arXiv subjects

Evelyn Lunasin

Publications and source records attributed to Evelyn Lunasin.

15 recordsLinked to original sources

arXiv2024

Data-Driven Model Identification Using Time Delayed Nonlinear Maps for Systems with Multiple Attractors

This study presents a method, along with its algorithmic and computational framework implementation, and performance verification for dynamical system identification. The approach incorporates insights from phase space structures, such as attractors and their basins. By understanding these structures, we have improved training and testing strategies for operator learning and system identification. Our method uses time delay and non-linear maps rather than embeddings, enabling the assessment of algorithmic accuracy and expressibility, particularly in systems exhibiting multiple attractors. This method, along with its associated algorithm and computational framework, offers broad applicability across various scientific and engineering domains, providing a useful tool for data-driven characterization of systems with complex nonlinear system dynamics.

math.DS↗

arXiv2024

Probabilistic Eddy Identification with Uncertainty Quantification

Mesoscale eddies are critical in ocean circulation and the global climate system. Standard eddy identification methods are usually based on deterministic optimal point estimates of the ocean flow field. However, uncertainty exists in estimating the flow field due to noisy, sparse, and indirect observations and turbulent flow models. Because of the intrinsic strong nonlinearity in the eddy identification diagnostics, even a small uncertainty in estimating the flow field can cause a significant error in the identified eddies. This paper presents a general probabilistic eddy identification framework that adapts existing identification methods to incorporate uncertainty into the diagnostic, emphasizing the interaction between the uncertainty in state estimation and the nonlinearity in diagnostics for affecting the identification results. The probabilistic eddy identification framework starts by sampling an ensemble of flow realizations from the probabilistic state estimation, followed by applying traditional nonlinear eddy diagnostics to individual realizations. The corresponding eddy statistics are then aggregated from the diagnostic results based on these realizations. The framework is applied to a scenario mimicking the Beaufort Gyre marginal ice zone, where large uncertainty appears in estimating the ocean field using Lagrangian data assimilation with sparse ice floe trajectories. The skills in counting the number of eddies and computing the probability of each eddy event are significantly improved under the probabilistic framework. Notably, incorporating the nonlinear propagation of uncertainty in diagnostics provides a more accurate mean estimate than standard deterministic methods in estimating eddy lifetime. It also facilitates uncertainty quantification in inferring such a crucial dynamical quantity.

math.DS↗

arXiv2023

Launching Drifter Observations in the Presence of Uncertainty

Determining the optimal locations for placing extra observational measurements has practical significance. However, the exact underlying flow field is never known in practice. Significant uncertainty appears when the flow field is inferred from a limited number of existing observations via data assimilation or statistical forecast. In this paper, a new computationally efficient strategy for deploying Lagrangian drifters that highlights the central role of uncertainty is developed. A nonlinear trajectory diagnostic approach that underlines the importance of uncertainty is built to construct a phase portrait map. It consists of both the geometric structure of the underlying flow field and the uncertainty in the estimated state from Lagrangian data assimilation. The drifters are deployed at the maxima of this map and are required to be separated enough. Such a strategy allows the drifters to travel the longest distances to collect both the local and global information of the flow field. It also facilitates the reduction of a significant amount of uncertainty. To characterize the uncertainty, the estimated state is given by a probability density function (PDF). An information metric is then introduced to assess the information gain in such a PDF, which is fundamentally different from the traditional path-wise measurements. The information metric also avoids using the unknown truth to quantify the uncertainty reduction, making the method practical. Mathematical analysis exploiting simple illustrative examples is used to validate the strategy. Numerical simulations based on multiscale turbulent flows are then adopted to demonstrate the advantages of this strategy over some other methods.

