Methods For Data Assimilation In Chaotic Systems
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| Author | : William G. Whartenby |
| Publisher | : |
| Total Pages | : 92 |
| Release | : 2012 |
| Genre | : |
| ISBN | : 9781267412324 |
Data assimilation has wide ranging applications, including neuroscience, oceanography and climate science. In this dissertation we will examine data assimilation as a tool for systems of partial differential equations on a discretized spacial grid, using simple geophysical models as a twin for our study. We will use the 1 layer shallow water equations (SWE), and describe how to extend the method to a 2 layer SWE. Although we only used the SWE for this dissertation, we examine how we would use the barotropic vorticity equations (BVE) as the twin in the same study. We will examine two different methods for performing data assimilation on chaotic systems. The first method relies on the measurements to smooth the synchronization manifold, allowing a nonlinear optimizer to correctly determine the most likely path, or the path which minimizes the cost function. The second method we call Metropolis-Hastings Monte Carlo (MHMC) integration scheme. MHMC also allows retention of a group of path samples whose statistics reflect the probability of each path, allowing histograms of state vector values for analysis or inputs to particle filter methods for prediction. The study uses MHMC with the SWE as twin. in this chapter we will examine a data set used for the study. We then describe he various numbers of state vectors needed as data, and the increase in the quality of the fit. We determine the number of state vectors needed as measurements to accurately predict the unmeasured ones.
| Author | : Mark Asch |
| Publisher | : SIAM |
| Total Pages | : 310 |
| Release | : 2016-12-29 |
| Genre | : Mathematics |
| ISBN | : 1611974534 |
Data assimilation is an approach that combines observations and model output, with the objective of improving the latter. This book places data assimilation into the broader context of inverse problems and the theory, methods, and algorithms that are used for their solution. It provides a framework for, and insight into, the inverse problem nature of data assimilation, emphasizing why and not just how. Methods and diagnostics are emphasized, enabling readers to readily apply them to their own field of study. Readers will find a comprehensive guide that is accessible to nonexperts; numerous examples and diverse applications from a broad range of domains, including geophysics and geophysical flows, environmental acoustics, medical imaging, mechanical and biomedical engineering, economics and finance, and traffic control and urban planning; and the latest methods for advanced data assimilation, combining variational and statistical approaches.
| Author | : Axel Hutt |
| Publisher | : Frontiers Media SA |
| Total Pages | : 116 |
| Release | : 2019-08-16 |
| Genre | : |
| ISBN | : 2889459853 |
The understanding of complex systems is a key element to predict and control the system’s dynamics. To gain deeper insights into the underlying actions of complex systems today, more and more data of diverse types are analyzed that mirror the systems dynamics, whereas system models are still hard to derive. Data assimilation merges both data and model to an optimal description of complex systems’ dynamics. The present eBook brings together both recent theoretical work in data assimilation and control and demonstrates applications in diverse research fields.
| Author | : Rodolfo Guzzi |
| Publisher | : Springer |
| Total Pages | : 140 |
| Release | : 2015-09-16 |
| Genre | : Computers |
| ISBN | : 3319224107 |
This book endeavours to give a concise contribution to understanding the data assimilation and related methodologies. The mathematical concepts and related algorithms are fully presented, especially for those facing this theme for the first time. The first chapter gives a wide overview of the data assimilation steps starting from Gauss' first methods to the most recent as those developed under the Monte Carlo methods. The second chapter treats the representation of the physical system as an ontological basis of the problem. The third chapter deals with the classical Kalman filter, while the fourth chapter deals with the advanced methods based on recursive Bayesian Estimation. A special chapter, the fifth, deals with the possible applications, from the first Lorenz model, passing trough the biology and medicine up to planetary assimilation, mainly on Mars. This book serves both teachers and college students, and other interested parties providing the algorithms and formulas to manage the data assimilation everywhere a dynamic system is present.
| Author | : Henry D. I. Abarbanel |
| Publisher | : Cambridge University Press |
| Total Pages | : 207 |
| Release | : 2022-02-17 |
| Genre | : Computers |
| ISBN | : 1316519635 |
The theory of data assimilation and machine learning is introduced in an accessible manner for undergraduate and graduate students.
| Author | : Kangbo Hao |
| Publisher | : |
| Total Pages | : 99 |
| Release | : 2020 |
| Genre | : |
| ISBN | : |
In the study of data assimilation, people focus on estimating state variables and parameters of dynamical models, and make predictions forward in time, using given observations. It is a method that has been applied to many different fields, such as numerical weather prediction and neurobiology. To make successful estimations and predictions using data assimilation methods, there are a few difficulties that are often encountered. First is the quantity and quality of the data. In some of the typical problems in data assimilation, the number of observations are usually a few order of magnitude smaller than the number of total variables. Considering this and the fact that almost all the data gathered are noisy, how to estimate the observed and unobserved state variables and make good predictions using the noisy and incomplete data is one of the key challenge in data assimilation. Another issue arises from the dynamical model. Most of the interesting models are non-linear, and usually chaotic, which means that a small error in the estimation will grow exponentially over time. This property of the chaotic system addresses the necessity of accurate estimations of variables. In this thesis, I will start with an overview of data assimilation, by formulating the problem that data assimilation tries to solve, and introducing several widely used methods. Then I will explain the Precision Annealing Monte Carlo method that has been developed in the group, as well as its variation using Hamiltonian Monte Carlo. Finally I will demonstrate a few example problems that can be solved using data assimilation methods, varying from a simple but instructional 20-dimension Lorenz 96 model, to a complicated ocean model named Regional Ocean Modeling System.
