Tensor Spaces and Numerical Tensor Calculus

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.


Author: Wolfgang Hackbusch

Publisher: Springer Science & Business Media

ISBN: 9783642280276

Category: Mathematics

Page: 500

View: 282

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. ​

Tensor Spaces and Numerical Tensor Calculus

Abstract In this chapter, isomorphisms between the tensor space of order d and vector spaces or other tensor spaces are considered. The vectorisation from Section 5.1 ignores the tensor structure and treats the tensor space as a usual ...


Author: Wolfgang Hackbusch

Publisher: Springer Nature

ISBN: 9783030355548

Category: Mathematics

Page: 605

View: 564

Special numerical techniques are already needed to deal with n × n matrices for large n. Tensor data are of size n × n ×...× n=nd, where nd 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. This monograph describes the methods by which tensors can be practically treated and shows how numerical operations can be performed. Applications include problems from quantum chemistry, approximation of multivariate functions, solution of partial differential equations, for example with stochastic coefficients, and more. In addition to containing corrections of the unavoidable misprints, this revised second edition includes new parts ranging from single additional statements to new subchapters. The book is mainly addressed to numerical mathematicians and researchers working with high-dimensional data. It also touches problems related to Geometric Algebra.

Introduction to Tensor Analysis and the Calculus of Moving Surfaces

The definition of a tensor comes only in Chapter 6 – when the reader is ready for it. While this text maintains a consistent level of rigor, it takes great care to avoid formalizing the subject.


Author: Pavel Grinfeld

Publisher: Springer Science & Business Media

ISBN: 9781461478676

Category: Mathematics

Page: 302

View: 638

This textbook is distinguished from other texts on the subject by the depth of the presentation and the discussion of the calculus of moving surfaces, which is an extension of tensor calculus to deforming manifolds. Designed for advanced undergraduate and graduate students, this text invites its audience to take a fresh look at previously learned material through the prism of tensor calculus. Once the framework is mastered, the student is introduced to new material which includes differential geometry on manifolds, shape optimization, boundary perturbation and dynamic fluid film equations. The language of tensors, originally championed by Einstein, is as fundamental as the languages of calculus and linear algebra and is one that every technical scientist ought to speak. The tensor technique, invented at the turn of the 20th century, is now considered classical. Yet, as the author shows, it remains remarkably vital and relevant. The author’s skilled lecturing capabilities are evident by the inclusion of insightful examples and a plethora of exercises. A great deal of material is devoted to the geometric fundamentals, the mechanics of change of variables, the proper use of the tensor notation and the discussion of the interplay between algebra and geometry. The early chapters have many words and few equations. The definition of a tensor comes only in Chapter 6 – when the reader is ready for it. While this text maintains a consistent level of rigor, it takes great care to avoid formalizing the subject. The last part of the textbook is devoted to the Calculus of Moving Surfaces. It is the first textbook exposition of this important technique and is one of the gems of this text. A number of exciting applications of the calculus are presented including shape optimization, boundary perturbation of boundary value problems and dynamic fluid film equations developed by the author in recent years. Furthermore, the moving surfaces framework is used to offer new derivations of classical results such as the geodesic equation and the celebrated Gauss-Bonnet theorem.

Tensor Numerical Methods in Quantum Chemistry

Tensor-product approximation to elliptic and parabolic solution operators in higher dimensions. Computing, 74, 131–157, 2005. ... Tensor Spaces and Numerical Tensor Calculus. Springer, [111] [112] [113] [114] [115] [116] [117] [118] ...


Author: Venera Khoromskaia

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 9783110391374

Category: Mathematics

Page: 297

View: 674

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.

Sparse Grids and Applications

This gives a connection between sparse grids and the tensor space setting. On the other hand, in the tensor calculus one is interested in suitable representations of tensors (cf. Hackbusch, Tensor spaces and numerical tensor calculus, ...


Author: Jochen Garcke

Publisher: Springer Science & Business Media

ISBN: 9783642317033

Category: Mathematics

Page: 286

View: 351

In the recent decade, there has been a growing interest in the numerical treatment of high-dimensional problems. It is well known that classical numerical discretization schemes fail in more than three or four dimensions due to the curse of dimensionality. The technique of sparse grids helps overcome this problem to some extent under suitable regularity assumptions. This discretization approach is obtained from a multi-scale basis by a tensor product construction and subsequent truncation of the resulting multiresolution series expansion. This volume of LNCSE is a collection of the papers from the proceedings of the workshop on sparse grids and its applications held in Bonn in May 2011. The selected articles present recent advances in the mathematical understanding and analysis of sparse grid discretization. Aspects arising from applications are given particular attention.

