DOCSLIB.ORG

Performance Tensor Transpose Library for Gpus

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Performance Tensor Transpose Library for Gpus S7255: CUTT: A HIGH- PERFORMANCE TENSOR TRANSPOSE LIBRARY FOR GPUS Antti-Pekka Hynninen, 5/10/2017, GTC2017, San Jose CA MOTIVATION Tensor contractions Tensor contractions are the most computationally intensive part of quantum many- body methods used in NWCHEM, DIRAC, LS-DALTON, and ACES-IV 퐷 푎, 푏, += 퐿 푎, 푐, , 푘 푅 푘, 푏, 푐 Sum over repeated indices 푐 and 푘 Evaluating tensor contractions directly requires implementing a lot of hard-to-write custom code Indirect approach transposes tensors and uses efficient linear algebra libraries (such as cuBLAS) to perform matrix multiply 2 TENSOR CONTRACTIONS Indirect approach Reduction over a pair of indices shared by two tensors, e.g. 퐷 푎, 푏, += 퐿 푎, 푐, , 푘 푅 푘, 푏, 푐 This can be evaluated as 퐿 푎, 푐, , 푘 → 퐿 푎, , 푘, 푐 # tensor transpose 푅 푘, 푏, 푐 → 푅 푘, 푐, 푏 # tensor transpose 퐷 푎, , 푏 += 퐿 푎, , 푘, 푐 푅 푘, 푐, 푏 # matrix multiply 퐷 푎, , 푏 → 퐷 푎, 푏, # tensor transpose Able to take advantage of the high-performance matrix multiply routines provided by cuBLAS 3 PREVIOUS WORK No runtime high-performance tensor transpose library exists for GPUs Previous implementation by my co-author [1] was sub-optimal on GPU platforms Work in [2] relies on compiler to build custom kernels e.g. not runtime [1] Dmitry I. Lyakh. 2015. An efficient tensor transpose algorithm for multicore CPU, Intel Xeon Phi, and NVidia Tesla GPU. Computer Physics Communications 189, (2015), 84-91. DOI: http://dx.doi.org/10.1016/j.cpc.2014.12.013 [2] Paul Springer, Aravind Sankaran, and Paolo Bientinesi. 2016. TTC: A Tensor Transposition Compiler for Multiple Architectures 4 TENSOR TRANSPOSE ALGORITHMS 5 MATRIX TRANSPOSE: TILED ALGORITHM () syncthreads __ Step 1: Step 2: Read 32x32 tile from global Read shared memory in transposed memory order to shared memory and write to global memory 6 Mark Harris “An Efficient Matrix Transpose in CUDA C/C+”, Parallel Forall Blog: https://devblogs.nvidia.com/parallelforall/efficient-matrix-transpose-cuda-cc TILED ALGORITHM 1 2 3 4 5 6 Constant shared memory usage (~32x32) Performs well when d1 and d5 are 1 5 2 3 4 6 fairly large (~32) shared memory volume looped over using TB Poor performance for small (2-8) dimensions Would it be possible to pack 5 4 1 6 3 2 multiple small dimensions into shared memory? 7 PACKED ALGORITHM 1 2 3 4 5 6 No longer uses 32x32 shared memory tile Loads entire dimensions into shared memory (not tiled) 1 2 5 3 4 6 As much shared memory is allocated as it shared memory TB loop volume takes to store the elements Must choose which dimensions to pack 5 4 1 6 3 2 New problem: What if e.g. d5 is very large? 