Multigrid accelerated tensor approximation of function related multidimensional arrays

SIAM Journal on Scientific Computing

Volumn 31, Issue 4, 2009, Pages 3002-3026

  Khoromskij, B N ^a   Khoromskaia, V ^a  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

Adaptive orthogonal tensor product basis; Canonical model; Discrete convolution; Multigrid acceleration; Orthogonal Tucker tensor decomposition

Indexed keywords

APPROXIMATION THEORY; CONVOLUTION; ELECTRONIC STRUCTURE; ORTHOGONAL FUNCTIONS;

ADAPTIVE ORTHOGONAL TENSOR-PRODUCT BASE; CANONICAL MODELS; DISCRETE CONVOLUTION; MULTIGRID ACCELERATION; MULTIGRIDS; ORTHOGONAL TENSORS; ORTHOGONAL TUCK TENSOR DECOMPOSITION; TENSOR APPROXIMATION; TENSOR DECOMPOSITION; TENSOR PRODUCTS;

TENSORS;

EID: 84897586218     PISSN: 10648275     EISSN: 10957200     Source Type: Journal    
DOI: 10.1137/080730408     Document Type: Article

References (28)

1. G. BEYLKIN AND M. J. MOHLENKAMP, Numerical operator calculus in higher dimensions, Proc. Natl. Acad. Sci. USA, 99 (2002), pp. 10246-10251.
2. G. BEYLKIN AND M. J. MOHLENKAMP, Algorithms for numerical analysis in high dimensions, SIAM J. Sci. Comput., 26 (2005), pp. 2133-2159.
3. J. D. CARROL, S. PRUZANSKY, AND J. B. KRUSKAL, CANDELINC: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters, Psychometrika, 45 (1980), pp. 3-24.
4. S. R. CHINNAMSETTY, M. ESPIG, W. HACKBUSCH, B. N. KHOROMSKIJ, AND H.-J. FLAD, Kronecker tensor product approximation in quantum chemistry, J. Chem. Phys., 127 (2007), 084110.
5. L. DE LATHAUWER, B. DE MOOR, AND J. VANDEWALLE, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1253-1278.
6. N) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1324-1342.
7. H.-J. FLAD, W. HACKBUSCH, B. N. KHOROMSKIJ, AND R. SCHNEIDER, Concept of data-sparse tensor-product approximation in many-particle modeling, in Matrix Methods: Theory, Algorithms, and Applications, V. Olshevsky and E. Tyrtyshnikov, eds., World Scientific, Singapore, 2008, pp. 313-347.
8. H.-J. FLAD, W. HACKBUSCH, AND R. SCHNEIDER, Best N-term approximation in electronic structure calculations : I. One-electron reduced density matrix, M2AN Math. Model. Numer. Anal., 40 (2006), pp. 49-61.
9. S. FRIEDLAND AND V. MEHRMANN, Best Subspace Tensor Approximations, http:/arxiv.org/abs/0805.4220v1 (2008).
10. M. GRIEBEL AND J. HAMAEKERS, Sparse grids for the Schrödinger equation, M2AN Math. Model. Numer. Anal., 41 (2007), pp. 215-247.
11. W. HACKBUSCH, Fast and exact projected convolution for non-equidistant grids, Computing, 80 (2007), pp. 137-168.
12. W. HACKBUSCH AND B. N. KHOROMSKIJ, Low-rank Kronecker product approximation to multidimensional nonlocal operators. Part I. Separable approximation of multi-variate functions, Computing, 76 (2006), pp. 177-202.
13. W. HACKBUSCH AND B. N. KHOROMSKIJ, Tensor-product approximation to operators and functions in high dimensions, J. Complexity, 23 (2007), pp. 697-714.
14. W. HACKBUSCH, B. N. KHOROMSKIJ, AND E. TYRTYSHNIKOV, Approximate iteration for structured matrices, Numer. Math., 109 (2008), pp. 365-383.
15. W. HACKBUSCH, B. N. KHOROMSKIJ, AND E. E. TYRTYSHNIKOV, Hierarchical Kronecker tensor-product approximations, J. Numer. Math., 13 (2005), pp. 119-156.
16. d, Comput. Methods Appl. Math., 6 (2006), pp. 194-220.
17. B. N. KHOROMSKIJ, Fast and Accurate Tensor Approximation of Multivariate Convolution with Linear Scaling in Dimension, Preprint MPI MIS 36, MPI MIS, Leipzig, Germany, 2008; J. Comput. Appl. Math., to appear.
18. B. N. KHOROMSKIJ, V. KHOROMSKAIA, S. R. CHINNAMSETTY, AND H.-J. FLAD, Tensor decomposition in electronic structure calculations on 3D Cartesian grids, J. Comput. Phys., 228 (2009), pp. 5749-5762.
19. B. N. KHOROMSKIJ AND V. KHOROMSKAIA, Low rank Tucker tensor approximation to the classical potentials, Cent. Eur. J. Math., 5 (2007), pp. 1-28.
20. T. G. KOLDA, Orthogonal tensor decompositions, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 243-255.
21. T. G. KOLDA AND B. W. BADER, Tensor decompositions and applications, SIAM Rev., to appear.
22. C. LE BRIS, ED., Handbook of Numerical Analysis, Vol. X, Computational Chemistry, North-Holland, Amsterdam, 2003.
23. I. V. OSELEDETS, D. V. SAVOSTIANOV, AND E. E. TYRTYSHNIKOV, Tucker dimensionality reduction of three-dimensional arrays in linear time, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 939-956.
24. C. SCHWAB AND R. TODOR, Sparse finite elements for elliptic problems with stochastic loading, Numer. Math., 95 (2003), pp. 707-713.
25. A. SMILDE, R. BRO, AND P. GELADI, Multiway Analysis, Wiley, New York, 2004.
26. L. R. TUCKER, Some mathematical notes on three-mode factor analysis, Psychometrika, 31 (1966), pp. 279-311.
27. E. E. TYRTYSHNIKOV, Kronecker-product approximations for some function-related matrices, Linear Algebra Appl., 379 (2004), pp. 423-437.
28. H. YSERENTANT, The hyperbolic cross space approximation of electronic wavefunctions, Numer. Math., 105 (2007), pp. 659-690.