Tensors Structured Numerical Methods In Scientific Computing
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Tensor Numerical Methods In Scientific Computing
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Author : Boris N. Khoromskij
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2018-06-11
Tensor Numerical Methods In Scientific Computing written by Boris N. Khoromskij and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-06-11 with Mathematics categories.
The most difficult computational problems nowadays are those of higher dimensions. This research monograph offers an introduction to tensor numerical methods designed for the solution of the multidimensional problems in scientific computing. These methods are based on the rank-structured approximation of multivariate functions and operators by using the appropriate tensor formats. The old and new rank-structured tensor formats are investigated. We discuss in detail the novel quantized tensor approximation method (QTT) which provides function-operator calculus in higher dimensions in logarithmic complexity rendering super-fast convolution, FFT and wavelet transforms. This book suggests the constructive recipes and computational schemes for a number of real life problems described by the multidimensional partial differential equations. We present the theory and algorithms for the sinc-based separable approximation of the analytic radial basis functions including Green’s and Helmholtz kernels. The efficient tensor-based techniques for computational problems in electronic structure calculations and for the grid-based evaluation of long-range interaction potentials in multi-particle systems are considered. We also discuss the QTT numerical approach in many-particle dynamics, tensor techniques for stochastic/parametric PDEs as well as for the solution and homogenization of the elliptic equations with highly-oscillating coefficients. Contents Theory on separable approximation of multivariate functions Multilinear algebra and nonlinear tensor approximation Superfast computations via quantized tensor approximation Tensor approach to multidimensional integrodifferential equations
Tensors Structured Numerical Methods In Scientific Computing
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Author : Boris N. Choromskij
language : en
Publisher:
Release Date : 2010
Tensors Structured Numerical Methods In Scientific Computing written by Boris N. Choromskij and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with categories.
In the present paper, we give a survey of the recent results and outline future prospects of the tensor-structured numerical methods in applications to multidimensional problems in scientific computing. The guiding principle of the tensor methods is an approximation of multivariate functions and operators relying on certain separation of variables. Along with the traditional canonical and Tucker models, we focus on the recent quantics-TT tensor approximation method that allows to represent N-d tensors with log-volume complexity, O(dlog N). We outline how these methods can be applied in the framework of tensor truncated iteration for the solution of the high-dimensional elliptic/parabolic equations and parametric PDEs. Numerical examples demonstrate that the tensor-structured methods have proved their value in application to various computational problems arising in quantum chemistry and in the multi-dimensional/parametric FEM/BEM modelingthe tool apparently works and gives the promise for future use in challenging high-dimensional applications.
High Performance Tensor Computations In Scientific Computing And Data Science
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Author : Edoardo Angelo Di Napoli
language : en
Publisher: Frontiers Media SA
Release Date : 2022-11-08
High Performance Tensor Computations In Scientific Computing And Data Science written by Edoardo Angelo Di Napoli and has been published by Frontiers Media SA this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-11-08 with Science categories.
Model Reduction And Approximation
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Author : Peter Benner
language : en
Publisher: SIAM
Release Date : 2017-07-06
Model Reduction And Approximation written by Peter Benner and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-07-06 with Science categories.
Many physical, chemical, biomedical, and technical processes can be described by partial differential equations or dynamical systems. In spite of increasing computational capacities, many problems are of such high complexity that they are solvable only with severe simplifications, and the design of efficient numerical schemes remains a central research challenge. This book presents a tutorial introduction to recent developments in mathematical methods for model reduction and approximation of complex systems. Model Reduction and Approximation: Theory and Algorithms contains three parts that cover (I) sampling-based methods, such as the reduced basis method and proper orthogonal decomposition, (II) approximation of high-dimensional problems by low-rank tensor techniques, and (III) system-theoretic methods, such as balanced truncation, interpolatory methods, and the Loewner framework. It is tutorial in nature, giving an accessible introduction to state-of-the-art model reduction and approximation methods. It also covers a wide range of methods drawn from typically distinct communities (sampling based, tensor based, system-theoretic).?? This book is intended for researchers interested in model reduction and approximation, particularly graduate students and young researchers.