8 PACKED-SPLIT ALGORITHM 1 2 3 4 5 6 Split largest dimension Number of splits is determined by the shared memory size 1 2 5 3 4 6 Must choose which dimensions to pack, and shared memory TB loop volume number of splits 5 4 1 6 3 2 9 MEMORY POSITION CALCULATION 10 GLOBAL MEMORY POSITION CALCULATION glRead H = Number of elements in shared memory 1 2 3 4 5 6 M = Number of elements in loop volume s= 0, ..., H-1 p= 0, ..., M-1 Need to convert scalar positions s and p to global memory positions: 1 2 5 3 4 6 glRead = Global memory read shRead glWrite = Global memory write Global memory position is split into: 5 4 1 6 3 2 glWrite glRead = glMinorRead(s) + glMajorRead(p) glWrite = glMinorWrite(s) + glMajorWrite(p) 11 MAJOR POSITION CALCULATION p= 0, ..., M-1 3 4 6 // int p =0,...,M-1 // int c[n] = {1, d3, d3*d4} glMajorRead(p) // int d[n] = {d3, d4, d6} 1 2 3 4 5 6 // int t[n] = {d1*d2, d1*d2*d3, d1*d2*d3*d4*d5} 푛 푝 int glMajorRead = 0; glMajorRead 푝 = ෍ 푚표푑 , 푑푖 푡푖 푐푖 푖=1 for (int i=0;i < n;i++) { glMajorRead += ((p / c[i]) % d[i]) * t[i]; } O(n) Observation: p is constant within thread block (and therefore 12warp) WARP-PARALLEL POSITION CALCULATION p= 0, ..., M-1 // int p = 0,...,M-1 3 4 6 // int c = {1, d3, d3*d4, 1, ..., 1} glMajorRead(p) // int d = {d3, d4, d6 , 1, ..., 1} 1 2 3 4 5 6 // int t = {d1*d2, d1*d2*d3, d1*d2*d3*d4*d5,...} int glMajorRead = ((p / c) % d) * t; 푛 for (int i=16;i >= 1;i/=2) { 푝 ෍ 푚표푑 , 푑푖 푡푖 푐푖 glMajorRead += __shfl_xor(glMajorRead, i); 푖=1 } Single divide, modulo, and multiply O(1) i.e. performance independent of tensor rank Works up to n=32 13 MINOR POSITION CALCULATION For Tiled algorithm this is trivial 1 2 3 4 5 6 For Packed and Packed-Split, pre-compute positions and store into registers glMinorRead(s) s= 0, ..., H-1 Number of registers per thread: 1 2 5 shared memory numReg = (H - 1)/blockDim.x + 1 shRead(s) int glMinorRead[numReg] int shRead[numReg] 5 4 1 6 3 2 int glMinorWrite[numReg] glMinorWrite(s) Template kernel with numReg 14 ALGORITHM & PARAMETER CHOICE 15 CHOOSING THE BEST ALGORITHM Algorithm choice: Tiled, Packed, Packed-Split Tiled: no free parameters 1 2 3 4 5 6 5 4 1 6 3 2 Packed: input and output ranks 1 2 3 4 5 6 5 4 1 6 3 2 Packed-Split: input and output ranks, number of splits 1 2 3 4 5 6 5 4 1 6 3 2 Large performance differences between different algorithm and parameter choices16 CUTT PLANS cuttResult cuttPlanMeasure(cuttHandle* handle, int rank, int* dim, int* permutation, size_t sizeofType, cudaStream_t stream, void* idata, void * odata); cuttResult cuttPlan(cuttHandle* handle, int rank, int* dim, int* permutation, size_t sizeofType, cudaStream_t stream); Measure –plans perform all possible tensor transposes and choose the best performing plan. LARGE overhead Heuristic –plans choose best plan by estimating the transpose runtime based on analytical GPU performance model. SMALL overhead Heuristic plans must be used in QM calculations Getting the heuristic planning to work accurately was a major hurdle Better approach is needed for choosing the heuristic plans (Machine Learning?) 17 BENCHMARKS 18 Tensor ranks 2 to 7 Ratio between largest and smallest tensor dimensions 1:1, 5:1, and 15:1 Tensor volume normally distributed with average BENCHMARK 1 200M elements and standard deviation of 20M elements 500 random permutations for each tensor rank and ratio 9000 tensor transposes in total 19 TESLA K20X * * maximum bandwidth measured using GPU-STREAM: Tom Deakin, James Price, Matt J. Martineau M, and Simon N. McIntosh-Smith. 