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Krister Åhlander - One of the best experts on this subject based on the ideXlab platform.

  • Supporting tensor symmetries in EinSum
    Computers & Mathematics with Applications, 2003
    Co-Authors: Krister Åhlander
    Abstract:

    Abstract Exploiting symmetries are important in numerical mathematics, both with respect to efficient memory usage and with respect to symmetry exploiting algorithms. In this paper, the symmetries of tensors are in focus. A convenient notation for describing coordinate-free tensor symmetries is established, based on sets of permutations. Completely symmetric and antisymmetric tensors are included as special cases. The extensions to multidimensional arrays with other kinds of symmetries or invariant features are also treated. The symmetry information is used to represent tensors with symmetries more economically with respect to memory. In addition, three algorithms that exploit symmetries are presented. First, a Frobenius norm computation is derived. Second, a projection to an index space with general symmetries is shown, and proven to be optimal in the Frobenius norm. Third, a symmetry utilizing formula for a dual mapping between completely antisymmetric index spaces is shown. The implementation of symmetry support in EinSum is discussed. EinSum is a C++ package primarily intended for tensor algebra, capable of supporting the Einstein Summation Convention. Details on the symmetry part of the implementation are explained. Code for the implementation of the Frobenius norm, the general projection, and the dual mapping is shown, illustrating how symmetry aware software may decrease both the memory usage and the number of arithmetic operations.

  • on software support for finite difference schemes based on index notation
    International Conference on Computational Science, 2002
    Co-Authors: Krister Åhlander, Kurt Otto
    Abstract:

    A formulation of finite difference schemes based on the index notation of tensor algebra is advocated. Finite difference operators on regular grids may be described as sparse, banded, "tensors". Especially for 3D, it is claimed that index notation better corresponds to the inherent problem structure than does Conventional matrix notation. The transition from mathematical index notation to implementation is discussed. Software support for index notation that obeys the Einstein Summation Convention has been implemented in the C++ package Ein-Sum. The extension of EinSum to support typical data structures of finite difference schemes is outlined. A combination of general index notation software and special-purpose routines for instance for fast transforms is envisioned.

  • Einstein Summation for multidimensional arrays
    Computers & Mathematics with Applications, 2002
    Co-Authors: Krister Åhlander
    Abstract:

    Abstract One of the most common data structures, at least in scientific computing, is the multidimensional array. Some numerical algorithms may conveniently be expressed as a generalized matrix multiplication, which computes a multidimensional array from two other multidimensional arrays. By adopting index notation with the Einstein Summation Convention, an elegant tool for expressing generalized matrix multiplications is obtained. Index notation is the succinct and compact notation primarily used in tensor calculus. In this paper, we develop computer support for index notation as a domain specific language. Grammar and semantics are proposed, yielding an unambiguous interpretation algorithm. An object-oriented implementation of a C++ library that supports index notation is described. A key advantage with computer support of index notation is that the notational gap between a mathematical index notation algorithm and its implementation in a computer language is avoided. This facilitates program construction as well as program understanding. Program examples that demonstrate the close resemblance between code and the original mathematical formulation are presented.

  • Einstein Summation for multi-dimensional arrays
    2000
    Co-Authors: Krister Åhlander
    Abstract:

    One of the most common data abstractions, at least in scientific computing, is the multi-dimensional array. A numerical algorithm may sometimes conveniently be expressed as a generalized matrix multiplication, which computes a multi-dimensional array from two other multi-dimensional arrays. By adopting index notation with the Einstein Summation Convention, an elegant tool for expressing generalized matrix multiplications is obtained. Index notation is the succinct and compact notation primarily used in tensor calculus. In this paper, we develop computer support for index notation as a domain specific language. Grammar and semantics are proposed, yielding an unambiguous interpretation algorithm. An object-oriented implementation of a C++ library that supports index notation is described. A ke

  • Einstein Summation for Multi-Dimensional Arrays
    2000
    Co-Authors: Krister Åhlander
    Abstract:

