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Author: Venera Khoromskaia
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110391376
Category : Mathematics
Languages : en
Pages : 343
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Book Description
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: Venera Khoromskaia
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110391376
Category : Mathematics
Languages : en
Pages : 343
Get Book Here
Book Description
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: Boris N. Khoromskij
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110391392
Category : Mathematics
Languages : en
Pages : 475
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Book Description
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: Vadim Olshevsky
Publisher: World Scientific
ISBN: 9814469556
Category : Mathematics
Languages : en
Pages : 604
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Book Description
Compared to other books devoted to matrices, this volume is unique in covering the whole of a triptych consisting of algebraic theory, algorithmic problems and numerical applications, all united by the essential use and urge for development of matrix methods. This was the spirit of the 2nd International Conference on Matrix Methods and Operator Equations from 23-27 July 2007 in Moscow that was organized by Dario Bini, Gene Golub, Alexander Guterman, Vadim Olshevsky, Stefano Serra-Capizzano, Gilbert Strang and Eugene Tyrtyshnikov.Matrix methods provide the key to many problems in pure and applied mathematics. However, linear algebra theory, numerical algorithms and matrices in FEM/BEM applications usually live as if in three separate worlds. In this volume, maybe for the first time ever, they are compiled together as one entity as it was at the Moscow meeting, where the algebraic part was impersonated by Hans Schneider, algorithms by Gene Golub, and applications by Guri Marchuk. All topics intervened in plenary sessions are specially categorized into three sections of this volume.The soul of the meeting was Gene Golub, who rendered a charming “Golub's dimension” to the three main axes of the conference topics. This volume is dedicated in gratitude to his memory.
Author:
Publisher: Academic Press
ISBN: 0128161752
Category : Science
Languages : en
Pages : 362
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Book Description
State of the Art of Molecular Electronic Structure Computations: Correlation Methods, Basis Sets and More, Volume 79 in the Advances in Quantum Chemistry series, presents surveys of current topics in this rapidly developing field that has emerged at the cross section of the historically established areas of mathematics, physics, chemistry and biology. Chapters in this new release include Computing accurate molecular properties in real space using multiresolution analysis, Self-consistent electron-nucleus cusp correction for molecular orbitals, Correlated methods for computational spectroscopy, Potential energy curves for the NaH molecule and its cation with the cock space coupled cluster method, and much more. - Presents surveys of current topics in this rapidly-developing field that has emerged at the cross section of the historically established areas of mathematics, physics, chemistry and biology - Features detailed reviews written by leading international researchers
Author: Edoardo Angelo Di Napoli
Publisher: Frontiers Media SA
ISBN: 2832504256
Category : Science
Languages : en
Pages : 192
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Book Description
Author: Olivier Ouari
Publisher: Royal Society of Chemistry
ISBN: 1788019660
Category : Science
Languages : en
Pages : 611
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Book Description
Nitroxides are versatile small organic molecules possessing a stabilised free radical. With their unpaired electron spin they display a unique reactivity towards various environmental factors, enabling a diverse range of applications. They have uses as synthetic tools, such as catalysts or building blocks; imaging agents and probes in biomedicine and materials science; for medicinal antioxidant applications; and in energy storage. Polynitroxides (polymers bearing pendant nitroxide sidechains) have been used in organic radical batteries, oxidation catalysts and in exchange reactions for constructing complex architectures. Chapters in this book cover the synthesis of nitroxides, EPR studies and magnetic resonance applications, physiochemical studies, and applications including in batteries, imaging and organic synthesis. With contributions from leaders in the field, Nitroxides will be of interest to graduate students and researchers across chemistry, physics, biology and materials science.