| Author | : Seon Ki Park |
| Publisher | : Cambridge University Press |
| Total Pages | : 413 |
| Release | : 2022-09-29 |
| Genre | : Science |
| ISBN | : 1108831761 |
A unique combination of both theoretical and practical aspects of data assimilation with examples and exercises for students.
| Author | : Seon Ki Park |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 736 |
| Release | : 2013-05-22 |
| Genre | : Science |
| ISBN | : 3642350887 |
This book contains the most recent progress in data assimilation in meteorology, oceanography and hydrology including land surface. It spans both theoretical and applicative aspects with various methodologies such as variational, Kalman filter, ensemble, Monte Carlo and artificial intelligence methods. Besides data assimilation, other important topics are also covered including targeting observation, sensitivity analysis, and parameter estimation. The book will be useful to individual researchers as well as graduate students for a reference in the field of data assimilation.
| Author | : Zhe An |
| Publisher | : |
| Total Pages | : 111 |
| Release | : 2019 |
| Genre | : |
| ISBN | : |
The problem of forecasting the behavior of a complex dynamical system through analysis of observational time-series data becomes difficult when the system expresses chaotic behavior and the measurements are sparse, in both space and/or time. Despite the fact that this situation is quite typical across many fields, including numerical weather prediction, the issue of whether the available observations are 'sufficient' for generating successful forecasts is still not well understood. The goal of this study is to develop the assimilation schemes using the time series observation. Several approaches to Lorenz 96 chaotic dynamics and geophysical fluid dynamics have been proposed. Chapter 1 and 2 contain the problem statement of data assimilation and description of two nonlinear models, the Lorenz 96 model, a toy model commonly used in data assimilation, and the shallow water model, a fast but relatively realistic model in geophysical fluid dynamics. The details of the simulations are provided. In Chapter 3, we will systematically compare the data assimilation methodology in literature. We will focus on the advantages and disadvantages of each method. In Chapter 4, the time delayed nudging method with the time delayed observation data is presented. We show that in certain circumstances, it provides a sizable reduction in the number of observations required to construct accurate estimates and high-quality predictions. In particular, we find that this estimate of 70% can be reduced to about 33% using time delays, and even further if Lagrangian drifter locations are also used as measurements. Chapter 5, we apply the sequential adversarial method on Lorenz 96 model. We find that the adversarial method can highly reduce the task burden of the parameter tuning in the data assimilation algorithm. We obtain a substantial improvement in the traditional sequential data assimilation scheme. Chapter 6 will be the conclusion and discussion on future work.
| Author | : Aleksandra Shirman |
| Publisher | : |
| Total Pages | : 101 |
| Release | : 2018 |
| Genre | : |
| ISBN | : |
Data Assimilation (DA) is a method through which information is extracted from measured quantities and with the help of a mathematical model is transferred through a probability distribution to unknown or unmeasured states and parameters characterizing the system of study. With an estimate of the model paramters, quantitative predictions may be made and compared to subsequent data. Many recent DA efforts rely on an probability distribution optimization that locates the most probable state and parameter values given a set of data. The procedure developed and demonstrated here extends the optimization by appending a biased random walk around the states and parameters of high probability to generate an estimate of the structure in state space of the probability density function (PDF). The estimate of the structure of the PDF will facilitate more accurate estimates of expectation values of means, standard deviations and higher moments of states and parameters that characterize the behavior of the system of study. The ability to calculate these expectation values will allow for an error bar or tolerance interval to be attached to each estimated state or parameter, in turn giving significance to any results generated. The estimation method's merits will be demonstrated on a simulated well known chaotic system, the Lorenz 96 system, and on a toy model of a neuron. In both situations the model system provides unique challenges for estimation: In chaotic systems any small error in estimation generates extremely large prediction errors while in neurons only one of the (at minimum) four dynamical variables can be measured leading to a small amount of data with which to work. This thesis will conclude with an exploration of the equivalence of machine learning and the formulation of statistical DA. The application of previous DA methods are demonstrated on the classic machine learning problem: the characterization of handwritten images from the MNIST data set. The results of this work are used to validate common assumptions in machine learning work such as the dependence of the quality of results on the amount of data presented and the size of the network used. Finally DA is proposed as a method through which to discern an 'ideal' network size for a set of given data which optimizes predictive capabilities while minimizing computational costs.