    One of the most common data abstractions, at least in scientific computing, is the multi-dimensional array. A numerical algorithm may sometimes conveniently be expressed as a generalized matrix multiplication, which computes a multi-dimensional array from two other multi-dimensional arrays. By adopting index notation with the Einstein Summation Convention, an elegant tool for expressing generalized matrix multiplications is obtained. Index notation is the succinct and compact notation primarily used in tensor calculus. In this paper, we develop computer support for index notation as a domain specific language. Grammar and semantics are proposed, yielding an unambiguous interpretation. An object-oriented implementation of a C++ library that supports index notation is described. A key advantage with computer support of index notation is that the notational gap between a mathematical index notation algorithm and its implementation in a computer language is avoided. This facilitates program construction as well as program understanding. Program examples that demonstrate the resemblance between code and the original mathematical formulation are presented. Key words: Index notation, domain specific language, tensor calculus

Ahmad Rashid - One of the best experts on this subject based on the ideXlab platform.

  • Star product on non(anti)commutative superspace
    2010
    Co-Authors: Ahmad Rashid
    Abstract:

    Diese Arbeit widmet sich der expliziten Berechnung des Sternprodukts im bosonischen als auch im Superraum und dessen Verallgemeinerung auf den nicht-assoziativen Fall. Koordinatenfunktionen bilden in der klassischen Geometrie Generatoren einer kommutativen Algebra. Diese werden in der nicht-kommutativen Geometrie mit Elementen einer nicht-kommutativen aber üblicherweise assoziativen Algebra ersetzt, während in dieser Arbeit auch der nicht-assoziative Fall behandelt wird. Koordinaten-Monome als Basis der kommutativen Funktionenalgebra lassen sich isomorph abbilden auf Weyl-geordnete Monome der Generatoren der nicht-kommutativen Algebra. Über diese Einbettung induziert die nicht-kommutative Algebra ein nicht-kommutatives Produkt im Funktionenraum, eben das Sternprodukt. Vor Kurzem wurde eine effektive Methode für die explizite iterative Berechnung des assoziativen bosonischen Sternprodukts vorgeschlagen. Es basiert auf die Darstellung der nicht-kommutativen Algebra über Polydifferentialoperatoren. Diese Herangehensweise wird in der vorliegenden Arbeit bis zur dritten Ordnung im Entwicklungsparameter vorgestellt und auf den Superraum erweitert. Auch Überlegungen zu einer möglichen Erweiterung dieser Methode auf den nicht-assoziativen Fall werden angestellt. In einer der zwei vorgeschlagenen Verallgemeinerungen wird das nicht-assoziative Sternprodukt bis zu zweiter Ordnung berechnet und eine Zyklizitäts-Bedingung wird untersucht. Hat man einmal das Sternprodukt gegeben, ist es naheliegend, seine diversen Eigenschaften zu studieren. Diffeomorphismen auf kommutativen Koordinatenräumen können über die Komultiplikation der Hopfalgebra definiert werden. Entsprechend kann man deformierte Diffeomorphismen auf dem nicht-kommutativen Koordinatenraum über die Deformation der Komultiplikation und somit der Hopfalgebra definieren. In der vorliegenden Arbeit wird vorgeschlagen, Quantenkorrekturen der klassischen Transformationen des Sternprodukts unter der Lie-Ableitung über den Formality-Satz zu berechnen. Dies wird dann bis zu zweiter Ordnung im Entwicklungsparameter durchgeführt. Sämtliche Rechnungen dieser Arbeit werden für graduierte Objekte ausgeführt, sehen aber aufgrund der verwendeten graduierten Einsteinschen Summationskonvention aus wie im bosonischen Fall. Am Ende werden jedoch auch einige Ergebnisse explizit in bosonische und fermionische Anteile getrennt. Die Twist-Darstellung des Sternproduktes auf nicht-(anti-)kommutativem Superraum wird im Anhang erläutert.This thesis is devoted to the explicit calculation of the star product in bosonic space as well as in superspace and its generalization to the nonassociative case. Promoting the coordinate functions as elements of a commutative algebra to elements of a noncommutative associative algebra is carefully reviewed and further generalized to a nonassociative algebra. The coordinate monomials as basis of the commutative algebra of functions are naturally mapped to Weyl ordered monomials of the generators of the noncommutative algebra. Via this embedding the noncommutative algebra product induces a noncommutative product on the space of functions, namely the star product. Recently an effective method for the explicit calculation of the star product to higher derivative orders has been presented, based on a representation of the non commutative algebra via polydifferential operators. This approach is reviewed up to third order in the expansion parameter and generalized to the superspace. Comments on a possible extension of the method to the nonassociative case are given. In one of the proposed approaches the non associative star product is calculated to the second order and a cyclicity condition is imposed. Once we have the star product at every order, it is compelling to look for its different properties. Diffeomorphisms on commutative coordinate space are defined with the help of comultiplication of the Hopf algebra and deformed diffeomorphisms are introduced on noncommutative coordinate space by deforming the comultiplication and hence the Hopf algebra. Quantum corrections to the classical transformation of the star product under Lie derivative are proposed via the formality theorem and computed to the second order in the star product expansion parameter. Although all the calculations are done with graded objects, equations look as in the bosonic case due to the use of a graded Einstein Summation Convention. However, different components of the star product on the non(anti)commutative superspace are explicitly computed at the end. The twist representation of the star product on non(anti)superspace is given in the appendix.Rashid AhmadWien, Techn. Univ., Diss., 2010OeBB(VLID)161119