Tensors Asymptotic Geometry and Developments 2016 2018

MR3865047 Wolfgang Hackbusch, Tensor spaces and numerical tensor calculus, Springer Series in Computational Mathematics, vol. 42, Springer, Heidelberg, 2012. MR3236394 Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol.


Author: J.M. Landsberg

Publisher: American Mathematical Soc.

ISBN: 9781470451363

Category: Calculus of tensors

Page: 144

View: 435

Tensors are used throughout the sciences, especially in solid state physics and quantum information theory. This book brings a geometric perspective to the use of tensors in these areas. It begins with an introduction to the geometry of tensors and provides geometric expositions of the basics of quantum information theory, Strassen's laser method for matrix multiplication, and moment maps in algebraic geometry. It also details several exciting recent developments regarding tensors in general. In particular, it discusses and explains the following material previously only available in the original research papers: (1) Shitov's 2017 refutation of longstanding conjectures of Strassen on rank additivity and Common on symmetric rank; (2) The 2017 Christandl-Vrana-Zuiddam quantum spectral points that bring together quantum information theory, the asymptotic geometry of tensors, matrix multiplication complexity, and moment polytopes in geometric invariant theory; (3) the use of representation theory in quantum information theory, including the solution of the quantum marginal problem; (4) the use of tensor network states in solid state physics, and (5) recent geometric paths towards upper bounds for the complexity of matrix multiplication. Numerous open problems appropriate for graduate students and post-docs are included throughout.

From Algebraic Structures to Tensors

Tensor Spaces and Numerical Tensor Calculus (Springer series in Computational Mathematics). Springer, Berlin and Heidelberg, 2012. Harshman, R.A. (1970). Foundations of the Parafac procedure: models and conditions for an explanatory ...


Author: Gérard Favier

Publisher: John Wiley & Sons

ISBN: 9781119681090

Category: Technology & Engineering

Page: 318

View: 619

Nowadays, tensors play a central role for the representation, mining, analysis, and fusion of multidimensional, multimodal, and heterogeneous big data in numerous fields. This set on Matrices and Tensors in Signal Processing aims at giving a self-contained and comprehensive presentation of various concepts and methods, starting from fundamental algebraic structures to advanced tensor-based applications, including recently developed tensor models and efficient algorithms for dimensionality reduction and parameter estimation. Although its title suggests an orientation towards signal processing, the results presented in this set will also be of use to readers interested in other disciplines. This first book provides an introduction to matrices and tensors of higher-order based on the structures of vector space and tensor space. Some standard algebraic structures are first described, with a focus on the hilbertian approach for signal representation, and function approximation based on Fourier series and orthogonal polynomial series. Matrices and hypermatrices associated with linear, bilinear and multilinear maps are more particularly studied. Some basic results are presented for block matrices. The notions of decomposition, rank, eigenvalue, singular value, and unfolding of a tensor are introduced, by emphasizing similarities and differences between matrices and tensors of higher-order.

Hierarchical Matrices Algorithms and Analysis

Springer, Berlin (2009) Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus, SSCM, Vol. 42. Springer, Berlin (2012) Hackbusch, W.: Numerical tensor calculus. Acta Numerica 23, 651–742 (2014) Hackbusch, W.: The Concept of ...


Author: Wolfgang Hackbusch

Publisher: Springer

ISBN: 9783662473245

Category: Mathematics

Page: 511

View: 490

This self-contained monograph presents matrix algorithms and their analysis. The new technique enables not only the solution of linear systems but also the approximation of matrix functions, e.g., the matrix exponential. Other applications include the solution of matrix equations, e.g., the Lyapunov or Riccati equation. The required mathematical background can be found in the appendix. The numerical treatment of fully populated large-scale matrices is usually rather costly. However, the technique of hierarchical matrices makes it possible to store matrices and to perform matrix operations approximately with almost linear cost and a controllable degree of approximation error. For important classes of matrices, the computational cost increases only logarithmically with the approximation error. The operations provided include the matrix inversion and LU decomposition. Since large-scale linear algebra problems are standard in scientific computing, the subject of hierarchical matrices is of interest to scientists in computational mathematics, physics, chemistry and engineering.