Author: Robin K. Harris
Publisher: John Wiley & Sons
ISBN: 1118587324
Category : Science
Languages : en
Pages : 523
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Book Description
The content of this volume has been added to eMagRes (formerly Encyclopedia of Magnetic Resonance) - the ultimate online resource for NMR and MRI. The term "NMR Crystallography" has only recently come into common usage, and even now causes raised eyebrows within some parts of the diffraction community. The power of solid-state NMR to give crystallographic information has considerably increased since the CPMAS suite of techniques was introduced in 1976. In the first years of the 21st century, the ability of NMR to provide information to support and facilitate the analysis of single-crystal and powder diffraction patterns has become widely accepted. Indeed, NMR can now be used to refine diffraction results and, in favorable cases, to solve crystal structures with minimal (or even no) diffraction data. The increasing ability to relate chemical shifts (including the tensor components) to the crystallographic location of relevant atoms in the unit cell via computational methods has added significantly to the practice of NMR crystallography. Diffraction experts will increasingly welcome NMR as an allied technique in their structural analyses. Indeed, it may be that in the future crystal structures will be determined by simultaneously fitting diffraction patterns and NMR spectra. This Handbook is organised into six sections. The first contains an overview and some articles on fundamental NMR topics, followed by a section concentrating on chemical shifts, and one on coupling interactions. The fourth section contains articles describing how NMR results relate to fundamental crystallography concepts and to diffraction methods. The fifth section concerns specific aspects of structure, such as hydrogen bonding. Finally, four articles in the sixth section give applications of NMR crystallography to structural biology, organic & pharmaceutical chemistry, inorganic & materials chemistry, and geochemistry. About EMR Handbooks / eMagRes Handbooks The Encyclopedia of Magnetic Resonance (up to 2012) and eMagRes (from 2013 onward) publish a wide range of online articles on all aspects of magnetic resonance in physics, chemistry, biology and medicine. The existence of this large number of articles, written by experts in various fields, is enabling the publication of a series of EMR Handbooks / eMagRes Handbooks on specific areas of NMR and MRI. The chapters of each of these handbooks will comprise a carefully chosen selection of articles from eMagRes. In consultation with the eMagRes Editorial Board, the EMR Handbooks / eMagRes Handbooks are coherently planned in advance by specially-selected Editors, and new articles are written (together with updates of some already existing articles) to give appropriate complete coverage. The handbooks are intended to be of value and interest to research students, postdoctoral fellows and other researchers learning about the scientific area in question and undertaking relevant experiments, whether in academia or industry. Have the content of this Handbook and the complete content of eMagRes at your fingertips! Visit: www.wileyonlinelibrary.com/ref/eMagRes View other eMagRes publications here
Author: Stephan Dahlke
Publisher: Springer
ISBN: 3319081594
Category : Mathematics
Languages : en
Pages : 446
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Book Description
In April 2007, the Deutsche Forschungsgemeinschaft (DFG) approved the Priority Program 1324 “Mathematical Methods for Extracting Quantifiable Information from Complex Systems.” This volume presents a comprehensive overview of the most important results obtained over the course of the program. Mathematical models of complex systems provide the foundation for further technological developments in science, engineering and computational finance. Motivated by the trend toward steadily increasing computer power, ever more realistic models have been developed in recent years. These models have also become increasingly complex, and their numerical treatment poses serious challenges. Recent developments in mathematics suggest that, in the long run, much more powerful numerical solution strategies could be derived if the interconnections between the different fields of research were systematically exploited at a conceptual level. Accordingly, a deeper understanding of the mathematical foundations as well as the development of new and efficient numerical algorithms were among the main goals of this Priority Program. The treatment of high-dimensional systems is clearly one of the most challenging tasks in applied mathematics today. Since the problem of high-dimensionality appears in many fields of application, the above-mentioned synergy and cross-fertilization effects were expected to make a great impact. To be truly successful, the following issues had to be kept in mind: theoretical research and practical applications had to be developed hand in hand; moreover, it has proven necessary to combine different fields of mathematics, such as numerical analysis and computational stochastics. To keep the whole program sufficiently focused, we concentrated on specific but related fields of application that share common characteristics and as such, they allowed us to use closely related approaches.
Author: Franz Michael Dolg
Publisher: Oldenbourg Verlag
ISBN: 3486711636
Category : Science
Languages : en
Pages : 431
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Book Description
Progress in Physical Chemistry is a collection of recent »Review Articles« published in the »Zeitschrift für Physikalische Chemie«. The third volume of the series "Progress in Physical Chemistry" comprises 27 articles, most of them with review character, written by the members of the Priority Program (SPP) 1145 of the German Research Foundation (DFG).
Author: Wolfgang Hackbusch
Publisher: Springer Science & Business Media
ISBN: 3642280277
Category : Mathematics
Languages : en
Pages : 525
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Book Description
Special numerical techniques are already needed to deal with nxn matrices for large n.Tensor data are of size nxnx...xn=n^d, where n^d exceeds the computer memory by far. They appear for problems of high spatial dimensions. Since standard methods fail, a particular tensor calculus is needed to treat such problems. The monograph describes the methods how tensors can be practically treated and how numerical operations can be performed. Applications are problems from quantum chemistry, approximation of multivariate functions, solution of pde, e.g., with stochastic coefficients, etc.