  • Star product on non(anti)commutative superspace
    2010
    Co-Authors: Ahmad Rashid
    Abstract:

    Diese Arbeit widmet sich der expliziten Berechnung des Sternprodukts im bosonischen als auch im Superraum und dessen Verallgemeinerung auf den nicht-assoziativen Fall. Koordinatenfunktionen bilden in der klassischen Geometrie Generatoren einer kommutativen Algebra. Diese werden in der nicht-kommutativen Geometrie mit Elementen einer nicht-kommutativen aber üblicherweise assoziativen Algebra ersetzt, während in dieser Arbeit auch der nicht-assoziative Fall behandelt wird. Koordinaten-Monome als Basis der kommutativen Funktionenalgebra lassen sich isomorph abbilden auf Weyl-geordnete Monome der Generatoren der nicht-kommutativen Algebra. Über diese Einbettung induziert die nicht-kommutative Algebra ein nicht-kommutatives Produkt im Funktionenraum, eben das Sternprodukt.Vor Kurzem wurde eine effektive Methode für die explizite iterative Berechnung des assoziativen bosonischen Sternprodukts vorgeschlagen. Es basiert auf die Darstellung der nicht-kommutativen Algebra über Polydifferentialoperatoren. Diese Herangehensweise wird in der vorliegenden Arbeit bis zur dritten Ordnung im Entwicklungsparameter vorgestellt und auf den Superraum erweitert. Auch Überlegungen zu einer möglichen Erweiterung dieser Methode auf den nicht-assoziativen Fall werden angestellt. In einer der zwei vorgeschlagenen Verallgemeinerungen wird das nicht-assoziative Sternprodukt bis zu zweiter Ordnung berechnet und eine Zyklizitäts-Bedingung wird untersucht.Hat man einmal das Sternprodukt gegeben, ist es naheliegend, seine diversen Eigenschaften zu studieren. Diffeomorphismen auf kommutativen Koordinatenräumen können über die Komultiplikation der Hopfalgebra definiert werden. Entsprechend kann man deformierte Diffeomorphismen auf dem nicht-kommutativen Koordinatenraum über die Deformation der Komultiplikation und somit der Hopfalgebra definieren. In der vorliegenden Arbeit wird vorgeschlagen, Quantenkorrekturen der klassischen Transformationen des Sternprodukts unter der Lie-Ableitung über den Formality-Satz zu berechnen. Dies wird dann bis zu zweiter Ordnung im Entwicklungsparameter durchgeführt.Sämtliche Rechnungen dieser Arbeit werden für graduierte Objekte ausgeführt, sehen aber aufgrund der verwendeten graduierten Einsteinschen Summationskonvention aus wie im bosonischen Fall. Am Ende werden jedoch auch einige Ergebnisse explizit in bosonische und fermionische Anteile getrennt.Die Twist-Darstellung des Sternproduktes auf nicht-(anti-)kommutativem Superraum wird im Anhang erläutert.This thesis is devoted to the explicit calculation of the star product in bosonic space as well as in superspace and its generalization to the nonassociative case.Promoting the coordinate functions as elements of a commutative algebra to elements of a noncommutative associative algebra is carefully reviewed and further generalized to a nonassociative algebra. The coordinate monomials as basis of the commutative algebra of functions are naturally mapped to Weyl ordered monomials of the generators of the noncommutative algebra. Via this embedding the noncommutative algebra product induces a noncommutative product on the space of functions, namely the star product. Recently an effective method for the explicit calculation of the star product to higher derivative orders has been presented, based on a representation of the non commutative algebra via polydifferential operators. This approach is reviewed up to third order in the expansion parameter and generalized to the superspace. Comments on a possible extension of the method to the nonassociative case are given. In one of the proposed approaches the non associative star product is calculated to the second order and a cyclicity condition is imposed.Once we have the star product at every order, it is compelling to look for its different properties.Diffeomorphisms on commutative coordinate space are defined with the help of comultiplication of the Hopf algebra and deformed diffeomorphisms are introduced on noncommutative coordinate space by deforming the comultiplication and hence the Hopf algebra.Quantum corrections to the classical transformation of the star product under Lie derivative are proposed via the formality theorem and computed to the second order in the star product expansion parameter. Although all the calculations are done with graded objects, equations look as in the bosonic case due to the use of a graded Einstein Summation Convention. However, different components of the star product on the non(anti)commutative superspace are explicitly computed at the end. The twist representation of the star product on non(anti)superspace is given in the appendix.7

