct-replacement filter and the nudging filter from continuous data assimilation (CDA), and then subsequently invoked to show that convergence in these filters implies that the 2D NSE possesses finitely many determining modes. The main achievement of this article is to therefore to develop a new conceptual framework, that of self-synchronous intertwinement, through which the precise inter-relationship between the determining modes property and synchronization phenomenon in these CDA filters is rigorously established and made decisively clear. The theoretical results are then complemented by numerical experiments that confirm the conclusions of the theorems.

math.AP↗

arXiv2023

Large-time behavior of the 2D thermally non-diffusive Boussinesq equations with Navier-slip boundary conditions

This paper investigates the large-time behavior of a buoyancy-driven fluid without thermal diffusion under Navier-slip boundary conditions in a bounded domain with Lipschitz-continuous second derivatives. After establishing improved regularity for classical solutions, we analyze their large-time asymptotics. Specifically, we show that the solutions converge to a state where, as $t \rightarrow \infty$, $\|u\|_{W^{1,p}} \rightarrow 0$, and hydrostatic balance is achieved in the weak topology of $L^2$. Furthermore, we identify the necessary conditions under which stable stratification and hydrostatic balance can be achieved in the strong topology as time approaches infinity. We then analyze a particular steady state, the hydrostatic equilibrium, characterized by $ u = 0 $, $ \theta = \beta x_2 + \gamma $, and $ p = \frac{\beta}{2}x_2^2 + \gamma x_2 + \delta $. In a periodic strip, we establish the linear stability of this state for $\beta > 0$, indicating that the temperature is vertically stably stratified. This work builds upon the results in [Doering et al.], which focus on free-slip boundary conditions, as well as recent studies [Ayd{\i}n, Kukavica, Ziane; Ayd{\i}n, Jayanti] that address no-slip boundary conditions. Notably, the novelty of this study lies in the ability to directly bound the pressure term, made possible by the Navier-slip boundary conditions.

math.AP↗

arXiv2023

Super-exponential convergence rate of a nonlinear continuous data assimilation algorithm: The 2D Navier-Stokes equations paradigm

We study a nonlinear-nudging modification of the Azouani-Olson-Titi continuous data assimilation (downscaling) algorithm for the 2D incompressible Navier-Stokes equations. We give a rigorous proof that the nonlinear-nudging system is globally well-posed, and moreover that its solutions converge to the true solution exponentially fast in time. Furthermore, we also prove that, once the error has decreased below a certain order one threshold, the convergence becomes double-exponentially fast in time, up until a precision determined by the sparsity of the observed data. In addition, we demonstrate the applicability of the analytical and sharpness of the results computationally.

math.AP↗

arXiv2021

Dynamically learning the parameters of a chaotic system using partial observations

Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we rigorously establish the convergence of this algorithm to the correct parameters when the system in question is the classic three-dimensional Lorenz system. Computationally, we demonstrate the efficacy of this algorithm on the Lorenz system by recovering any proper subset of the three non-dimensional parameters of the system, so long as a corresponding subset of the state is observable. We also provide computational evidence that this algorithm works well beyond the hypotheses required in the rigorous analysis, including in the presence of noisy observations, stochastic forcing, and the case where the observations are discrete and sparse in time.

math.CA↗

arXiv2020

Sensitivity Analysis for the 2D Navier-Stokes Equations with Applications to Continuous Data Assimilation

We rigorously prove the well-posedness of the formal sensitivity equations with respect to the Reynolds number corresponding to the 2D incompressible Navier-Stokes equations. Moreover, we do so by showing a sequence of difference quotients converges to the unique solution of the sensitivity equations for both the 2D Navier-Stokes equations and the related data assimilation equations, which utilize the continuous data assimilation algorithm proposed by Azouani, Olson, and Titi. As a result, this method of proof provides uniform bounds on difference quotients, demonstrating parameter recovery algorithms that change parameters as the system evolves will not blow-up. We also note that this appears to be the first such rigorous proof of global existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier-Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the Reynolds number.

math.AP↗

arXiv2018

Parameter Recovery and Sensitivity Analysis for the 2D Navier-Stokes Equations Via Continuous Data Assimilation

We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown Reynolds number. We determine the large-time error between the true solution of the 2D Navier-Stokes equations and the assimilated solution due to discrepancy between an approximate Reynolds number and the physical Reynolds number. Additionally, we develop an algorithm that can be run in tandem with the AOT algorithm to recover both the true solution and the Reynolds number (or equivalently the true viscosity) using only spatially discrete velocity measurements. The algorithm we propose involves changing the viscosity mid-simulation. Therefore, we also examine the sensitivity of the equations with respect to the Reynolds number. We prove that a sequence of difference quotients with respect to the Reynolds number converges to the unique solution of the sensitivity equations for both the 2D Navier-Stokes equations and the assimilated equations. We also note that this appears to be the first such rigorous proof of existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier-Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the Reynolds number.

math.AP↗