2016. GPU-STREAM v2.0: Benchmarking the achievable memory bandwidth of many-core processors across diverse parallel programming models. 2016. Paper presented at P^3MA Workshop at ISC High Performance, Frankfurt, Germany 20 TESLA M40 21 TESLA P100 22 Tensor ranks 8 and 12 Rank 8: (5, 3, 2, 4, 35, 33, 37, 40) 200M elements Rank 12: (2, 3, 4, 3, 2, 2, 3, 2, 20, 18, 22, 24) BENCHMARK 2 328M elements 500 random permutations for both tensor ranks Simulates realistic workload in Quantum Chemistry calculations 23 TESLA K20X 24 TESLA M40 25 TESLA P100 26 PERFORMANCE DISTRIBUTION 27 Set of 57 tensor transposes from (TTC): P. Springer, J. R. Hammond, and P. Bientinesi. TTC: A high performance compiler for tensor BENCHMARK 3 transpositions. CoRR, 2016. http://arxiv.org/abs/1603.02297 Somewhat “easy” benchmark due to small number of permutations 28 TESLA K40M TTC average 140 GiB/s cuTT average 144 GiB/s TTC data from: Paul Springer, Aravind Sankaran, and Paolo Bientinesi. 2016. TTC: A Tensor Transposition Compiler for Multiple Architectures. https://arxiv.org/abs/1607.01249 29 Real world tensor contractions performed on TAL- SH (Tensor Algebra Library for Shared Memory Computers) Dmitry I. Lyakh at Oak Ridge National Laboratory BENCHMARK 4 9306 random permutations on tensors up to rank 8 Matrix multiply performed using cuBLAS 30 TESLA K20X GPU (a) 100 (b) 100 90 1200 90 3500 Best Best 80 80 3000 Average 1000 Average 70 70 2500 Worst Worst 60 800 60 2000 50 50 600 1500 40 40 GFlop/s GFlop/s 30 400 30 1000 20 20 500 200 Percentage Percentage of max. performance 10 10 Percentage of max. performance 0 0 0 0 1 10 100 1000 10000 1 10 100 1000 10000 Arithmetic Intensity Arithmetic Intensity Single precision Double precision 2 vol 퐷 vol 퐿 vol 푅 퐴푟푡ℎ푚푒푡푐 퐼푛푡푒푛푠푡푦 = 퐷 = 퐷 + 퐿 ∙ 푅 vol 퐷 + vol 퐿 + vol 푅 31 TESLA M40 Single precision 32 TESLA P100 100 100 9000 4500 (a) (b) Best 90 90 8000 Best 4000 80 Average 80 7000 Average 3500 70 Worst 70 6000 Worst 3000 60 60 5000 2500 50 50 4000 40 2000 40 GFlop/s GFlop/s 3000 30 1500 30 2000 20 1000 20 1000 10 500 10 Percentage of max. performance Percentage of max. performance 0 0 0 0 1 10 100 1000 10000 1 10 100 1000 10000 Arithmetic Intensity Arithmetic Intensity Single precision Double precision 33 CONCLUSIONS & ACKNOWLEDGEMENTS 34 CONCLUSIONS Fully runtime library for high-performance tensor transposing on NVIDIA GPUs Extensive benchmarking Achieves median of 70-80% of the maximum achievable memory bandwidth Performance equals or exceeds the performance of compiler-based approach (TTC) Enables close to peak FLOP tensor contractions on P100 Integrated as part of TAL-SH (https://github.com/DmitryLyakh/TAL_SH) Work underway to be used in NWCHEM, DIRAC, LS-DALTON, and ACES-IV Source code available at: https://github.com/ap-hynninen/cutt Manuscript available at: https://arxiv.org/abs/1705.01598 35 ACKNOWLEDGEMENTS Dmitry I.
