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Author : Boris Khoromskij
Publisher : Unknown
Page : 290 pages
File Size : 46,8 Mb
Release : 2016
Category : Electronic
ISBN : 3110365928
Author : Boris N. Khoromskij
Publisher : Walter de Gruyter GmbH & Co KG
Page : 379 pages
File Size : 41,7 Mb
Release : 2018-06-11
Category : Mathematics
ISBN : 9783110365917
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
Author : Venera Khoromskaia,Boris N. Khoromskij
Publisher : Walter de Gruyter GmbH & Co KG
Page : 297 pages
File Size : 41,8 Mb
Release : 2018-06-11
Category : Mathematics
ISBN : 9783110391374
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.
Author : Germund Dahlquist,Ake Bjorck
Publisher : SIAM
Page : 741 pages
File Size : 44,5 Mb
Release : 2008-09-04
Category : Mathematics
ISBN : 9780898716443
This work addresses the increasingly important role of numerical methods in science and engineering. It combines traditional and well-developed topics with other material such as interval arithmetic, elementary functions, operator series, convergence acceleration, and continued fractions.
Author : Venera Khoromskaia,Boris Khoromskij
Publisher : Unknown
Page : 240 pages
File Size : 55,6 Mb
Release : 2016
Category : Electronic
ISBN : 3110365847
Author : Edoardo Angelo Di Napoli,Paolo Bientinesi,Jiajia Li,André Uschmajew
Publisher : Frontiers Media SA
Page : 192 pages
File Size : 45,8 Mb
Release : 2022-11-08
Category : Science
ISBN : 9782832504253
Author : Friedrich W Hehl,Roland A. Puntigam,Hanns Ruder
Publisher : Springer Science & Business Media
Page : 405 pages
File Size : 47,5 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783642957321
For this set of lectures we assumed that the reader has a reasonable back ground in physics and some knowledge of general relativity, the modern theory of gravity in macrophysics, and cosmology. Computer methods are present ed by leading experts in the three main domains: in numerics, in computer algebra, and in visualization. The idea was that each of these subdisciplines is introduced by an extended set of main lectures and that each is conceived as being of comparable 'importance. Therefpre we believe that the book represents a good introduction into scientific I computing for any student who wants to specialize in relativity, gravitation, and/or astrophysics. We took great care to select lecturers who teach in a comprehensible way and who are, at the same time, at the research front of their respective field. In numerics we had the privilege of having a lecturer from the National Center for Supercomputing Applications (NCSA, Champaign, IL, USA) and some from other leading institutions of the world; visualization was taught by a visualization expert from Boeing; and in com puter algebra we took recourse to practitioners of different computer algebra systems as applied to classical general relativity up to quantum gravity and differential geometry.
Author : Ivan Oseledets
Publisher : de Gruyter
Page : 216 pages
File Size : 49,9 Mb
Release : 2018-01-31
Category : Mathematics
ISBN : 3110461625
Covering both theoretical foundations and applications in mathematics and engineering, this graduate textbook introduces numerical, tensor-based methods for tackling high-dimensional problems. Concepts known as tensor trains, matrix product states or hierarchical tensor networks have a range of applications in solving differential equations, multidimensional integration, machine learning, condensed matter physics, and theoretical chemistry.
Author : Michael W. Berry,Kyle A. Gallivan,Efstratios Gallopoulos,Ananth Grama,Bernard Philippe,Yousef Saad,Faisal Saied
Publisher : Springer Science & Business Media
Page : 350 pages
File Size : 52,8 Mb
Release : 2012-01-18
Category : Computers
ISBN : 9781447124375
This book presents the state of the art in parallel numerical algorithms, applications, architectures, and system software. The book examines various solutions for issues of concurrency, scale, energy efficiency, and programmability, which are discussed in the context of a diverse range of applications. Features: includes contributions from an international selection of world-class authorities; examines parallel algorithm-architecture interaction through issues of computational capacity-based codesign and automatic restructuring of programs using compilation techniques; reviews emerging applications of numerical methods in information retrieval and data mining; discusses the latest issues in dense and sparse matrix computations for modern high-performance systems, multicores, manycores and GPUs, and several perspectives on the Spike family of algorithms for solving linear systems; presents outstanding challenges and developing technologies, and puts these in their historical context.
Author : Zhangxin Chen,R. Glowinski,Kaitai Li,American Mathematical Society
Publisher : American Mathematical Soc.
Page : 372 pages
File Size : 40,8 Mb
Release : 2003
Category : Science
ISBN : 9780821832615
This volume contains 36 research papers written by prominent researchers. The papers are based on a large satellite conference on scientific computing held at the International Congress of Mathematicians (ICM) in Xi'an, China. Topics covered include a variety of subjects in modern scientific computing and its applications, such as numerical discretization methods, linear solvers, parallel computing, high performance computing, and applications to solid and fluid mechanics, energy, environment, and semiconductors. The book will serve as an excellent reference work for graduate students and researchers working with scientific computing for problems in science and engineering.
