The Experts below are selected from a list of 240 Experts worldwide ranked by ideXlab platform
Krister Åhlander - One of the best experts on this subject based on the ideXlab platform.
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Supporting tensor symmetries in EinSum
Computers & Mathematics with Applications, 2003Co-Authors: Krister ÅhlanderAbstract: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.
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on software support for finite difference schemes based on index notation
International Conference on Computational Science, 2002Co-Authors: Krister Åhlander, Kurt OttoAbstract: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.
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International Conference on Computational Science (3) - On Software Support for Finite Difference Schemes Based on Index Notation
Lecture Notes in Computer Science, 2002Co-Authors: Krister Åhlander, Kurt OttoAbstract: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.
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Einstein Summation for multidimensional arrays
Computers & Mathematics with Applications, 2002Co-Authors: Krister ÅhlanderAbstract: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.
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Einstein Summation for multi-dimensional arrays
2000Co-Authors: Krister ÅhlanderAbstract: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
Kurt Otto - One of the best experts on this subject based on the ideXlab platform.
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on software support for finite difference schemes based on index notation
International Conference on Computational Science, 2002Co-Authors: Krister Åhlander, Kurt OttoAbstract: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.
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International Conference on Computational Science (3) - On Software Support for Finite Difference Schemes Based on Index Notation
Lecture Notes in Computer Science, 2002Co-Authors: Krister Åhlander, Kurt OttoAbstract: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.
Luis Vega - One of the best experts on this subject based on the ideXlab platform.
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The Cauchy problem for quasi-linear Schrödinger equations
Inventiones mathematicae, 2004Co-Authors: Carlos E. Kenig, Gustavo Ponce, Luis VegaAbstract:with x ∈ Rn, t ∈ R with ∇ = (∂x1, .., ∂xn) and Summation Convention. One may think of the equation in (1.1) as a non-linear Schrodinger equation where the operator modeling the dispersion relation is non-isotropic and depends also on the unknown function, its conjugate and their gradients in the space variables. Under appropriate assumptions on the function F the method presented here extends to fully non-linear Schrodinger equations of the form
Carlos E. Kenig - One of the best experts on this subject based on the ideXlab platform.
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The Cauchy problem for quasi-linear Schrödinger equations
Inventiones mathematicae, 2004Co-Authors: Carlos E. Kenig, Gustavo Ponce, Luis VegaAbstract:with x ∈ Rn, t ∈ R with ∇ = (∂x1, .., ∂xn) and Summation Convention. One may think of the equation in (1.1) as a non-linear Schrodinger equation where the operator modeling the dispersion relation is non-isotropic and depends also on the unknown function, its conjugate and their gradients in the space variables. Under appropriate assumptions on the function F the method presented here extends to fully non-linear Schrodinger equations of the form
Andrew D. Lewis - One of the best experts on this subject based on the ideXlab platform.
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Linear and multilinear algebra
Texts in Applied Mathematics, 2005Co-Authors: Francesco Bullo, Andrew D. LewisAbstract:The study of the geometry of Lagrangian mechanics requires that one be familiar with basic concepts in abstract linear and multilinear algebra. The reader is expected to have encountered at least some of these concepts before, so this chapter serves primarily as a refresher. We also use our discussion of linear algebra as a means of introducing the Summation Convention in a systematic manner. Since this gets used in computations, the reader may wish to take the opportunity to become familiar with it.
