Kronecker-product approximations for some function-related matrices

By E. E. Tyrtyshnikov. 2004

75 citations recorded by the source corpus.

Subfields: Atomic and Molecular Physics, and Optics, Computational Mathematics, Computational Theory and Mathematics

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In Atomic and Molecular Physics, and Optics

In Computational Mathematics

In Computational Theory and Mathematics

Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Other techniques include applying numerical quadrature to an integral representation of the function, see, e.g., [24, 77, 87, 110, 115], Taylor series expansion [249, 250], and polynomial interpolation [19, 33].”
Algorithms for Numerical Analysis in High Dimensions (2005, background): “The class of complete asymptotically smooth functions is considered in [16, 43, 42].”
Tensor properties of multilevel Toeplitz and related matrices (2006, methodology): “Then, one can apply general results for the so-called asymptotically smooth functions [26,27].”
Tensor approximations of matrices generated by asymptotically smooth functions (2003, unknown): “In [ 18 ] we described for the first time a certain broad class of matrices and proved for them the existence of fairly close tensor approximations of small rank r .H owever, in our constructions in that paper we contented ourselves with the case m =2 and the condition that the entries of the matrix”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, background): “For the class of (physically relevant) analytic multivariate functions and Green’s kernels the exponential convergence in the separation rank R can be proved [25, 93, 49, 53, 54], that leads to the asymptotically optimal bound on the canonical rank, R = O(logN), implying the approximation rate O(N−α”
Structured data-sparse approximation to high order tensors arising from the deterministic Boltzmann equation (2007, methodology): “The latter will be used in our particular application in § 3.”
Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods (2011, unknown): “6.1 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140”
A least-squares approximation of partial differential equations with high-dimensional random inputs (2009, methodology): “Separated representations and the ALS algorithm Separation-of-variable techniques have been widely used to approximate high-dimensional functions using one-dimensional operations, thus virtually eliminating the curse of dimensionality, see [23,16,15,3,25,13,4] and references therein.”
Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time (2008, background): “The rank estimates obtained in [15, 19, 20] have form r 6 C(log n log(2) ε), where ε is an error of the approximation, so the rank grows only logarithmically with n and ε.”
Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling (2008, background): “[83, 84]) it is possible to show that tensor-product approximations with low separation rank exist.”