Simulation and Modeling Methodologies Technologies and Applications

Moreover, the original N-agents system can be modeled exactly by a hybrid tensor system with a small rank of 2N. ... W (2012) Tensor spaces and numerical tensor calculus, vol 42 of Springer series in computational mathematics.


Author: Mohammad S. Obaidat

Publisher: Springer

ISBN: 9783319114576

Category: Computers

Page: 272

View: 991

This book includes extended and revised versions of a set of selected papers from the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2013) which was co-organized by the Reykjavik University (RU) and sponsored by the Institute for Systems and Technologies of Information, Control and Communication (INSTICC). SIMULTECH 2013 was held in cooperation with the ACM SIGSIM - Special Interest Group (SIG) on SImulation and Modeling (SIM), Movimento Italiano Modellazione e Simulazione (MIMOS) and AIS Special Interest Group on Modeling and Simulation (AIS SIGMAS) and technically co-sponsored by the Society for Modeling & Simulation International (SCS), Liophant Simulation, Simulation Team and International Federation for Information Processing (IFIP). This proceedings brings together researchers, engineers, applied mathematicians and practitioners working in the advances and applications in the field of system simulation.

Latent Variable Analysis and Signal Separation

Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Series in Computational Mathematics. Springer, Heidelberg (2012) 4. Comon, P.: Tensors: a brief introduction. IEEE Sig. Proc. Mag. 31(3), 44–53 (2014).


Author: Yannick Deville

Publisher: Springer

ISBN: 9783319937649

Category: Computers

Page: 580

View: 100

This book constitutes the proceedings of the 14th International Conference on Latent Variable Analysis and Signal Separation, LVA/ICA 2018, held in Guildford, UK, in July 2018.The 52 full papers were carefully reviewed and selected from 62 initial submissions. As research topics the papers encompass a wide range of general mixtures of latent variables models but also theories and tools drawn from a great variety of disciplines such as structured tensor decompositions and applications; matrix and tensor factorizations; ICA methods; nonlinear mixtures; audio data and methods; signal separation evaluation campaign; deep learning and data-driven methods; advances in phase retrieval and applications; sparsity-related methods; and biomedical data and methods.

Handbook of Variational Methods for Nonlinear Geometric Data

36(3), 1167–1192 (2016) Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer, Heidelberg (2012) Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15(5), 706–722 (2009) Haegeman, ...


Author: Philipp Grohs

Publisher: Springer Nature

ISBN: 9783030313517

Category: Mathematics

Page: 701

View: 960

This book covers different, current research directions in the context of variational methods for non-linear geometric data. Each chapter is authored by leading experts in the respective discipline and provides an introduction, an overview and a description of the current state of the art. Non-linear geometric data arises in various applications in science and engineering. Examples of nonlinear data spaces are diverse and include, for instance, nonlinear spaces of matrices, spaces of curves, shapes as well as manifolds of probability measures. Applications can be found in biology, medicine, product engineering, geography and computer vision for instance. Variational methods on the other hand have evolved to being amongst the most powerful tools for applied mathematics. They involve techniques from various branches of mathematics such as statistics, modeling, optimization, numerical mathematics and analysis. The vast majority of research on variational methods, however, is focused on data in linear spaces. Variational methods for non-linear data is currently an emerging research topic. As a result, and since such methods involve various branches of mathematics, there is a plethora of different, recent approaches dealing with different aspects of variational methods for nonlinear geometric data. Research results are rather scattered and appear in journals of different mathematical communities. The main purpose of the book is to account for that by providing, for the first time, a comprehensive collection of different research directions and existing approaches in this context. It is organized in a way that leading researchers from the different fields provide an introductory overview of recent research directions in their respective discipline. As such, the book is a unique reference work for both newcomers in the field of variational methods for non-linear geometric data, as well as for established experts that aim at to exploit new research directions or collaborations. Chapter 9 of this book is available open access under a CC BY 4.0 license at link.springer.com.

Many Electron Approaches in Physics Chemistry and Mathematics

Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM. J. Matrix Anal. Appl. 31, 2029 (2010) 9. Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus, SSCM, vol. 42. Springer, Heidelberg (2012) Hackbusch, W., ...