physics.flu-dyn↗

arXiv2023

Lagrangian Descriptors with Uncertainty

Lagrangian descriptors provide a global dynamical picture of the geometric structures for arbitrarily time-dependent flows with broad applications. This paper develops a mathematical framework for computing Lagrangian descriptors when uncertainty appears. The uncertainty originates from estimating the underlying flow field as a natural consequence of data assimilation or statistical forecast. It also appears in the resulting Lagrangian trajectories. The uncertainty in the flow field directly affects the path integration of the crucial nonlinear positive scalar function in computing the Lagrangian descriptor, making it fundamentally different from many other diagnostic methods. Despite being highly nonlinear and non-Gaussian, closed analytic formulae are developed to efficiently compute the expectation of such a scalar function due to the uncertain velocity field by exploiting suitable approximations. A rapid and accurate sampling algorithm is then built to assist the forecast of the probability density function (PDF) of the Lagrangian trajectories. Such a PDF provides the weight to combine the Lagrangian descriptors along different paths. Simple but illustrative examples are designed to show the distinguished behavior of using Lagrangian descriptors in revealing the flow field when uncertainty appears. Uncertainty can either completely erode the coherent structure or barely affect the underlying geometry of the flow field. The method is also applied for eddy identification, indicating that uncertainty has distinct impacts on detecting eddies at different time scales. Finally, when uncertainty is incorporated into the Lagrangian descriptor for inferring the source target, the likelihood criterion provides a very different conclusion from the deterministic methods.

physics.flu-dyn↗

arXiv2021

An Efficient Continuous Data Assimilation Algorithm for the Sabra Shell Model of Turbulence

Complex nonlinear turbulent dynamical systems are ubiquitous in many areas. Recovering unobserved state variables is an important topic for the data assimilation of turbulent systems. In this article, an efficient continuous in time data assimilation scheme is developed, which exploits closed analytic formulae for updating the unobserved state variables. Therefore, it is computationally efficient and accurate. The new data assimilation scheme is combined with a simple reduced order modeling technique that involves a cheap closure approximation and a noise inflation. In such a way, many complicated turbulent dynamical systems can satisfy the requirements of the mathematical structures for the proposed efficient data assimilation scheme. The new data assimilation scheme is then applied to the Sabra shell model, which is a conceptual model for nonlinear turbulence. The goal is to recover the unobserved shell velocities across different spatial scales. It has been shown that the new data assimilation scheme is skillful in capturing the nonlinear features of turbulence including the intermittency and extreme events in both the chaotic and the turbulent dynamical regimes. It has also been shown that the new data assimilation scheme is more accurate and computationally cheaper than the standard ensemble Kalman filter and nudging data assimilation schemes for assimilating the Sabra shell model.

physics.flu-dyn↗

arXiv2017

Finite determining parameters feedback control for distributed nonlinear dissipative systems - a computational study

We present a computational study of a simple finite-dimensional feedback control algorithm for stabilizing solutions of infinite-dimensional dissipative evolution equations such as reaction-diffusion systems, the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. This feedback control scheme takes advantage of the fact that such systems possess finite number of determining parameters or degrees of freedom, namely, finite number of determining Fourier modes, determining nodes, and determining interpolants and projections. In particular, the feedback control scheme uses finitely many of such observables and controllers that are acting on the coarse spatial scales. We demonstrate our numerical results for the stabilization of the unstable zero solution of the 1D Chafee-Infante equation and 1D Kuramoto-Sivashinksky equation. We give rigorous stability analysis for the feedback control algorithm and derive sufficient conditions relating the control parameters and model parameter values to attune to the control objective.

math.AP↗

arXiv2017

A data assimilation algorithm: the paradigm of the 3D Leray-alpha model of turbulence