Advances In Data Analysis With Computational Intelligence Methods
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Author : Adam E Gawęda
language : en
Publisher: Springer
Release Date : 2017-09-21
Advances In Data Analysis With Computational Intelligence Methods written by Adam E Gawęda and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-21 with Technology & Engineering categories.
This book is a tribute to Professor Jacek Żurada, who is best known for his contributions to computational intelligence and knowledge-based neurocomputing. It is dedicated to Professor Jacek Żurada, Full Professor at the Computational Intelligence Laboratory, Department of Electrical and Computer Engineering, J.B. Speed School of Engineering, University of Louisville, Kentucky, USA, as a token of appreciation for his scientific and scholarly achievements, and for his longstanding service to many communities, notably the computational intelligence community, in particular neural networks, machine learning, data analyses and data mining, but also the fuzzy logic and evolutionary computation communities, to name but a few. At the same time, the book recognizes and honors Professor Żurada’s dedication and service to many scientific, scholarly and professional societies, especially the IEEE (Institute of Electrical and Electronics Engineers), the world’s largest professional technical professional organization dedicated to advancing science and technology in a broad spectrum of areas and fields. The volume is divided into five major parts, the first of which addresses theoretic, algorithmic and implementation problems related to the intelligent use of data in the sense of how to derive practically useful information and knowledge from data. In turn, Part 2 is devoted to various aspects of neural networks and connectionist systems. Part 3 deals with essential tools and techniques for intelligent technologies in systems modeling and Part 4 focuses on intelligent technologies in decision-making, optimization and control, while Part 5 explores the applications of intelligent technologies.
Tensor Numerical Methods In Quantum Chemistry
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Author : Venera Khoromskaia
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2018-06-11
Tensor Numerical Methods In Quantum Chemistry written by Venera Khoromskaia and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-06-11 with Mathematics categories.
The conventional numerical methods when applied to multidimensional problems suffer from the so-called "curse of dimensionality", that cannot be eliminated by using parallel architectures and high performance computing. The novel tensor numerical methods are based on a "smart" rank-structured tensor representation of the multivariate functions and operators discretized on Cartesian grids thus reducing solution of the multidimensional integral-differential equations to 1D calculations. We explain basic tensor formats and algorithms and show how the orthogonal Tucker tensor decomposition originating from chemometrics made a revolution in numerical analysis, relying on rigorous results from approximation theory. Benefits of tensor approach are demonstrated in ab-initio electronic structure calculations. Computation of the 3D convolution integrals for functions with multiple singularities is replaced by a sequence of 1D operations, thus enabling accurate MATLAB calculations on a laptop using 3D uniform tensor grids of the size up to 1015. Fast tensor-based Hartree-Fock solver, incorporating the grid-based low-rank factorization of the two-electron integrals, serves as a prerequisite for economical calculation of the excitation energies of molecules. Tensor approach suggests efficient grid-based numerical treatment of the long-range electrostatic potentials on large 3D finite lattices with defects.The novel range-separated tensor format applies to interaction potentials of multi-particle systems of general type opening the new prospects for tensor methods in scientific computing. This research monograph presenting the modern tensor techniques applied to problems in quantum chemistry may be interesting for a wide audience of students and scientists working in computational chemistry, material science and scientific computing.
Tensor Numerical Methods In Quantum Chemistry
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Author : Venera Khoromskaia
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2018-06-11
Tensor Numerical Methods In Quantum Chemistry written by Venera Khoromskaia and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-06-11 with Mathematics categories.
The conventional numerical methods when applied to multidimensional problems suffer from the so-called "curse of dimensionality", that cannot be eliminated by using parallel architectures and high performance computing. The novel tensor numerical methods are based on a "smart" rank-structured tensor representation of the multivariate functions and operators discretized on Cartesian grids thus reducing solution of the multidimensional integral-differential equations to 1D calculations. We explain basic tensor formats and algorithms and show how the orthogonal Tucker tensor decomposition originating from chemometrics made a revolution in numerical analysis, relying on rigorous results from approximation theory. Benefits of tensor approach are demonstrated in ab-initio electronic structure calculations. Computation of the 3D convolution integrals for functions with multiple singularities is replaced by a sequence of 1D operations, thus enabling accurate MATLAB calculations on a laptop using 3D uniform tensor grids of the size up to 1015. Fast tensor-based Hartree-Fock solver, incorporating the grid-based low-rank factorization of the two-electron integrals, serves as a prerequisite for economical calculation of the excitation energies of molecules. Tensor approach suggests efficient grid-based numerical treatment of the long-range electrostatic potentials on large 3D finite lattices with defects.The novel range-separated tensor format applies to interaction potentials of multi-particle systems of general type opening the new prospects for tensor methods in scientific computing. This research monograph presenting the modern tensor techniques applied to problems in quantum chemistry may be interesting for a wide audience of students and scientists working in computational chemistry, material science and scientific computing.