Load more

Recommended publications
  • On Manifolds of Tensors of Fixed Tt-Rank
    ON MANIFOLDS OF TENSORS OF FIXED TT-RANK SEBASTIAN HOLTZ, THORSTEN ROHWEDDER, AND REINHOLD SCHNEIDER Abstract. Recently, the format of TT tensors [19, 38, 34, 39] has turned out to be a promising new format for the approximation of solutions of high dimensional problems. In this paper, we prove some new results for the TT representation of a tensor U ∈ Rn1×...×nd and for the manifold of tensors of TT-rank r. As a first result, we prove that the TT (or compression) ranks ri of a tensor U are unique and equal to the respective seperation ranks of U if the components of the TT decomposition are required to fulfil a certain maximal rank condition. We then show that d the set T of TT tensors of fixed rank r forms an embedded manifold in Rn , therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices [7], we introduce certain gauge conditions to obtain a unique representation of the tangent space TU T of T and deduce a local parametrization of the TT manifold. The parametrisation of TU T is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems [33]. We conclude with remarks on those applications and present some numerical examples. 1. Introduction The treatment of high-dimensional problems, typically of problems involving quantities from Rd for larger dimensions d, is still a challenging task for numerical approxima- tion. This is owed to the principal problem that classical approaches for their treatment normally scale exponentially in the dimension d in both needed storage and computa- tional time and thus quickly become computationally infeasable for sensible discretiza- tions of problems of interest.
    [Show full text]
  • Matrices and Tensors
    APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM..
    [Show full text]
  • A Some Basic Rules of Tensor Calculus
    A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e.
    [Show full text]
  • 1.3 Cartesian Tensors a Second-Order Cartesian Tensor Is Defined As A
    1.3 Cartesian tensors A second-order Cartesian tensor is defined as a linear combination of dyadic products as, T Tijee i j . (1.3.1) The coefficients Tij are the components of T . A tensor exists independent of any coordinate system. The tensor will have different components in different coordinate systems. The tensor T has components Tij with respect to basis {ei} and components Tij with respect to basis {e i}, i.e., T T e e T e e . (1.3.2) pq p q ij i j From (1.3.2) and (1.2.4.6), Tpq ep eq TpqQipQjqei e j Tij e i e j . (1.3.3) Tij QipQjqTpq . (1.3.4) Similarly from (1.3.2) and (1.2.4.6) Tij e i e j Tij QipQjqep eq Tpqe p eq , (1.3.5) Tpq QipQjqTij . (1.3.6) Equations (1.3.4) and (1.3.6) are the transformation rules for changing second order tensor components under change of basis. In general Cartesian tensors of higher order can be expressed as T T e e ... e , (1.3.7) ij ...n i j n and the components transform according to Tijk ... QipQjqQkr ...Tpqr... , Tpqr ... QipQjqQkr ...Tijk ... (1.3.8) The tensor product S T of a CT(m) S and a CT(n) T is a CT(m+n) such that S T S T e e e e e e . i1i2 im j1j 2 jn i1 i2 im j1 j2 j n 1.3.1 Contraction T Consider the components i1i2 ip iq in of a CT(n).
