Verification of the cross 3D algorithm on quantum chemistry data

By Heinz-Jürgen Flad, Boris N. Khoromskij, Dmitry V. Savostyanov, E. E. Tyrtyshnikov. 2008

33 citations recorded by the source corpus.

Subfields: Atomic and Molecular Physics, and Optics, Computational Mathematics, Nuclear and High Energy Physics

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “In [89, 212], this method has been combined with multilevel ideas and applied to quantum chemistry.”
Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity (2013, methodology): “Alternatively, tensor methods (in the Tucker [44, 45], CP, TT or QTT format) may be applied directly to electronic density functions, potentials and other data and operators involved in computations (see [22, 23, 25, 24] for details and further references).”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, methodology): “5 On tensor-structured solution of multidimensional equations Tensor numerical methods are proved to be efficient for data-sparse representation of functions and operators in the Hartree-Fock and Kohn-Sham models in electronic structure calculations [7, 39, 59, 57, 23, 76, 47].”
Cross approximation in tensor electron density computations (2010, methodology): “Instead, we can apply the recently developed Cross3D algorithm [6, 16] and its multilevel version [9].”
Structure of the Hessian matrix and an economical implementation of Newton’s method in the problem of canonical approximation of tensors (2010, methodology): “In [ 24 ], an approximate Tucker decomposition of S was constructed with the use of three�dimensional cross approximations on a sequence of grids.”
Linear algebra for tensor problems (2009, background): “[10,11] for the 2D case) and developed in [7,9] making a Tucker approximation feasible even for huge-scale tensors.”
O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling (2011, methodology): “3 Toward asymptotically optimal numerical schemes Tensor-structured numerical methods of the linear scaling O(dN), are proved to be efficient for representation of functions and operators in the Hartree-Fock and Kohn-Sham models in electronic structure calculations [3, 18, 29, 27, 12, 38, 22].”
Superfast Fourier Transform Using QTT Approximation (2012, background): “We refer also to [8, 12, 27, 29, 34, 43, 47, 51] for the detailed discussions on recent progress in this ﬁeld.”
Quasi-Newton Methods on Grassmannians and Multilinear Approximations of Tensors (2009, methodology): “Yet a third approach to obtain low multilinear rank tensor approximations are the cross methods in [48, 47, 25].”
Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods (2011, unknown): “6.1 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140”
Adaptive Cross Approximation of Multivariate Functions (2011, methodology): “A method that is similar to the kind of matrix approximation treated in this article (at least for the trivariate case) was presented in [ 14 , 23].”
Black Box Low Tensor-Rank Approximation Using Fiber-Crosses (2009, background): “The only reference—to our knowledge—for an alternative approach in the literature is the three-dimensional cross approximation [13] (see also [2] for low rank approximation on given point sets and [8] for an application).”
Use of tensor formats in elliptic eigenvalue problems (2012, unknown): “Copyright © 2011 John Wiley & Sons, Ltd. KEY WORDS: elliptic operators; spectra; eigenfunctions; separation of variables; separable approximations; matrix approximations; low-rank matrices; Kronecker products; multi-dimensional matrices; tensors; numerical methods; iterative algorithms”
Wedderburn Rank Reduction and Krylov Subspace Method for Tensor Approximation. Part 1: Tucker Case (2010, methodology): “Such recompression was done in [4] using the Tucker-ALS algorithm [22, 7], in [20] by the Tucker-ALS with the initial guess obtained from the coarser grids, in [11] by the Cross3D algorithm [25], in [26] by the individual cross approximation of canonical factors and in [32] by the cross approximatio”
Fast truncation of mode ranks for bilinear tensor operations (2010, unknown): “The Cross3D algorithm was veriﬁed in [23, 12] by exhaustive check on parallel memory platforms, and can be considered as reliable answer.”