Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels.(English)Zbl 1308.65213
Summary: Tensor-product approximation provides a convenient tool for efficient numerical treatment of high-dimensional problems that arise, in particular, in electronic structure calculations in \(\mathbb R^{d}\). In this work we apply tensor approximation to the Galerkin representation of the Newton and Yukawa potentials for a set of tensor-product, piecewise polynomial basis functions. To construct tensor-structured representations, we make use of the well-known Gaussian transform of the potentials, and then approximate the resulting univariate integral in \(\mathbb R\) by special sinc-quadratures. The novelty of the approach lies on the optimisation of the quadrature parameters that allows to reduce dramatically the initial tensor-rank obtained by the standard sinc-quadratures. The numerical experiments show that this approach gives tensor-ranks close to the optimal in 3D computations on large spatial grids and with linear complexity in the univariate grid size. Particularly, this scheme becomes attractive for the multiple calculation of the Yukawa potential when the exponents in Gaussian functions vary during the computational process.
Keywords:
tensor-product approximation;Newton/Yukawa potentials;Gaussian integral transform;sinc-quadrature;electronic structure calculationsCite
Full Text:DOI
References:
| [1] | Harrison, R.; Fann, G.; Yanai, T.; Gan, Z.; Beylkin, G., Multiresolution quantum chemistry: Basic theory and initial applications, J. Chem. Phys., 121, 23, 11587-11598 (2004) |
| [2] | Khoromskij, B.; Khoromskaia, V., Multigrid accelerated tensor approximation of function related multidimensional arrays, SIAM J. Sci. Comput., 31, 3002-3026 (2009) ·Zbl 1197.65215 |
| [3] | Khoromskij, B., On tensor approximation of Green iterations for Kohn-Sham equations, Comput. Vis. Sci., 11, 259-271 (2008) ·Zbl 1522.65249 |
| [4] | Hackbusch, W.; Khoromskij, B., Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. I. Separable approximation of multi-variate functions, Computing, 76, 3-4, 177-202 (2006) ·Zbl 1087.65049 |
| [5] | Khoromskij, B., Fast and accurate tensor approximation of multivariate convolution with linear scaling in dimension, J. Comput. Appl. Math., 234, 3122-3139 (2010) ·Zbl 1197.65216 |
| [6] | Khoromskij, B.; Khoromskaia, V., Low rank Tucker-type tensor approximation to classical potentials, Cent. Eur. J. Math., 5, 3, 1-28 (2007) ·Zbl 1130.65060 |
| [7] | Langer, U.; Steinbach, O., Coupled Finite and Boundary Element Domain Decomposition Method, Lecture Notes in Applied and Computational Mechanics, vol. 29 (2007), pp. 29-59 |
| [8] | Khoromskij, B.; Khoromskaia, V.; Flad, H., Numerical solution of the Hartree-Fock equation in multilevel tensor-structured format, SIAM J. Sci. Comput., 33, 1, 45-65 (2011) ·Zbl 1227.65113 |
| [9] | Le Bris, C., Computational chemistry from the perspective of numerical analysis, Acta Numer., 363-444 (2005) ·Zbl 1119.65390 |
| [10] | Oseledets, I. V., Approximation of \(2^d \times 2^d\) matrices using tensor decomposition, SIAM J. Matrix Anal. Appl., 31, 2130-2145 (2010) ·Zbl 1200.15005 |
| [11] | Khoromskij, B., \(O(d \log N)\)-quantics approximation of \(N\)-d tensors in high-dimensional numerical modelling, J. Constr. Approx., 34, 2, 257-289 (2011) ·Zbl 1228.65069 |
| [12] | Khoromskij, B. N.; Oseledets, I. V., Quantics-TT approximation of elliptic solution operators in higher dimensions, Russ. J. Numer. Anal. Math. Modelling, 26, 3, 303-322 (2011) ·Zbl 1221.65288 |
| [13] | Gavrilyuk, I.; Hackbusch, W.; Khoromskij, B., Data-sparse approximation to a class of operator-valued functions, Math. Comp., 74, 681-708 (2005) ·Zbl 1066.65060 |
| [14] | Gavrilyuk, I.; Hackbusch, W.; Khoromskij, B., Tensor-product approximation to elliptic and parabolic solution operators in higher dimensions, Computing, 74, 131-157 (2005) ·Zbl 1071.65032 |
| [15] | Stenger, F., Numerical Methods Based on Sinc and Analytic Functions (1993), Springer-Verlag ·Zbl 0803.65141 |
| [16] | Braess, D.; Hackbusch, W., Approximation of \(1 / x\) by exponential sums in \([1, \infty]\), IMA J. Numer. Anal., 25, 4, 685 (2005) ·Zbl 1082.65025 |
| [17] | Khoromskij, B.; Khoromskaia, V.; Chinnamsetty, S.; Flad, H.-J., Tensor decomposition in electronic structure calculations on 3d cartesian grids, J. Comput. Phys., 228, 5749-5762 (2009) ·Zbl 1171.82017 |
| [18] | Hackbusch, W.; Khoromskij, B., Tensor-product approximation to operators and functions in high dimension, J. Complexity, 23, 1, 697-714 (2007) ·Zbl 1141.65032 |
| [19] | Braess, D.; Hackbusch, W., On the efficient computation of high-dimensional integrals and the approximation by exponential sums, (Multiscale, Nonlinear and Adaptive Approximation (2009)), 39-74 ·Zbl 1190.65036 |
| [20] | Lund, J.; Bowers, K., Sinc Methods for Quadrature and Differential Equations (1992), SIAM: SIAM Philadelphia ·Zbl 0753.65081 |
| [21] | V. Khoromskaia, Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods, PhD thesis, TU Berlin, 2011.; V. Khoromskaia, Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods, PhD thesis, TU Berlin, 2011. ·Zbl 1227.65113 |
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