Tensor-Based Dynamical Systems
Author | : Can Chen |
Publisher | : Springer Nature |
Total Pages | : 115 |
Release | : |
Genre | : |
ISBN | : 3031545052 |
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Author | : Can Chen |
Publisher | : Springer Nature |
Total Pages | : 115 |
Release | : |
Genre | : |
ISBN | : 3031545052 |
Author | : Can Chen |
Publisher | : Springer |
Total Pages | : 0 |
Release | : 2024-04-11 |
Genre | : Technology & Engineering |
ISBN | : 9783031545047 |
This book provides a comprehensive review on tensor algebra, including tensor products, tensor unfolding, tensor eigenvalues, and tensor decompositions. Tensors are multidimensional arrays generalized from vectors and matrices, which can capture higher-order interactions within multiway data. In addition, tensors have wide applications in many domains such as signal processing, machine learning, and data analysis, and the author explores the role of tensors/tensor algebra in tensor-based dynamical systems where system evolutions are captured through various tensor products. The author provides an overview of existing literature on the topic and aims to inspire readers to learn, develop, and apply the framework of tensor-based dynamical systems.
Author | : André H. Erhardt |
Publisher | : Frontiers Media SA |
Total Pages | : 209 |
Release | : 2023-02-15 |
Genre | : Science |
ISBN | : 2832514588 |
Author | : Péter Baranyi |
Publisher | : CRC Press |
Total Pages | : 262 |
Release | : 2018-09-03 |
Genre | : Technology & Engineering |
ISBN | : 1439818177 |
Tensor Product Model Transformation in Polytopic Model-Based Control offers a new perspective of control system design. Instead of relying solely on the formulation of more effective LMIs, which is the widely adopted approach in existing LMI-related studies, this cutting-edge book calls for a systematic modification and reshaping of the polytopic convex hull to achieve enhanced performance. Varying the convexity of the resulting TP canonical form is a key new feature of the approach. The book concentrates on reducing analytical derivations in the design process, echoing the recent paradigm shift on the acceptance of numerical solution as a valid form of output to control system problems. The salient features of the book include: Presents a new HOSVD-based canonical representation for (qLPV) models that enables trade-offs between approximation accuracy and computation complexity Supports a conceptually new control design methodology by proposing TP model transformation that offers a straightforward way of manipulating different types of convexity to appear in polytopic representation Introduces a numerical transformation that has the advantage of readily accommodating models described by non-conventional modeling and identification approaches, such as neural networks and fuzzy rules Presents a number of practical examples to demonstrate the application of the approach to generate control system design for complex (qLPV) systems and multiple control objectives. The authors’ approach is based on an extended version of singular value decomposition applicable to hyperdimensional tensors. Under the approach, trade-offs between approximation accuracy and computation complexity can be performed through the singular values to be retained in the process. The use of LMIs enables the incorporation of multiple performance objectives into the control design problem and assurance of a solution via convex optimization if feasible. Tensor Product Model Transformation in Polytopic Model-Based Control includes examples and incorporates MATLAB® Toolbox TPtool. It provides a reference guide for graduate students, researchers, engineers, and practitioners who are dealing with nonlinear systems control applications.
Author | : J. Nathan Kutz |
Publisher | : SIAM |
Total Pages | : 241 |
Release | : 2016-11-23 |
Genre | : Science |
ISBN | : 1611974496 |
Data-driven dynamical systems is a burgeoning field?it connects how measurements of nonlinear dynamical systems and/or complex systems can be used with well-established methods in dynamical systems theory. This is a critically important new direction because the governing equations of many problems under consideration by practitioners in various scientific fields are not typically known. Thus, using data alone to help derive, in an optimal sense, the best dynamical system representation of a given application allows for important new insights. The recently developed dynamic mode decomposition (DMD) is an innovative tool for integrating data with dynamical systems theory. The DMD has deep connections with traditional dynamical systems theory and many recent innovations in compressed sensing and machine learning. Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, the first book to address the DMD algorithm, presents a pedagogical and comprehensive approach to all aspects of DMD currently developed or under development; blends theoretical development, example codes, and applications to showcase the theory and its many innovations and uses; highlights the numerous innovations around the DMD algorithm and demonstrates its efficacy using example problems from engineering and the physical and biological sciences; and provides extensive MATLAB code, data for intuitive examples of key methods, and graphical presentations.
Author | : Michel Verhaegen |
Publisher | : Cambridge University Press |
Total Pages | : 287 |
Release | : 2022-05-12 |
Genre | : Technology & Engineering |
ISBN | : 1316515702 |
A comprehensive introduction to identifying network-connected systems, covering models and methods, and applications in adaptive optics.