Author: Volker Bach

Publisher: Springer

ISBN: 9783319063799

Category: Science

Page: 417

View: 873

This book provides a broad description of the development and (computational) application of many-electron approaches from a multidisciplinary perspective. In the context of studying many-electron systems Computer Science, Chemistry, Mathematics and Physics are all intimately interconnected. However, beyond a handful of communities working at the interface between these disciplines, there is still a marked separation of subjects. This book seeks to offer a common platform for possible exchanges between the various fields and to introduce the reader to perspectives for potential further developments across the disciplines. The rapid advances of modern technology will inevitably require substantial improvements in the approaches currently used, which will in turn make exchanges between disciplines indispensable. In essence this book is one of the very first attempts at an interdisciplinary approach to the many-electron problem.

Multilinear Subspace Learning

Guo, W., Kotsia, I., and Patras, I. Tensor learning for regression. IEEE Transactions on Image Processing, 21(2):816–827, 2012. Hackbusch, W. Tensor Spaces and Numerical Tensor Calculus, volume 42. Springer, Berlin, 2012.


Author: Haiping Lu

Publisher: CRC Press

ISBN: 9781439857243

Category: Computers

Page: 296

View: 544

Due to advances in sensor, storage, and networking technologies, data is being generated on a daily basis at an ever-increasing pace in a wide range of applications, including cloud computing, mobile Internet, and medical imaging. This large multidimensional data requires more efficient dimensionality reduction schemes than the traditional techniques. Addressing this need, multilinear subspace learning (MSL) reduces the dimensionality of big data directly from its natural multidimensional representation, a tensor. Multilinear Subspace Learning: Dimensionality Reduction of Multidimensional Data gives a comprehensive introduction to both theoretical and practical aspects of MSL for the dimensionality reduction of multidimensional data based on tensors. It covers the fundamentals, algorithms, and applications of MSL. Emphasizing essential concepts and system-level perspectives, the authors provide a foundation for solving many of today’s most interesting and challenging problems in big multidimensional data processing. They trace the history of MSL, detail recent advances, and explore future developments and emerging applications. The book follows a unifying MSL framework formulation to systematically derive representative MSL algorithms. It describes various applications of the algorithms, along with their pseudocode. Implementation tips help practitioners in further development, evaluation, and application. The book also provides researchers with useful theoretical information on big multidimensional data in machine learning and pattern recognition. MATLAB® source code, data, and other materials are available at www.comp.hkbu.edu.hk/~haiping/MSL.html

Tensor Calculus and Analytical Dynamics

1.3.9 Tensor Calculus ( TC ) ............... 1.3.10 Subspaces in a ... Chapter 2 Tensor Algebra 2.1 Introduction : Affine and Euclidean , or Metric , Vector Spaces . ............. 31 2.1.1 Vectors 31 ... 51 2.5.9 Numerical Tensors .


Author: John G. Papastavridis

Publisher: Routledge

ISBN: 9781351411622

Category: Mathematics

Page: 448

View: 361

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.

Computational Methods for Solids and Fluids

John Wiley & Sons, Chichester (2007) Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer, Berlin (2012) Hida, T., Kuo, H.H., Potthoff, J., Streit, L.: White Noise—An Infinite Dimensional Calculus.


Author: Adnan Ibrahimbegovic

Publisher: Springer

ISBN: 9783319279961

Category: Technology & Engineering

Page: 493

View: 325

This volume contains the best papers presented at the 2nd ECCOMAS International Conference on Multiscale Computations for Solids and Fluids, held June 10-12, 2015. Topics dealt with include multiscale strategy for efficient development of scientific software for large-scale computations, coupled probability-nonlinear-mechanics problems and solution methods, and modern mathematical and computational setting for multi-phase flows and fluid-structure interaction. The papers consist of contributions by six experts who taught short courses prior to the conference, along with several selected articles from other participants dealing with complementary issues, covering both solid mechanics and applied mathematics.

Elliptic Differential Equations

Pitman, Boston (1985) Großmann, C., Roos, H.G., Stynes: Numerical Treatment of Partial Differential Equations. ... Springer, Berlin (2003) Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus, SSCM, Vol. 42.