    [Show full text]
  • Tensor-Spinor Theory of Gravitation in General Even Space-Time Dimensions
    Physics Letters B 817 (2021) 136288 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Tensor-spinor theory of gravitation in general even space-time dimensions ∗ Hitoshi Nishino a, ,1, Subhash Rajpoot b a Department of Physics, College of Natural Sciences and Mathematics, California State University, 2345 E. San Ramon Avenue, M/S ST90, Fresno, CA 93740, United States of America b Department of Physics & Astronomy, California State University, 1250 Bellflower Boulevard, Long Beach, CA 90840, United States of America a r t i c l e i n f o a b s t r a c t Article history: We present a purely tensor-spinor theory of gravity in arbitrary even D = 2n space-time dimensions. Received 18 March 2021 This is a generalization of the purely vector-spinor theory of gravitation by Bars and MacDowell (BM) in Accepted 9 April 2021 4D to general even dimensions with the signature (2n − 1, 1). In the original BM-theory in D = (3, 1), Available online 21 April 2021 the conventional Einstein equation emerges from a theory based on the vector-spinor field ψμ from a Editor: N. Lambert m lagrangian free of both the fundamental metric gμν and the vierbein eμ . We first improve the original Keywords: BM-formulation by introducing a compensator χ, so that the resulting theory has manifest invariance = =− = Bars-MacDowell theory under the nilpotent local fermionic symmetry: δψ Dμ and δ χ . We next generalize it to D Vector-spinor (2n − 1, 1), following the same principle based on a lagrangian free of fundamental metric or vielbein Tensors-spinors rs − now with the field content (ψμ1···μn−1 , ωμ , χμ1···μn−2 ), where ψμ1···μn−1 (or χμ1···μn−2 ) is a (n 1) (or Metric-less formulation (n − 2)) rank tensor-spinor.
    [Show full text]
  • Low-Level Image Processing with the Structure Multivector
    Low-Level Image Processing with the Structure Multivector Michael Felsberg Bericht Nr. 0202 Institut f¨ur Informatik und Praktische Mathematik der Christian-Albrechts-Universitat¨ zu Kiel Olshausenstr. 40 D – 24098 Kiel e-mail: [email protected] 12. Marz¨ 2002 Dieser Bericht enthalt¨ die Dissertation des Verfassers 1. Gutachter Prof. G. Sommer (Kiel) 2. Gutachter Prof. U. Heute (Kiel) 3. Gutachter Prof. J. J. Koenderink (Utrecht) Datum der mundlichen¨ Prufung:¨ 12.2.2002 To Regina ABSTRACT The present thesis deals with two-dimensional signal processing for computer vi- sion. The main topic is the development of a sophisticated generalization of the one-dimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance – equivariance constraint, and by its relation to complex analysis and potential theory, a two-dimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of one-dimensional functions to two dimensions. For general two-dimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the struc- ture multivector are combined with an appropriate scale-space approach, resulting in generalized quadrature filters.
    [Show full text]
  • The Riemann Curvature Tensor
    The Riemann Curvature Tensor Jennifer Cox May 6, 2019 Project Advisor: Dr. Jonathan Walters Abstract A tensor is a mathematical object that has applications in areas including physics, psychology, and artificial intelligence. The Riemann curvature tensor is a tool used to describe the curvature of n-dimensional spaces such as Riemannian manifolds in the field of differential geometry. The Riemann tensor plays an important role in the theories of general relativity and gravity as well as the curvature of spacetime. This paper will provide an overview of tensors and tensor operations. In particular, properties of the Riemann tensor will be examined. Calculations of the Riemann tensor for several two and three dimensional surfaces such as that of the sphere and torus will be demonstrated. The relationship between the Riemann tensor for the 2-sphere and 3-sphere will be studied, and it will be shown that these tensors satisfy the general equation of the Riemann tensor for an n-dimensional sphere. The connection between the Gaussian curvature and the Riemann curvature tensor will also be shown using Gauss's Theorem Egregium. Keywords: tensor, tensors, Riemann tensor, Riemann curvature tensor, curvature 1 Introduction Coordinate systems are the basis of analytic geometry and are necessary to solve geomet- ric problems using algebraic methods. The introduction of coordinate systems allowed for the blending of algebraic and geometric methods that eventually led to the development of calculus. Reliance on coordinate systems, however, can result in a loss of geometric insight and an unnecessary increase in the complexity of relevant expressions. Tensor calculus is an effective framework that will avoid the cons of relying on coordinate systems.
