Multilevel toeplitz matrices generated by tensor-structured vectors and convolution with logarithmic complexity

SIAM Journal on Scientific Computing

Volumn 35, Issue 3, 2013, Pages

  Kazeev, Vladimir A ^a   Khoromskij, Boris N ^b   Tyrtyshnikov, Eugene E ^c  

^a ETH ZURICH   (Switzerland)

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

^c INSTITUTE OF NUMERICAL MATHEMATICS   (Russian Federation)

Author keywords

Circulant matrices; Convolution; Low rank representation; Newton potential; QTT; Quantized tensor train; Tensor decompositions; Tensor rank; Tensor train; Toeplitz matrices; TT; Virtual levels

Indexed keywords

LINEAR TRANSFORMATIONS; MATRIX ALGEBRA; TENSORS; VECTORS;

CIRCULANT MATRIX; LOW-RANK REPRESENTATIONS; NEWTON POTENTIAL; QUANTIZED TENSOR TRAIN; TENSOR DECOMPOSITION; TENSOR RANKS; TENSOR TRAINS; TOEPLITZ MATRICES; TT; VIRTUAL LEVELS;

CONVOLUTION;

EID: 84884918548     PISSN: 10648275     EISSN: 10957200     Source Type: Journal    
DOI: 10.1137/110844830     Document Type: Article

References (32)

1. J.-P. Calliess, M. Mai, and S. Pfeiffer, On the computational benefit of tensor separation for high-dimensional discrete convolutions, Multidimens. Syst. Signal Process., 23 (2012), pp. 255-279.
2. S. Serra Capizzano and E. Tyrtyshnikov, Any circulant-like preconditioner for multilevel matrices is not superlinear, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 431-439.
3. P. J. Davis, Circulant Matrices, AMS Chelsea, New York, 1994.
4. L. De Lathauwer, A survey of tensor methods, in Proceedings of the 2009 IEEE International Symposium on Circuits and Systems, 2009, pp. 2773-2776.
5. S. Dolgov, B. Khoromskij, and D. Savostyanov, Superfast Fourier transform using QTT approximation, J. Fourier Anal. Appl., 18 (2012), pp. 915-953.
6. S. Dolgov, B. N. Khoromskij, and D. Savostyanov, Multidimensional Fourier Transform in Logarithmic Complexity Using QTT Approximation, Preprint 1, Institute of Numerical Mathematics of RAS, Moscow, Russia, 2011.
7. H.-J. Flad, B. N. Khoromskij, D. V. Savostyanov, and E. E. Tyrtyshnikov, Verification of the cross 3D algorithm on quantum chemistry data, Russian J. Numer. Anal. Math. Modelling, 23 (2008), pp. 329-344.
8. W. Hackbusch and B. Khoromskij, Low-rank Kronecker-product approximation to multidimensional nonlocal operators. Part I. Separable approximation of multi-variate func-tions, Computing, 76 (2006), pp. 177-202.
9. T. Helgaker and P. R. Taylor, Gaussian basis sets and molecular integrals, in Modern Electronic Structure Theory: Part II, World Scientific, Singapore, 1995, pp. 725-856.
10. V. A. Kazeev and B. N. Khoromskij, Low-rank explicit QTT representation of the Laplace operator and its inverse, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 742-758.
11. V. Khoromskaia, Numerical Solution of the Hartree-Fock Equation by Multilevel Tensor-Structured Methods, Ph.D. thesis, Technische Universiẗat Berlin, Berlin, Germany, 2010.
12. V. Khoromskaia, B. N. Khoromskij, and R. Schneider, Tensor-Structured Factorized Calculation of Two-Electron Integrals in a General Basis, Preprint 29, Max-Planck-Institut f̈ur Mathematik in den Naturwissenschaften, Leipzig, Germany, 2012.
13. B. N. Khoromskij, O(d logN)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling, Preprint 55, Max-Planck-Institut f̈ur Mathematik in den Naturwissenschaften, Leipzig, Germany, 2009.
14. B. N. Khoromskij, Fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension, J. Comput. Appl. Math., 234 (2010), pp. 3122-3139.
15. B. N. Khoromskij, O(d logN)-quantics approximation of N-d tensors in high-dimensional numerical modeling, Constr. Approx., 34 (2011), pp. 257-280.
16. B. N. Khoromskij, Tensors-structured numerical methods in scientific computing: Survey on recent advances, Chemometr. Intell. Lab. Syst., 110 (2011), pp. 1-19.
17. 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.
18. B. N. Khoromskij, V. Khoromskaia, and H.-J. Flad, Numerical solution of the Hartre-Fock equation in multilevel tensor-structured format, SIAM J. Sci. Comput., 33 (2011), pp. 45-65.
19. T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455-500.
20. W. D. Launey and J. Seberry, The strong Kronecker product, J. Combin. Theory Ser. A, 66 (1994), pp. 192-213.
21. V. Olshevsky, I. Oseledets, and E. Tyrtyshnikov, Tensor properties of multilevel Toeplitz and related matrices, Linear Algebra Appl., 412 (2006), pp. 1-21.
22. I. Oseledets, Approximation of matrices with logarithmic number of parameters, Dokl. Math., 80 (2009), pp. 653-654.
23. I. Oseledets, A new tensor decomposition, Dokl. Math., 80 (2009), pp. 495-496.
24. I. Oseledets and E. Tyrtyshnikov, Recursive decomposition of multidimensional tensors, Dokl. Math., 80 (2009), pp. 460-462.
25. I. V. Oseledets, Approximation of 2d × 2d matrices using tensor decomposition, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2130-2145.
26. I. V. Oseledets, DMRG Approach to Fast Linear Algebra in the TT-Format, preprint, Hausdorff Research Institute for Mathematics, Bonn, Germany, 2011.
27. I. V. Oseledets, Tensor-train decomposition, SIAM J. Sci. Comput., 33 (2011), pp. 2295-2317.
28. I. V. Oseledets and E. E. Tyrtyshnikov, Breaking the curse of dimensionality, or how to use SVD in many dimensions, SIAM J. Sci. Comput., 31 (2009), pp. 3744-3759.
29. F. Verstraete, D. Porras, and J. I. Cirac, Density matrix renormalization group and periodic boundary conditions: A quantum information perspective, Phys. Rev. Lett., 93 (2004), 227205.
30. G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys.Rev. Lett., 91 (2003), 147902.
31. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett., 69 (1992), pp. 2863-2866.
32. S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B, 48 (1993), pp. 10345-10356.