Machine Learning Low Rank Approximations And Reduced Order Modeling In Computational Mechanics
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Author : Felix Fritzen
language : en
Publisher: MDPI
Release Date : 2019-09-18
Machine Learning Low Rank Approximations And Reduced Order Modeling In Computational Mechanics written by Felix Fritzen and has been published by MDPI this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-09-18 with Technology & Engineering categories.
The use of machine learning in mechanics is booming. Algorithms inspired by developments in the field of artificial intelligence today cover increasingly varied fields of application. This book illustrates recent results on coupling machine learning with computational mechanics, particularly for the construction of surrogate models or reduced order models. The articles contained in this compilation were presented at the EUROMECH Colloquium 597, « Reduced Order Modeling in Mechanics of Materials », held in Bad Herrenalb, Germany, from August 28th to August 31th 2018. In this book, Artificial Neural Networks are coupled to physics-based models. The tensor format of simulation data is exploited in surrogate models or for data pruning. Various reduced order models are proposed via machine learning strategies applied to simulation data. Since reduced order models have specific approximation errors, error estimators are also proposed in this book. The proposed numerical examples are very close to engineering problems. The reader would find this book to be a useful reference in identifying progress in machine learning and reduced order modeling for computational mechanics.
Curves And Surfaces
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Author : Jean-Daniel Boissonnat
language : en
Publisher: Springer
Release Date : 2015-08-13
Curves And Surfaces written by Jean-Daniel Boissonnat and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-08-13 with Computers categories.
This volume constitutes the thoroughly refereed post-conference proceedings of the 8th International Conference on Curves and Surfaces, held in Paris, France, in June 2014. The conference had the overall theme: "Representation and Approximation of Curves and Surfaces and Applications". The 32 revised full papers presented were carefully reviewed and selected from 39 submissions. The scope of the conference was on following topics: approximation theory, computer-aided geometric design, computer graphics and visualization, computational geometry and topology, geometry processing, image and signal processing, interpolation and smoothing, mesh generation, finite elements and splines, scattered data processing and learning theory, sparse and high-dimensional approximation, subdivision, wavelets and multi-resolution method.
Tensor Numerical Methods In Scientific Computing
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Author : Boris N. Khoromskij
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2018-06-11
Tensor Numerical Methods In Scientific Computing written by Boris N. Khoromskij and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-06-11 with Mathematics categories.
The most difficult computational problems nowadays are those of higher dimensions. This research monograph offers an introduction to tensor numerical methods designed for the solution of the multidimensional problems in scientific computing. These methods are based on the rank-structured approximation of multivariate functions and operators by using the appropriate tensor formats. The old and new rank-structured tensor formats are investigated. We discuss in detail the novel quantized tensor approximation method (QTT) which provides function-operator calculus in higher dimensions in logarithmic complexity rendering super-fast convolution, FFT and wavelet transforms. This book suggests the constructive recipes and computational schemes for a number of real life problems described by the multidimensional partial differential equations. We present the theory and algorithms for the sinc-based separable approximation of the analytic radial basis functions including Green’s and Helmholtz kernels. The efficient tensor-based techniques for computational problems in electronic structure calculations and for the grid-based evaluation of long-range interaction potentials in multi-particle systems are considered. We also discuss the QTT numerical approach in many-particle dynamics, tensor techniques for stochastic/parametric PDEs as well as for the solution and homogenization of the elliptic equations with highly-oscillating coefficients. Contents Theory on separable approximation of multivariate functions Multilinear algebra and nonlinear tensor approximation Superfast computations via quantized tensor approximation Tensor approach to multidimensional integrodifferential equations