arXiv subjects

Collin Victor

Publications and source records attributed to Collin Victor.

12 recordsLinked to original sources

arXiv2026

A critical note on back-and-forth Data Assimilation Nudging Algorithm

This work investigates the effectiveness of the Back-and-Forth Nudging (BFN) data assimilation algorithm, specifically its performance when employing the Azouani-Olson-Titi (AOT) continuous data assimilation downscaling nudging algorithm, for recovering initial conditions of dissipative dynamical systems. Contrary to previous reports in the literature, we show that, for several systems of interest, one can construct initial conditions that BFN cannot reliably recover. Our key finding is the construction of infinitely many distinct solutions for certain dissipative systems that share identical spatially sparse observational data. Since these observations are indistinguishable, no data assimilation method relying only on them can differentiate between these solutions or recover the correct initial condition. We illustrate these pathological initial conditions for the Lorenz 1963 model and several 1D partial differential equations: the heat equation, viscous linear transport, and viscous and inviscid Burgers equations. Our analytical results are supported by numerical simulations. To address the numerical ill-posedness of backward-in-time iterations, an essential step of BFN for dissipative models, we introduce a regularized backward step, the Voigt-regularized BFN (VBFN). We investigate its performance for the 2D Navier-Stokes equations and the viscous 1D KdV equation, comparing it with standard BFN and Diffusive BFN (dBFN). While VBFN improves numerical stability and reduces model bias relative to dBFN, it still cannot reconstruct the unobserved fine spatial scales of the reference solution. This reinforces our main conclusion: even with regularization, BFN-type algorithms are limited in recovering the full state, and in particular the initial data, from sparse spatial observations.

math.AP↗

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

On the inadequacy of nudging data assimilation algorithms for non-dissipative systems

In this work, we study the applicability of the Azouani-Olson-Titi (AOT) nudging algorithm for continuous data assimilation to evolutionary dynamical systems that are not dissipative. Specifically, we apply the AOT algorithm to a partially dissipative variant of the Lorenz 1963 system, the Korteweg-de Vries equation (KdV) in 1D, and the 2D incompressible Euler equations. Our analysis reveals that both the Euler and KdV equations lack the finitely many determining modes property, leading to the construction of infinitely many solutions with exactly the same sparse observational data, which data assimilation methods cannot distinguish between. Simultaneously, we numerically verify that the AOT algorithm successfully recovers these counterexamples for the damped and driven KdV equation, which is dissipative. Additionally, to further support our argument, we present numerical evidence showing that the AOT algorithm is ineffective at accurately recovering solutions for a partially dissipative variant of the Lorenz 1963 system.

math.OC↗

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↗

arXiv2024

An Assessment of Ensemble Kalman Filter and Azouani-Olson-Titi Algorithms for Data Assimilation: A Comparative Study

Continuous data assimilation (CDA) is a method that continuously integrates observational data into a dynamical system to improve model accuracy in real-time. The AOT algorithm is one of the most widely used methods in CDA due to its efficiency in incorporating observational data to enhance model accuracy. However, no research to date has evaluated the performance of the AOT algorithm compared to the most widely used DA method, the ensemble Kalman filter (EnKF). Hence, in this paper, we conduct an extensive numerical examination to evaluate and compare these two algorithms for CDA problems with measurement error, addressing this gap. By analyzing the one-dimensional Kuramoto-Sivashinsky equation and the two-dimensional Navier-Stokes equation, which are central to many applications and representative in CDA problems, we found a significant computational advantage of the AOT algorithm.

math.OC↗

arXiv2023

Application of Continuous Data Assimilation in High-Resolution Ocean Modeling

We demonstrate a formulation of the Azouani-Olson-Titi (AOT) algorithm in the MPAS-Ocean implementation of the primitive equations of the ocean, presenting global ocean simulations with realistic coastlines and bathymetry. We observe an exponentially fast decay in the error before reaching a certain error level, which depends on the terms involved and whether the AOT feedback control term was handled implicitly or explicitly. A wide range of errors was observed for both schemes, with the implicit scheme typically exhibiting lower error levels, depending on the specific physical terms included in the model. Several factors seem to be contributing to this wide range, but the vertical mixing term is demonstrated to be an especially problematic term. This study provides insight into the promises and challenges of adapting the AOT algorithm to the setting of high-resolution, realistic ocean models.

math.AP↗

arXiv2023

Continuous Data Assimilation for the 3D and Higher-Dimensional Navier--Stokes equations with Higher-Order Fractional Diffusion

