Tensor Numerical Methods In Electronic Structure Calculations
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Tensor Numerical Methods in Quantum Chemistry by Venera Khoromskaia,Boris N. Khoromskij Pdf
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 Page : 379 pages File Size : 42,6 Mb Release : 2018-06-11 Category : Mathematics ISBN : 9783110365917
Tensor Numerical Methods in Scientific Computing by Boris N. Khoromskij Pdf
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
Tensor Numerical Methods in Quantum Chemistry by Venera Khoromskaia,Boris N. Khoromskij Pdf
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.
A Mathematical Introduction to Electronic Structure Theory by Lin Lin,Jianfeng Lu Pdf
Based on first principle quantum mechanics, electronic structure theory is widely used in physics, chemistry, materials science, and related fields and has recently received increasing research attention in applied and computational mathematics. This book provides a self-contained, mathematically oriented introduction to the subject and its associated algorithms and analysis. It will help applied mathematics students and researchers with minimal background in physics understand the basics of electronic structure theory and prepare them to conduct research in this area. The book begins with an elementary introduction of quantum mechanics, including the uncertainty principle and the Hartree?Fock theory, which is considered the starting point of modern electronic structure theory. The authors then provide an in-depth discussion of two carefully selected topics that are directly related to several aspects of modern electronic structure calculations: density matrix based algorithms and linear response theory. Chapter 2 introduces the Kohn?Sham density functional theory with a focus on the density matrix based numerical algorithms, and Chapter 3 introduces linear response theory, which provides a unified viewpoint of several important phenomena in physics and numerics. An understanding of these topics will prepare readers for more advanced topics in this field. The book concludes with the random phase approximation to the correlation energy. The book is written for advanced undergraduate and beginning graduate students, specifically those with mathematical backgrounds but without a priori knowledge of quantum mechanics, and can be used for self-study by researchers, instructors, and other scientists. The book can also serve as a starting point to learn about many-body perturbation theory, a topic at the frontier of the study of interacting electrons.
Electronic Structure Calculations on Graphics Processing Units by Ross C. Walker,Andreas W. Goetz Pdf
Electronic Structure Calculations on Graphics Processing Units: From Quantum Chemistry to Condensed Matter Physics provides an overview of computing on graphics processing units (GPUs), a brief introduction to GPU programming, and the latest examples of code developments and applications for the most widely used electronic structure methods. The book covers all commonly used basis sets including localized Gaussian and Slater type basis functions, plane waves, wavelets and real-space grid-based approaches. The chapters expose details on the calculation of two-electron integrals, exchange-correlation quadrature, Fock matrix formation, solution of the self-consistent field equations, calculation of nuclear gradients to obtain forces, and methods to treat excited states within DFT. Other chapters focus on semiempirical and correlated wave function methods including density fitted second order Møller-Plesset perturbation theory and both iterative and perturbative single- and multireference coupled cluster methods. Electronic Structure Calculations on Graphics Processing Units: From Quantum Chemistry to Condensed Matter Physics presents an accessible overview of the field for graduate students and senior researchers of theoretical and computational chemistry, condensed matter physics and materials science, as well as software developers looking for an entry point into the realm of GPU and hybrid GPU/CPU programming for electronic structure calculations.
Computational Methods in Condensed Matter: Electronic Structure by A.A. Katsnelson,V.S. Stepanyuk,A. Szasz,O.V. Farberovich Pdf
"Blurb & Contents" This current and comprehensive treatment of the physics of small- amplitude waves in hot magnetized plasmas provides a thorough update of the author's classic Theory of Plasma Waves. New topics include quasi-linear theory, inhomogeneous plasmas, collisions, absolute and convective instability, and mode conversion. Valuable for graduates and advanced undergraduates and an indispensable reference work for researchers in plasmas, controlled fusion, and space science.
Semiempirical Methods of Electronic Structure Calculation by Gerald Segal Pdf
If one reflects upon the range of chemical problems accessible to the current quantum theoretical methods for calculations on the electronic structure of molecules, one is immediately struck by the rather narrow limits imposed by economic and numerical feasibility. Most of the systems with which experimental photochemists actually work are beyond the grasp of ab initio methods due to the presence of a few reasonably large aromatic ring systems. Potential energy surfaces for all but the smallest molecules are extremely expensive to produce, even over a restricted group of the possible degrees of freedom, and molecules containing the higher elements of the periodic table remain virtually untouched due to the large numbers of electrons involved. Almost the entire class of molecules of real biological interest is simply out of the question. In general, the theoretician is reduced to model systems of variable appositeness in most of these fields. The fundamental problem, from a basic computational point of view, is that large molecules require large numbers of basis functions, whether Slater type orbitals or Gaussian functions suitably contracted, to provide even a modestly accurate description of the molecular electronic environment. This leads to the necessity of dealing with very large matrices and numbers of integrals within the Hartree-Fock approximation and quickly becomes both numerically difficult and uneconomic.