    [Show full text]
  • Tensor, Exterior and Symmetric Algebras
    Tensor, Exterior and Symmetric Algebras Daniel Murfet May 16, 2006 Throughout this note R is a commutative ring, all modules are left R-modules. If we say a ring is noncommutative, we mean it is not necessarily commutative. Unless otherwise specified, all rings are noncommutative (except for R). If A is a ring then the center of A is the set of all x ∈ A with xy = yx for all y ∈ A. Contents 1 Definitions 1 2 The Tensor Algebra 5 3 The Exterior Algebra 6 3.1 Dimension of the Exterior Powers ............................ 11 3.2 Bilinear Forms ...................................... 14 3.3 Other Properties ..................................... 18 3.3.1 The determinant formula ............................ 18 4 The Symmetric Algebra 19 1 Definitions Definition 1. A R-algebra is a ring morphism φ : R −→ A where A is a ring and the image of φ is contained in the center of A. This is equivalent to A being an R-module and a ring, with r · (ab) = (r · a)b = a(r · b), via the identification of r · 1 and φ(r). A morphism of R-algebras is a ring morphism making the appropriate diagram commute, or equivalently a ring morphism which is also an R-module morphism. In this section RnAlg will denote the category of these R-algebras. We use RAlg to denote the category of commutative R-algebras. A graded ring is a ring A together with a set of subgroups Ad, d ≥ 0 such that A = ⊕d≥0Ad as an abelian group, and st ∈ Ad+e for all s ∈ Ad, t ∈ Ae.
    [Show full text]
  • Lecture 12. Tensors
    Lecture 12. Tensors In this lecture we define tensors on a manifold, and the associated bundles, and operations on tensors. 12.1 Basic definitions We have already seen several examples of the idea we are about to introduce, namely linear (or multilinear) operators acting on vectors on M. For example, the metric is a bilinear operator which takes two vectors to give a real number, i.e. gx : TxM × TxM → R for each x is defined by u, v → gx(u, v). The difference between two connections ∇(1) and ∇(2) is a bilinear op- erator which takes two vectors and gives a vector, i.e. a bilinear operator Sx : TxM × TxM → TxM for each x ∈ M. Similarly, the torsion of a connec- tion has this form. Definition 12.1.1 A covector ω at x ∈ M is a linear map from TxM to R. The set of covectors at x forms an n-dimensional vector space, which we ∗ denote Tx M.Atensor of type (k, l)atx is a multilinear map which takes k vectors and l covectors and gives a real number × × × ∗ × × ∗ → R Tx : .TxM .../0 TxM1 .Tx M .../0 Tx M1 . k times l times Note that a covector is just a tensor of type (1, 0), and a vector is a tensor of type (0, 1), since a vector v acts linearly on a covector ω by v(ω):=ω(v). Multilinearity means that i1 ik j1 jl T c vi1 ,..., c vik , aj1 ω ,..., ajl ω i i j j 1 k 1 l . i1 ik j1 jl = c ...c aj1 ...ajl T (vi1 ,...,vik ,ω ,...,ω ) i1,...,ik,j1,...,jl 110 Lecture 12.
    [Show full text]
  • Whitney Algebras and Grassmann's Regressive Products
    Whitney algebras and Grassmann’s regressive products Andrea Brini and Francesco Regonati Dipartimento di Matematica “Alma Mater Studiorum” Universit`adegli Studi di Bologna October 18, 2018 Abstract Geometric products on tensor powers Λ(V )⊗m of an exterior algebra and on Whitney algebras [13] provide a rigorous version of Grassmann’s regressive products of 1844 [17]. We study geometric products and their relations with other classical operators on exterior algebras, such as the Hodge ∗−operators and the join and meet products in Cayley-Grassmann algebras [2, 30]. We establish encodings of tensor powers Λ(V )⊗m and of Whitney algebras W m(M) in terms of letterplace algebras and of their geometric products in terms of divided powers of polarization operators. We use these encodings to provide simple proofs of the Crapo and Schmitt exchange relations in Whitney algebras and of two typical classes of identities in Cayley-Grassmann algebras . arXiv:1010.2964v1 [math.CO] 14 Oct 2010 We thank Henry Crapo and William Schmitt for their advice, encouragement, and invaluable suggestions Contents 1 Introduction 2 2 The algebras: generalities 5 3 The algebras: basic constructions 6 3.1 Cayley-Grassmannalgebras . .. .. 6 3.2 Tensorpowersofexterioralgebras. 7 3.3 Whitneyalgebras .............................. 11 1 4 Relations among the algebras 12 4.1 From tensor powers of exterior algebras to CG-Algebras . .... 12 4.2 From Whitney algebras to tensor powers of exterior algebras. .... 12 5 The Hodge operators 13 5.1 The Hodge ∗−operators .......................... 13 5.2 ThegeneralizedHodgeoperators . 15 6 Letterplace algebras and polarization operators 17 6.1 Skew-symmetric Letterplace algebras and place polarization operators.