Author: Wolfgang Hackbusch

Publisher: Springer

ISBN: 9783662549612

Category: Mathematics

Page: 455

View: 744

This book simultaneously presents the theory and the numerical treatment of elliptic boundary value problems, since an understanding of the theory is necessary for the numerical analysis of the discretisation. It first discusses the Laplace equation and its finite difference discretisation before addressing the general linear differential equation of second order. The variational formulation together with the necessary background from functional analysis provides the basis for the Galerkin and finite-element methods, which are explored in detail. A more advanced chapter leads the reader to the theory of regularity. Individual chapters are devoted to singularly perturbed as well as to elliptic eigenvalue problems. The book also presents the Stokes problem and its discretisation as an example of a saddle-point problem taking into account its relevance to applications in fluid dynamics.


(iii) The tensor is called the metric tensor, because, all essential metric properties of Euclidean space are ... Somtimes, ds2 I egijdzvidwj where the numerical factor e, called the indicator, equal to +1 or —1 so that ds2 is always ...



Publisher: PHI Learning Pvt. Ltd.

ISBN: 9788120345072

Category: Mathematics

Page: 552

View: 146

Primarily intended for the undergraduate and postgraduate students of mathematics, this textbook covers both geometry and tensor in a single volume. This book aims to provide a conceptual exposition of the fundamental results in the theory of tensors. It also illustrates the applications of tensors to differential geometry, mechanics and relativity. Organized in ten chapters, it provides the origin and nature of the tensor along with the scope of the tensor calculus. Besides this, it also discusses N-dimensional Riemannian space, characteristic peculiarity of Riemannian space, intrinsic property of surfaces, and properties and transformation of Christoffel’s symbols. Besides the students of mathematics, this book will be equally useful for the postgraduate students of physics. KEY FEATURES : Contains 250 worked out examples Includes more than 350 unsolved problems Gives thorough foundation in Tensors

Tensor Calculus

0(*"- *S*)(*n- *S), where 5" is unity if p and r have the same numerical value and 0 otherwise. ... J-r|mn -*r|nm- *- *Jim where 8 214 Xs - — - ^= -4- r* Vs - Vp Vs By 8.213, Ru,, is a tensor; we call it the 294 NON-RlEmANNIAN SPACES.


Author: John Lighton Synge

Publisher: Courier Corporation

ISBN: 0486636127

Category: Mathematics

Page: 324

View: 160

"This book is an excellent classroom text, since it is clearly written, contains numerous problems and exercises, and at the end of each chapter has a summary of the significant results of the chapter." — Quarterly of Applied Mathematics. Fundamental introduction for beginning student of absolute differential calculus and for those interested in applications of tensor calculus to mathematical physics and engineering. Topics include spaces and tensors; basic operations in Riemannian space, curvature of space, special types of space, relative tensors, ideas of volume, and more.

Matrix Calculus Kronecker Product And Tensor Product A Practical Approach To Linear Algebra Multilinear Algebra And Tensor Calculus With Software Implementations Third Edition

Chapter 4 Tensor Product 4.1 Hilbert Spaces In this section we introduce the concept of a Hilbert space (Young ... A linear space L is called a pre-Hilbert space if there is defined a numerical function called the scalar product (or ...


Author: Hardy Yorick

Publisher: World Scientific

ISBN: 9789811202537

Category: Mathematics

Page: 388

View: 911

Our self-contained volume provides an accessible introduction to linear and multilinear algebra as well as tensor calculus. Besides the standard techniques for linear algebra, multilinear algebra and tensor calculus, many advanced topics are included where emphasis is placed on the Kronecker product and tensor product. The Kronecker product has widespread applications in signal processing, discrete wavelets, statistical physics, Hopf algebra, Yang-Baxter relations, computer graphics, fractals, quantum mechanics, quantum computing, entanglement, teleportation and partial trace. All these fields are covered comprehensively.The volume contains many detailed worked-out examples. Each chapter includes useful exercises and supplementary problems. In the last chapter, software implementations are provided for different concepts. The volume is well suited for pure and applied mathematicians as well as theoretical physicists and engineers.New topics added to the third edition are: mutually unbiased bases, Cayley transform, spectral theorem, nonnormal matrices, Gâteaux derivatives and matrices, trace and partial trace, spin coherent states, Clebsch-Gordan series, entanglement, hyperdeterminant, tensor eigenvalue problem, Carleman matrix and Bell matrix, tensor fields and Ricci tensors, and software implementations.