AETA 2018 - Recent Advances in Electrical Engineering and Related Sciences: Theory and Application by Ivan Zelinka,Pavel Brandstetter,Tran Trong Dao,Vo Hoang Duy,Sang Bong Kim Pdf
These proceedings address a broad range of topic areas, including telecommunication, power systems, digital signal processing, robotics, control systems, renewable energy, power electronics, soft computing and more. Today’s world is based on vitally important technologies that combine e.g. electronics, cybernetics, computer science, telecommunication, and physics. However, since the advent of these technologies, we have been confronted with numerous technological challenges such as finding optimal solutions to various problems regarding controlling technologies, signal processing, power source design, robotics, etc. Readers will find papers on these and other topics, which share fresh ideas and provide state-of-the-art overviews. They will also benefit practitioners, who can easily apply the issues discussed here to solve real-life problems in their own work. Accordingly, the proceedings offer a valuable resource for all scientists and engineers pursuing research and applications in the above-mentioned fields.
Operators preserving primitivity for matrix pairs / L.B. Beasley, A.E. Guterman -- Decompositions of quaternions and their matrix equivalents / D. Janovská, G. Opfer -- Sensitivity analysis of Hamiltonian and reversible systems prone to dissipation-induced instabilities / O.N. Kirillov -- Block triangular miniversal deformations of matrices and matrix pencils / L. Klimenko, V.V. Sergeichuk -- Determining the Schein rank of boolean matrices / E.E. Marenich -- Lattices of matrix rows and matrix columns. Lattices of invariant column eigenvectors / V. Marenich -- Matrix algebras and their length / O.V. Markova -- On a new class of singular nonsymmetric matrices with nonnegative integer spectra / T. Nahtman, D. von Rosen -- Reduction of a set of matrices over a principal ideal domain to the Smith normal forms by means of the same one-sided transformation / V.M. Prokip -- Nonsymmetric algebraic Riccati equations associated with an M-matrix : recent advances and algorithms / D.A. Bini, B. Iannazzo, B. Meini, F. Poloni -- A generalized conjugate direction method for nonsymmetric large ill-conditioned linear systems / E.R. Boudinov, A.I. Manevich -- There exist normal Hankel ([symbol], [symbol])-circulants of any order [symbol] / V.N. Chugunov, Kh. D. Ikramov -- On the treatment of boundary artifacts in image restoration by reflection and/or anti-reflection / M. Donatelli, S. Serra-Capizzano -- Zeros of determinants of [symbol]-matrices / W. Gander -- How to find a good submatrix / S.A. Goreinov [und weiteren] -- Conjugate and semi-conjugate direction methods with preconditioning projectors / V.P. Il'in -- Some relationships between optimal preconditioner and superoptimal preconditioner / J.-B. Chen [und weiteren] -- Scaling, preconditioning, and superlinear convergence in GMRES-type iterations / I. Kaporin -- Toeplitz and Toeplitz-block-Toeplitz matrices and their correlation with syzygies of polynomials / H. Khalil, B. Mourrain, M. Schatzman -- Concepts of data-sparse tensor-product approximation in many-particle modelling / H.-J. Flad [und weiteren] -- Separation of variables in nonlinear fermi equation / Yu. I. Kuznetsov -- Faster multipoint polynomial evaluation via structured matrices / B. Murphy, R.E. Rosholt -- Testing pivoting policies in Gaussian elimination / B. Murphy [und weiteren] -- Newton's iteration for matrix inversion, advances and extensions / V.Y. Pan -- Truncated decompositions and filtering methods with reflective/antireflective boundary conditions : a comparison / C. Tablino Possio -- Discrete-time stability of a class of hermitian polynomial matrices with positive semidefinite coefficients / H.K. Wimmer -- Splitting algorithm for solving mixed variational inequalities with inversely strongly monotone operators / I. Badriev, O. Zadvornov -- Multilevel algorithm for graph partitioning / N.S. Bochkarev, O.V. Diyankov, V.Y. Pravilnikov -- 2D-extension of singular spectrum analysis : algorithm and elements of theory / N.E. Golyandina, K.D. Usevich -- Application of radon transform for fast solution of boundary value problems for elliptic PDE in domains with complicated geometry / A.I. Grebennikov -- Application of a multigrid method to solving diffusion-type equations / M.E. Ladonkina, O. Yu. Milukova, V.F. Tishkin -- Monotone matrices and finite volume schemes for diffusion problems preserving non-negativity of solution / I.V. Kapyrin -- Sparse approximation of FEM matrix for sheet current integro-differential equation / M. Khapaev, M. Yu. Kupriyanov -- The method of magnetic field computation in presence of an ideal conductive multiconnected surface by using the integro-differential equation of the first kind / T. Kochubey, V.I. Astakhov -- Spectral model order reduction preserving passivity for large multiport RCLM networks / Yu. M. Nechepurenko, A.S. Potyagalova, I.A. Karaseva -- New smoothers in multigrid methods for strongly nonsymmetric linear systems / G.V. Muratova, E.M. Andreeva -- Operator equations for eddy currents on singular carriers / J. Naumenko -- Matrix approach to modelling of polarized radiation transfer in heterogeneous systems / T.A. Sushkevich, S.A. Strelkov, S.V. Maksakova -- The Method of Regularization of Tikhonov Based on Augmented Systems / A.I. Zhdanov, T.G. Parchaikina
Introduction to Tensor Network Methods by Simone Montangero Pdf
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.