    [Show full text]
  • Appendix a Tensor Mathematics
    Appendix A Tensor Mathematics A.l Introduction In physics we are accustomed to relating quantities to other quantities by means of mathematical expressions. The quantities that represent physical properties of a point or of an infinitesimal volume in space may be of a single numerical value without direction, such as temperature, mass, speed, dist­ ance, specific gravity, energy, etc.; they are defined by a single magnitude and we call them scalars. Other physical quantities have direction as well and their magnitude is represented by an array of numerical values, their number corresponding to the number of dimensions of the space, three numerical values in a three-dimensional space; such quantities are called vectors, for example velocity, acceleration, force, etc., and the three numerical values are the components of the vector in the direction of the coordinates of the space. Still other physical quantities such as moment of inertia, stress, strain, permeability coefficients, electric flux, electromagnetic flux, etc., are repre­ sented in a three-dimensional space by nine numerical quantities, called components, and are known as tensors. We will introduce a slightly different definition. We shall call the scalars tensors of order zero, vectors - tensors of order one and tensors with nine components - tensors of order two. There are, of course, in physics and mathematics, tensors of higher order. Tensors of order three have an array of 27 components and tensors of order four have 81 components and so on. Couple stresses that arise in materials with polar forces are examples of tensors of order three, and the Riemann curvature tensor that appears in geodesics is an example of a tensor of order four.
    [Show full text]
  • Differential Forms on Riemannian (Lorentzian) and Riemann-Cartan
    Differential Forms on Riemannian (Lorentzian) and Riemann-Cartan Structures and Some Applications to Physics∗ Waldyr Alves Rodrigues Jr. Institute of Mathematics Statistics and Scientific Computation IMECC-UNICAMP CP 6065 13083760 Campinas SP Brazil e-mail: [email protected] or [email protected] May 28, 2018 Abstract In this paper after recalling some essential tools concerning the theory of differential forms in the Cartan, Hodge and Clifford bundles over a Riemannian or Riemann-Cartan space or a Lorentzian or Riemann-Cartan spacetime we solve with details several exercises involving different grades of difficult. One of the problems is to show that a recent formula given in [10] for the exterior covariant derivative of the Hodge dual of the torsion 2-forms is simply wrong. We believe that the paper will be useful for students (and eventually for some experts) on applications of differential geometry to some physical problems. A detailed account of the issues discussed in the paper appears in the table of contents. Contents 1 Introduction 3 arXiv:0712.3067v6 [math-ph] 4 Dec 2008 2 Classification of Metric Compatible Structures (M; g;D) 4 2.1 Levi-Civita and Riemann-Cartan Connections . 5 2.2 Spacetime Structures . 5 3 Absolute Differential and Covariant Derivatives 5 ∗published in: Ann. Fond. L. de Broglie 32 (special issue dedicate to torsion), 424-478 (2008). 1 ^ 4 Calculus on the Hodge Bundle ( T ∗M; ·; τg) 8 4.1 Exterior Product . 8 4.2 Scalar Product and Contractions . 9 4.3 Hodge Star Operator ? ........................ 9 4.4 Exterior derivative d and Hodge coderivative δ .
    [Show full text]