We study the use of the Azouani-Olson-Titi (AOT) continuous data assimilation algorithm to recover solutions of the Navier--Stokes equations modified to have higher-order fractional diffusion. The fractional diffusion case is of particular interest, as it is known to be globally well-posed for sufficiently large diffusion exponent $\alpha$. In this work, we prove that the assimilation equations are globally well-posed, and we demonstrate that the solutions produced by the AOT algorithm exhibit exponential convergence in time to the reference solution, given a sufficiently high spatial resolution of observations and a sufficiently large nudging parameter. We also note that the results hold in spatial dimensions $d$ where $2\leq d\leq 8$, so long as $\alpha\geq \frac12 +\frac{d}{4}$. Though the cases $3<d\leq8$ are likely only a mathematical curiosity, we include them as they cause no additional difficulty in the proof. Note that we show in a companion paper the $d=2$ case allows for $\alpha<1$.

math.AP↗

arXiv2023

The second-best way to do sparse-in-time continuous data assimilation: Improving convergence rates for the 2D and 3D Navier-Stokes equations

We study different approaches to implementing sparse-in-time observations into the the Azouani-Olson-Titi data assimilation algorithm. We propose a new method which introduces a "data assimilation window" separate from the observational time interval. We show that by making this window as small as possible, we can drastically increase the strength of the nudging parameter without losing stability. Previous methods used old data to nudge the solution until a new observation was made. In contrast, our method stops nudging the system almost immediately after an observation is made, allowing the system relax to the correct physics. We show that this leads to an order-of-magnitude improvement in the time to convergence in our 3D Navier-Stokes simulations. Moreover, our simulations indicate that our approach converges at nearly the same rate as the idealized method of direct replacement of low Fourier modes proposed by Hayden, Olson, and Titi (HOT). However, our approach can be readily adapted to non-idealized settings, such as finite element methods, finite difference methods, etc., since there is no need to access Fourier modes as our method works for general interpolants. It is in this sense that we think of our approach as ``second best;'' that is, the ``best'' method would be the direct replacement of Fourier modes as in HOT, but this idealized approach is typically not feasible in physically realistic settings. While our method has a convergence rate that is slightly sub-optimal compared to the idealized method, it is directly compatible with real-world applications. Moreover, we prove analytically that these new algorithms are globally well-posed, and converge to the true solution exponentially fast in time. In addition, we provide the first 3D computational validation of HOT algorithm.

math.AP↗

arXiv2021

The Bleeps, the Sweeps, and the Creeps: Convergence Rates for Dynamic Observer Patterns via Data Assimilation for the 2D Navier-Stokes Equations

We adapt a continuous data assimilation scheme, known as the Azouani-Olson-Titi (AOT) algorithm, to the case of moving observers for the 2D incompressible Navier-Stokes equations. We propose and test computationally several movement patterns (which we refer to as "the bleeps, the sweeps and the creeps"), as well as Lagrangian motion and combinations of these patterns, in comparison with static (i.e. non-moving) observers. In several cases, order-of-magnitude improvements in terms of the time-to-convergence are observed. We end with a discussion of possible applications to real-world data collection strategies that may lead to substantial improvements in predictive capabilities.

math.AP↗

arXiv2018

Continuous Data Assimilation with a Moving Cluster of Data Points for a Reaction Diffusion Equation: A Computational Study

Data assimilation is a technique for increasing the accuracy of simulations of solutions to partial differential equations by incorporating observable data into the solution as time evolves. Recently, a promising new algorithm for data assimilation based on feedback-control at the PDE level has been proposed in the pioneering work of Azouani, Olson, and Titi (2014). The standard version of this algorithm is based on measurement from data points that are fixed in space. In this work, we consider the scenario in which the data collection points move in space over time. We demonstrate computationally that, at least in the setting of the 1D Allen-Cahn reaction diffusion equations, the algorithm converges with significantly fewer measurement points, up to an order or magnitude in some cases. We also provide an application of the algorithm to an inverse problem in the case of a uniform static grid.

math.AP↗

arXiv2018

Gradient Sensing via Cell Communication

Experimental evidence lends support to the conjecture that the ability of chains of cells to sense the gradient of an external chemical concentration could rely on cell-to-cell communication. This is the basis for the gradient sensing nature of a specific model type of the Local Excitation, Global Inhibition (LEGI) principle, wherein the strength of the external chemical field is sensed through a comparison between a local exciting species and a global inhibitor that is shared via intra-cellular reactions in the cell chain. In this study we generalize the nearest neighbor communication mechanism in the above-mentioned LEGI model in order to explore how the chemical sensing characteristics depend on the parameterization of the communication itself, cell size, and the radius of influence of neighboring cells. It was found that the radius of influence was less important than the approximating model for communication. Higher order approximations to the communication mechanism were better able to sense an external gradient. However, an analysis of the signal to noise ratio established that higher order models for communication were more prone to noise and thus have a lower signal to noise ratio. The generalization as well as the tools used in the analysis of the dynamics can be extended to more heterogeneous networks and can thus prove useful in using models and observations in the process of understanding chemical gradient via LEGI models with a communication component.

q-bio.QM↗