Author | : Kevin Thomas Carlberg |
Publisher | : Stanford University |
Total Pages | : 130 |
Release | : 2011 |
Genre | : |
ISBN | : |
Despite the advent and maturation of high-performance computing, high-fidelity physics-based numerical simulations remain computationally intensive in many fields. As a result, such simulations are often impractical for time-critical applications such as fast-turnaround design, control, and uncertainty quantification. The objective of this thesis is to enable rapid, accurate analysis of high-fidelity nonlinear models to enable their use in time-critical settings. Model reduction presents a promising approach for realizing this goal. This class of methods generates low-dimensional models that preserves key features of the high-fidelity model. Such methods have been shown to generate fast, accurate solutions when applied to specialized problems such as linear time-invariant systems. However, model reduction techniques for highly nonlinear systems has been limited primarily to approaches based on the heuristic proper orthogonal decomposition (POD)--Galerkin approach. These methods often generate inaccurate responses because 1) POD--Galerkin does not generally minimize any measure of the system error, and 2) the POD basis is not constructed to minimize errors in the system's outputs of interest. Furthermore, simulation times for these models usually remain large, as reducing the dimension of a nonlinear system does not necessarily reduce its computational complexity. This thesis presents two model reduction techniques that addresses these shortcomings of the POD--Galerkin method. The first method is a `compact POD' approach for computing the small-dimensional trial basis; this approach is applicable to parameterized static systems. The compact POD basis is constructed using a goal-oriented framework that allows sensitivity derivatives to be employed as snapshots. The second method is a Gauss--Newton with approximated tensors (GNAT) method applicable to nonlinear systems. Similar to other POD-based approaches, the GNAT method first executes high-fidelity simulations during a costly `offline' stage; it computes a POD subspace that optimally represents the state as observed during these simulations. To compute fast, accurate `online' solutions, the method introduces two approximations that satisfy optimality and consistency conditions. First, the method decreases the system dimension by searching for the solutions in the low-dimensional POD subspace. As opposed to performing a Galerkin projection, the method handles the resulting overdetermined system of equations arising at each time step by formulating a least-squares problem; this ensures that a measure of the system error (i.e. the residual) is minimized. Second, the method decreases the model's computational complexity by approximating the residual and Jacobian using the `gappy POD' technique; this requires computing only a few rows of the approximated quantities. For computational mechanics problems, the GNAT method leads to the concept of a sample mesh: the subset of the mesh needed to compute the selected rows of the residual and Jacobian. Because the reduced-order model uses only the sample mesh for computations, the online stage requires minimal computational resources.
Author | : Daizhan Cheng |
Publisher | : Academic Press |
Total Pages | : 366 |
Release | : 2019-05-18 |
Genre | : Mathematics |
ISBN | : 0128178027 |
From Dimension-Free Matrix Theory to Cross-Dimensional Dynamic Systems illuminates the underlying mathematics of semi-tensor product (STP), a generalized matrix product that extends the conventional matrix product to two matrices of arbitrary dimensions. Dimension-varying systems feature prominently across many disciplines, and through innovative applications its newly developed theory can revolutionize large data systems such as genomics and biosystems, deep learning, IT, and information-based engineering applications. - Provides, for the first time, cross-dimensional system theory that is useful for modeling dimension-varying systems. - Offers potential applications to the analysis and control of new dimension-varying systems. - Investigates the underlying mathematics of semi-tensor product, including the equivalence and lattice structure of matrices and monoid of matrices with arbitrary dimensions.
Author | : Edoardo Angelo Di Napoli |
Publisher | : Frontiers Media SA |
Total Pages | : 192 |
Release | : 2022-11-08 |
Genre | : Science |
ISBN | : 2832504256 |
Author | : Shi-Ju Ran |
Publisher | : Springer Nature |
Total Pages | : 160 |
Release | : 2020-01-27 |
Genre | : Science |
ISBN | : 3030344894 |
Tensor network is a fundamental mathematical tool with a huge range of applications in physics, such as condensed matter physics, statistic physics, high energy physics, and quantum information sciences. This open access book aims to explain the tensor network contraction approaches in a systematic way, from the basic definitions to the important applications. This book is also useful to those who apply tensor networks in areas beyond physics, such as machine learning and the big-data analysis. Tensor network originates from the numerical renormalization group approach proposed by K. G. Wilson in 1975. Through a rapid development in the last two decades, tensor network has become a powerful numerical tool that can efficiently simulate a wide range of scientific problems, with particular success in quantum many-body physics. Varieties of tensor network algorithms have been proposed for different problems. However, the connections among different algorithms are not well discussed or reviewed. To fill this gap, this book explains the fundamental concepts and basic ideas that connect and/or unify different strategies of the tensor network contraction algorithms. In addition, some of the recent progresses in dealing with tensor decomposition techniques and quantum simulations are also represented in this book to help the readers to better understand tensor network. This open access book is intended for graduated students, but can also be used as a professional book for researchers in the related fields. To understand most of the contents in the book, only basic knowledge of quantum mechanics and linear algebra is required. In order to fully understand some advanced parts, the reader will need to be familiar with notion of condensed matter physics and quantum information, that however are not necessary to understand the main parts of the book. This book is a good source for non-specialists on quantum physics to understand tensor network algorithms and the related mathematics.