In this paper we survey the various implementations of a new data assimilation (downscaling) algorithm based on spatial coarse mesh measurements. As a paradigm, we demonstrate the application of this algorithm to the 3D Leray-$α$ subgrid scale turbulence model. Most importantly, we use this paradigm to show that it is not always necessary that one has to collect coarse mesh measurements of all the state variables, that are involved in the underlying evolutionary system, in order to recover the corresponding exact reference solution. Specifically, we show that in the case of the 3D Leray$-α$ model of turbulence the solutions of the algorithm, constructed using only coarse mesh observations of any two components of the three-dimensional velocity field, and without any information of the third component, converge, at an exponential rate in time, to the corresponding exact reference solution of the 3D Leray$-α$ model. This study serves as an addendum to our recent work on abridged continuous data assimilation for the 2D Navier-Stokes equations. Notably, similar results have also been recently established for the 3D viscous Planetary Geostrophic circulation model in which we show that coarse mesh measurements of the temperature alone are sufficient for recovering, through our data assimilation algorithm, the full solution; viz. the three components of velocity vector field and the temperature. Consequently, this proves the Charney conjecture for the 3D Planetary Geostrophic model; namely, that the history of the large spatial scales of temperature is sufficient for determining all the other quantities (state variables) of the model.

math.AP↗

arXiv2016

On the Charney Conjecture of Data Assimilation Employing Temperature Measurements Alone: The Paradigm of 3D Planetary Geostrophic Model

Analyzing the validity and success of a data assimilation algorithm when some state variable observations are not available is an important problem in meteorology and engineering. We present an improved data assimilation algorithm for recovering the exact full reference solution (i.e. the velocity and temperature) of the 3D Planetary Geostrophic model, at an exponential rate in time, by employing coarse spatial mesh observations of the temperature alone. This provides, in the case of this paradigm, a rigorous justification to an earlier conjecture of Charney which states that temperature history of the atmosphere, for certain simple atmospheric models, determines all other state variables.

math.AP↗

arXiv2016

Continuous Data Assimilation for a 2D Bénard Convection System through Horizontal Velocity Measurements Alone

In this paper we propose a continuous data assimilation (downscaling) algorithm for a two-dimensional Bénard convection problem. Specifically we consider the two-dimensional Boussinesq system of a layer of incompressible fluid between two solid horizontal walls, with no-normal flow and stress free boundary condition on the walls, and fluid is heated from the bottom and cooled from the top. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the horizontal velocity. We show that under an appropriate choice of the nudging parameter and the size of the spatial coarse mesh observables, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed data on the horizontal component of the velocity. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.

math.AP↗

arXiv2015

Data Assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements

In this paper we propose a continuous data assimilation (downscaling) algorithm for the Bénard convection in porous media using only coarse mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed (finite dimensional projection of) temperature data. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.

math.AP↗

arXiv2015

Abridged continuous data assimilation for the 2D Navier-Stokes equations utilizing measurements of only one component of the velocity field

We introduce a continuous data assimilation (downscaling) algorithm for the two-dimensional Navier-Stokes equations employing coarse mesh measurements of only one component of the velocity field. This algorithm can be implemented with a variety of finitely many observables: low Fourier modes, nodal values, finite volume averages, or finite elements. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate asymptotically in time, to the unique exact unknown reference solution, of the 2D Navier-Stokes equations, associated with the observed (finite dimensional projection of) velocity.

math.AP↗

arXiv2010

Global Well-posedness for The 2D Boussinesq System Without Heat Diffusion and With Either Anisotropic Viscosity or Inviscid Voigt-$α$ Regularization