State of The Art of Molecular Electronic Structure Computations: Correlation Methods, Basis Sets and More by Anonim Pdf
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
Electronic, Atomic and Molecular Calculations by Milan Trsic,Alberico da Silva Pdf
The Generator Coordinate Method (GCM) is a mathematical tool for the understanding of stable atomic nuclei. Electronic, Atomic and Molecular Calculations is designed to assist scientists applying GCM in the analysis of the electronic structure of atoms and molecules. There have been numerous publications covering nuclear physics and electronic structure of atoms and molecules, but this book is unique in the sense that it specifically addresses the application of GCM for such purposes. Using this book, researchers will be able to understand and calculate the electronic structure in a novel manner. * Only book that covers the Generator Coordinate Method and applications for atoms, molecules and nuclei * Clearly describes how the GCM can be used as a powerful tool for design of atomic basis sets * Reviews current literature on GCM in atomic and molecular fields and a large part of the literature of the method in nuclear physics
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.
Exascale Scientific Applications by Tjerk P. Straatsma,Katerina B. Antypas,Timothy J. Williams Pdf
From the Foreword: "The authors of the chapters in this book are the pioneers who will explore the exascale frontier. The path forward will not be easy... These authors, along with their colleagues who will produce these powerful computer systems will, with dedication and determination, overcome the scalability problem, discover the new algorithms needed to achieve exascale performance for the broad range of applications that they represent, and create the new tools needed to support the development of scalable and portable science and engineering applications. Although the focus is on exascale computers, the benefits will permeate all of science and engineering because the technologies developed for the exascale computers of tomorrow will also power the petascale servers and terascale workstations of tomorrow. These affordable computing capabilities will empower scientists and engineers everywhere." — Thom H. Dunning, Jr., Pacific Northwest National Laboratory and University of Washington, Seattle, Washington, USA "This comprehensive summary of applications targeting Exascale at the three DoE labs is a must read." — Rio Yokota, Tokyo Institute of Technology, Tokyo, Japan "Numerical simulation is now a need in many fields of science, technology, and industry. The complexity of the simulated systems coupled with the massive use of data makes HPC essential to move towards predictive simulations. Advances in computer architecture have so far permitted scientific advances, but at the cost of continually adapting algorithms and applications. The next technological breakthroughs force us to rethink the applications by taking energy consumption into account. These profound modifications require not only anticipation and sharing but also a paradigm shift in application design to ensure the sustainability of developments by guaranteeing a certain independence of the applications to the profound modifications of the architectures: it is the passage from optimal performance to the portability of performance. It is the challenge of this book to demonstrate by example the approach that one can adopt for the development of applications offering performance portability in spite of the profound changes of the computing architectures." — Christophe Calvin, CEA, Fundamental Research Division, Saclay, France "Three editors, one from each of the High Performance Computer Centers at Lawrence Berkeley, Argonne, and Oak Ridge National Laboratories, have compiled a very useful set of chapters aimed at describing software developments for the next generation exa-scale computers. Such a book is needed for scientists and engineers to see where the field is going and how they will be able to exploit such architectures for their own work. The book will also benefit students as it provides insights into how to develop software for such computer architectures. Overall, this book fills an important need in showing how to design and implement algorithms for exa-scale architectures which are heterogeneous and have unique memory systems. The book discusses issues with developing user codes for these architectures and how to address these issues including actual coding examples.’ — Dr. David A. Dixon, Robert Ramsay Chair, The University of Alabama, Tuscaloosa, Alabama, USA