Cimrman Robert - One of the best experts on this subject based on the ideXlab platform.

  • Fast Evaluation of Finite Element Weak Forms Using Python Tensor Contraction Packages
    'Elsevier BV', 2021
    Co-Authors: Cimrman Robert
    Abstract:

    In finite element calculations, the integral forms are usually evaluated using nested loops over elements, and over quadrature points. Many such forms (e.g. linear or multi-linear) can be expressed in a compact way, without the explicit loops, using a single tensor contraction expression by employing the Einstein Summation Convention. To automate this process and leverage existing high performance codes, we first introduce a notation allowing trivial differentiation of multi-linear finite element forms. Based on that we propose and describe a new transpiler from Einstein Summation based expressions, augmented to allow defining multi-linear finite element weak forms, to regular tensor contraction expressions. The resulting expressions are compatible with a number of Python scientific computing packages, that implement, optimize and in some cases parallelize the general tensor contractions. We assess the performance of those packages, as well as the influence of operand memory layouts and tensor contraction paths optimizations on the elapsed time and memory requirements of the finite element form evaluations. We also compare the efficiency of the transpiled weak form implementations to the C-based functions available in the finite element package SfePy

Alain Vautrin - One of the best experts on this subject based on the ideXlab platform.

  • Hybrid finite elements for the computation of sandwich plates
    2013
    Co-Authors: Vincent Manet, Woo-suck Han, Alain Vautrin
    Abstract:

    In this article, finite elements are developed on the basis of the displacements and the Pian and Tong functionals with help of Lagrange multipliers. These elements represent sandwich structure in an accurate way, especially at interfaces, where the equilibrium state is satisfied. Developed elements are assessed and compared on several examples of static linear problems. Notations In this article, Einstein Summation Convention is used, but does not apply between superior indices, which designate element, and inferior indices, which denotes components: it only applies between inferior indices. We note T = σn (or Ti = σijnj) the projection of σ on the outgoing normal n to a considered domain Ω on its boundary Γ: it is the normal trace of the stress tensor on Γ. K stands for the stiffness matrix of the whole system, Q the overall vector of nodal unknowns and F the overall vector of equivalent forces.

Stephen Bedding - One of the best experts on this subject based on the ideXlab platform.