We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with viscosity only in the horizontal direction, which arises in Ocean dynamics. This work improves the global well-posedness results established recently by R. Danchin and M. Paicu for the Boussinesq system with anisotropic viscosity and zero diffusion. Although we follow some of their ideas, in proving the uniqueness result, we have used an alternative approach by writing the transported temperature (density) as $θ= Δξ$ and adapting the techniques of V. Yudovich for the 2D incompressible Euler equations. This new idea allows us to establish uniqueness results with fewer assumptions on the initial data for the transported quantity $θ$. Furthermore, this new technique allows us to establish uniqueness results without having to resort to the paraproduct calculus of J. Bony. We also propose an inviscid $α$-regularization for the two-dimensional inviscid, non-diffusive Boussinesq system of equations, which we call the Boussinesq-Voigt equations. Global regularity of this system is established. Moreover, we establish the convergence of solutions of the Boussinesq-Voigt model to the corresponding solutions of the two-dimensional Boussinesq system of equations for inviscid flow without heat (density) diffusion on the interval of existence of the latter. Furthermore, we derive a criterion for finite-time blow-up of the solutions to the inviscid, non-diffusive 2D Boussinesq system based on this inviscid Voigt regularization. Finally, we propose a Voigt-$α$ regularization for the inviscid 3D Boussinesq equations with diffusion, and prove its global well-posedness. It is worth mentioning that our results are also valid in the presence of the $β$-plane approximation of the Coriolis force.

math.AP↗

arXiv2009

Analysis of a General Family of Regularized Navier-Stokes and MHD Models

We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n greater than or equal to 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-alpha-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. We give a unified analysis of the entire three-parameter family using only abstract mapping properties of the principle dissipation and smoothing operators, and then use specific parameterizations to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish results for singular perturbations, including the inviscid and alpha limits. Next we show existence of a global attractor for the general model, and give estimates for its dimension. We finish by establishing some results on determining operators for subfamilies of dissipative and non-dissipative models. In addition to establishing a number of results for all models in this general family, the framework recovers most of the previous results on existence, regularity, uniqueness, stability, attractor existence and dimension, and determining operators for well-known members of this family.

math.AP↗

arXiv2009

The Navier-Stokes-Voight Model for Image Inpainting

In 2001, Bertalmio et. al. drew an analogy between the image intensity function for the image inpainting problem and the stream function in a two-dimensional (2D) incompressible fluid. An approximate solution to the inpainting problem is obtained by numerically approximating the steady state solution of the 2D NSE vorticity transport equation, and simultaneously solving the Poisson problem between the vorticity and stream function, in the region to be inpainted. This elegant approach allows one to produce an approximate solution to the image inpainting problem by using techniques from computational fluid dynamics. Recently, the three-dimensional (3D) Navier-Stokes-Voight (NSV) model of viscoelastic fluid, was suggested by Cao et. al. as an inviscid regularization to the 3D Navier-Stokes equations (NSE). The NSV model is shown to be globally well-posed and has a finite-dimensional global attractor, making it an attractive sub-grid scale turbulence model for purposes of numerical simulation. In this paper we investigate the use of the 2D NSV model for use in algorithms for the inpainting problem. We also present some new theoretical results based on energy methods comparing the sufficient conditions for stability of the discretization scheme for the two model equations.

math.NA↗

arXiv2007

Spectral scaling of the Leray-$α$ model for two-dimensional turbulence

We present data from high-resolution numerical simulations of the Navier-Stokes-$α$ and the Leray-$α$ models for two-dimensional turbulence. It was shown previously (Lunasin et al., J. Turbulence, 8, (2007), 751-778), that for wavenumbers $k$ such that $kα\gg 1$, the energy spectrum of the smoothed velocity field for the two-dimensional Navier-Stokes-$α$ (NS-$α$) model scales as $k^{-7}$. This result is in agreement with the scaling deduced by dimensional analysis of the flux of the conserved enstrophy using its characteristic time scale. We therefore hypothesize that the spectral scaling of any $α$-model in the sub-$α$ spatial scales must depend only on the characteristic time scale and dynamics of the dominant cascading quantity in that regime of scales. The data presented here, from simulations of the two-dimensional Leray-$α$ model, confirm our hypothesis. We show that for $kα\gg 1$, the energy spectrum for the two-dimensional Leray-$α$ scales as $k^{-5}$, as expected by the characteristic time scale for the flux of the conserved enstrophy of the Leray-$α$ model. These results lead to our conclusion that the dominant directly cascading quantity of the model equations must determine the scaling of the energy spectrum.

physics.flu-dyn↗