Author : P.G. Ciarlet,Jacques-Louis Lions
Publisher : North-Holland is
Page : 814 pages
File Size : 44,8 Mb
Release : 1990
Category : Mathematics
ISBN : UCSC:32106011129522
Author : Sergio Pirozzoli,Tapan K. Sengupta
Publisher : Springer
Page : 250 pages
File Size : 43,7 Mb
Release : 2019-05-28
Category : Technology & Engineering
ISBN : 9783030170127
This book provides state-of-art information on high-accuracy scientific computing and its future prospects, as applicable to the broad areas of fluid mechanics and combustion, and across all speed regimes. Beginning with the concepts of space-time discretization and dispersion relation in numerical computing, the foundations are laid for the efficient solution of the Navier-Stokes equations, with special reference to prominent approaches such as LES, DES and DNS. The basis of high-accuracy computing is rooted in the concept of stability, dispersion and phase errors, which require the comprehensive analysis of discrete computing by rigorously applying error dynamics. In this context, high-order finite-difference and finite-volume methods are presented. Naturally, the coverage also includes fundamental notions of high-performance computing and advanced concepts on parallel computing, including their implementation in prospective hexascale computers. Moreover, the book seeks to raise the bar beyond the pedagogical use of high-accuracy computing by addressing more complex physical scenarios, including turbulent combustion. Tools like proper orthogonal decomposition (POD), proper generalized decomposition (PGD), singular value decomposition (SVD), recursive POD, and high-order SVD in multi-parameter spaces are presented. Special attention is paid to bivariate and multivariate datasets in connection with various canonical flow and heat transfer cases. The book mainly addresses the needs of researchers and doctoral students in mechanical engineering, aerospace engineering, and all applied disciplines including applied mathematics, offering these readers a unique resource.
Author : Ivan Lirkov,Svetozar Margenov
Publisher : Springer Nature
Page : 636 pages
File Size : 47,9 Mb
Release : 2020-02-13
Category : Computers
ISBN : 9783030410322
This book constitutes revised papers from the 12th International Conference on Large-Scale Scientific Computing, LSSC 2019, held in Sozopol, Bulgaria, in June 2019. The 70 papers presented in this volume were carefully reviewed and selected from 81 submissions. The book also contains two invited talks. The papers were organized in topical sections named as follows: control and optimization of dynamical systems; meshfree and particle methods; fractional diffusion problems: numerical methods, algorithms and applications; pore scale flow and transport simulation; tensors based algorithms and structures in optimization and applications; HPC and big data: algorithms and applications; large-scale models: numerical methods, parallel computations and applications; monte carlo algorithms: innovative applications in conjunctions with other methods; application of metaheuristics to large-scale problems; large scale machine learning: multiscale algorithms and performance guarantees; and contributed papers.
Author : Simone Montangero
Publisher : Springer
Page : 172 pages
File Size : 43,9 Mb
Release : 2018-11-28
Category : Science
ISBN : 9783030014094
This volume of lecture notes briefly introduces the basic concepts needed in any computational physics course: software and hardware, programming skills, linear algebra, and differential calculus. It then presents more advanced numerical methods to tackle the quantum many-body problem: it reviews the numerical renormalization group and then focuses on tensor network methods, from basic concepts to gauge invariant ones. Finally, in the last part, the author presents some applications of tensor network methods to equilibrium and out-of-equilibrium correlated quantum matter. The book can be used for a graduate computational physics course. After successfully completing such a course, a student should be able to write a tensor network program and can begin to explore the physics of many-body quantum systems. The book can also serve as a reference for researchers working or starting out in the field.
Author : Francis X. Giraldo
Publisher : Springer Nature
Page : 559 pages
File Size : 40,8 Mb
Release : 2020-10-30
Category : Mathematics
ISBN : 9783030550691
This book introduces the reader to solving partial differential equations (PDEs) numerically using element-based Galerkin methods. Although it draws on a solid theoretical foundation (e.g. the theory of interpolation, numerical integration, and function spaces), the book’s main focus is on how to build the method, what the resulting matrices look like, and how to write algorithms for coding Galerkin methods. In addition, the spotlight is on tensor-product bases, which means that only line elements (in one dimension), quadrilateral elements (in two dimensions), and cubes (in three dimensions) are considered. The types of Galerkin methods covered are: continuous Galerkin methods (i.e., finite/spectral elements), discontinuous Galerkin methods, and hybridized discontinuous Galerkin methods using both nodal and modal basis functions. In addition, examples are included (which can also serve as student projects) for solving hyperbolic and elliptic partial differential equations, including both scalar PDEs and systems of equations.
