Tensor-Based Dynamical Systems

Tensor-Based Dynamical Systems
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.

Tensor Calculus and Analytical Dynamics

Tensor Calculus and Analytical Dynamics
Author: John G. Papastavridis
Publisher: Routledge
Total Pages: 444
Release: 2018-12-12
Genre: Mathematics
ISBN: 1351411616

Tensor Calculus and Analytical Dynamics provides a concise, comprehensive, and readable introduction to classical tensor calculus - in both holonomic and nonholonomic coordinates - as well as to its principal applications to the Lagrangean dynamics of discrete systems under positional or velocity constraints. The thrust of the book focuses on formal structure and basic geometrical/physical ideas underlying most general equations of motion of mechanical systems under linear velocity constraints. Written for the theoretically minded engineer, Tensor Calculus and Analytical Dynamics contains uniquely accessbile treatments of such intricate topics as: tensor calculus in nonholonomic variables Pfaffian nonholonomic constraints related integrability theory of Frobenius The book enables readers to move quickly and confidently in any particular geometry-based area of theoretical or applied mechanics in either classical or modern form.

Dynamic Mode Decomposition

Dynamic Mode Decomposition
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.

Numerical Analysis meets Machine Learning

Numerical Analysis meets Machine Learning
Author:
Publisher: Elsevier
Total Pages: 590
Release: 2024-06-13
Genre: Mathematics
ISBN: 0443239851

Numerical Analysis Meets Machine Learning series, highlights new advances in the field, with this new volume presenting interesting chapters. Each chapter is written by an international board of authors. - Provides the authority and expertise of leading contributors from an international board of authors - Presents the latest release in the Handbook of Numerical Analysis series - Updated release includes the latest information on the Numerical Analysis Meets Machine Learning

Data-Driven Identification of Networks of Dynamic Systems

Data-Driven Identification of Networks of Dynamic Systems
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.

Dynamic Network Representation Based on Latent Factorization of Tensors

Dynamic Network Representation Based on Latent Factorization of Tensors
Author: Hao Wu
Publisher: Springer Nature
Total Pages: 89
Release: 2023-03-07
Genre: Computers
ISBN: 9811989346

A dynamic network is frequently encountered in various real industrial applications, such as the Internet of Things. It is composed of numerous nodes and large-scale dynamic real-time interactions among them, where each node indicates a specified entity, each directed link indicates a real-time interaction, and the strength of an interaction can be quantified as the weight of a link. As the involved nodes increase drastically, it becomes impossible to observe their full interactions at each time slot, making a resultant dynamic network High Dimensional and Incomplete (HDI). An HDI dynamic network with directed and weighted links, despite its HDI nature, contains rich knowledge regarding involved nodes’ various behavior patterns. Therefore, it is essential to study how to build efficient and effective representation learning models for acquiring useful knowledge. In this book, we first model a dynamic network into an HDI tensor and present the basic latent factorization of tensors (LFT) model. Then, we propose four representative LFT-based network representation methods. The first method integrates the short-time bias, long-time bias and preprocessing bias to precisely represent the volatility of network data. The second method utilizes a proportion-al-integral-derivative controller to construct an adjusted instance error to achieve a higher convergence rate. The third method considers the non-negativity of fluctuating network data by constraining latent features to be non-negative and incorporating the extended linear bias. The fourth method adopts an alternating direction method of multipliers framework to build a learning model for implementing representation to dynamic networks with high preciseness and efficiency.

Recent Advances in 3D Imaging, Modeling, and Reconstruction

Recent Advances in 3D Imaging, Modeling, and Reconstruction
Author: Voulodimos, Athanasios
Publisher: IGI Global
Total Pages: 396
Release: 2020-02-28
Genre: Computers
ISBN: 1522552952

3D image reconstruction is used in many fields, such as medicine, entertainment, and computer science. This highly demanded process comes with many challenges, such as images becoming blurry by atmospheric turbulence, getting snowed with noise, or becoming damaged within foreign regions. It is imperative to remain well-informed with the latest research in this field. Recent Advances in 3D Imaging, Modeling, and Reconstruction is a collection of innovative research on the methods and common techniques of image reconstruction as well as the accuracy of these methods. Featuring coverage on a wide range of topics such as ray casting, holographic techniques, and machine learning, this publication is ideally designed for graphic designers, computer engineers, medical professionals, robotics engineers, city planners, game developers, researchers, academicians, and students.

Manifolds, Tensor Analysis, and Applications

Manifolds, Tensor Analysis, and Applications
Author: Ralph Abraham
Publisher: Springer Science & Business Media
Total Pages: 666
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461210291

The purpose of this book is to provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors, and differential forms. Some applications to Hamiltonian mechanics, fluid me chanics, electromagnetism, plasma dynamics and control thcory arc given in Chapter 8, using both invariant and index notation. The current edition of the book does not deal with Riemannian geometry in much detail, and it does not treat Lie groups, principal bundles, or Morse theory. Some of this is planned for a subsequent edition. Meanwhile, the authors will make available to interested readers supplementary chapters on Lie Groups and Differential Topology and invite comments on the book's contents and development. Throughout the text supplementary topics are given, marked with the symbols ~ and {l:;J. This device enables the reader to skip various topics without disturbing the main flow of the text. Some of these provide additional background material intended for completeness, to minimize the necessity of consulting too many outside references. We treat finite and infinite-dimensional manifolds simultaneously. This is partly for efficiency of exposition. Without advanced applications, using manifolds of mappings, the study of infinite-dimensional manifolds can be hard to motivate.

Data-Driven Methods for Dynamic Systems

Data-Driven Methods for Dynamic Systems
Author: Jason Bramburger
Publisher: SIAM
Total Pages: 180
Release: 2024-11-05
Genre: Science
ISBN: 1611978165

As experimental data sets have grown and computational power has increased, new tools have been developed that have the power to model new systems and fundamentally alter how current systems are analyzed. This book brings together modern computational tools to provide an accurate understanding of dynamic data. The techniques build on pencil-and-paper mathematical techniques that go back decades and sometimes even centuries. The result is an introduction to state-of-the-art methods that complement, rather than replace, traditional analysis of time-dependent systems. Data-Driven Methods for Dynamic Systems provides readers with methods not found in other texts as well as novel ones developed just for this book; an example-driven presentation that provides background material and descriptions of methods without getting bogged down in technicalities; and examples that demonstrate the applicability of a method and introduce the features and drawbacks of their application. The online supplementary material includes a code repository that can be used to reproduce every example and that can be repurposed to fit a variety of applications not found in the book. This book is intended as an introduction to the field of data-driven methods for graduate students. It will also be of interest to researchers who want to familiarize themselves with the discipline. It can be used in courses on dynamical systems, differential equations, and data science.