Superfast Fourier Transform Using QTT Approximation

Journal of Fourier Analysis and Applications

Volumn 18, Issue 5, 2012, Pages 915-953

  Dolgov, Sergey ^a,b   Khoromskij, Boris ^b   Savostyanov, Dmitry ^a  

^a INSTITUTE OF NUMERICAL MATHEMATICS   (Russian Federation)

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

Author keywords

Convolution; Fourier transform; High dimensional problems; QTT; Quantum Fourier transform; Sparse Fourier transform; Tensor train format

Indexed keywords

EID: 84866519225     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00041-012-9227-4     Document Type: Article

References (55)

1. Bertoglio, C., Khoromskij, B. N.: Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels. Comput. Phys. Commun. 183(4), 904-912 (2012). doi: 10. 1016/j. cpc. 2011. 12. 016.
2. Cooley, J. W., Tukey, J. W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297-301 (1965). doi: 10. 2307/2003354.
3. de Lathauwer, L.: A survey of tensor methods. In: IEEE International Symposium on Circuits and Systems, May 2009, pp. 2773-2776 (2009). doi: 10. 1109/iscas. 2009. 5118377.
4. Dolgov, S., Khoromskij, B., Oseledets, I. V., Tyrtyshnikov, E. E.: Tensor structured iterative solution of elliptic problems with jumping coefficients (2010). Preprint 55. MPI MIS, Leipzig. www. mis. mpg. de/preprints/2010/preprint2010_55. pdf.
5. Dolgov, S., Khoromskij, B. N., Oseledets, I. V.: Fast solution of multi-dimensional parabolic problems in the TT/QTT-format with initial application to the Fokker-Planck equation (2011). Preprint 80. MPI MIS, Leipzig. http://www. mis. mpg. de/preprints/2011/preprint2011_80. pdf.
6. Ekert, A., Jozsa, R.: Quantum algorithms: entanglement-enhanced information processing. Philos. Trans. R. Soc. Lond. 356, 1769-1782 (1998).
7. Fenn, M., Kunis, S., Potts, D.: Fast evaluation of trigonometric polynomials from hyperbolic crosses. Numer. Algorithms 41, 339-352 (2006). doi: 10. 1007/s11075-006-9017-7.
8. Flad, H.-J., Khoromskij, B. N., Savostyanov, D. V., Tyrtyshnikov, E. E.: Verification of the cross 3D algorithm on quantum chemistry data. Russ. J. Numer. Anal. Math. Model. 23(4), 329-344 (2008). doi: 10. 1515/RJNAMM. 2008. 020.
9. Gauss, C. F.: Nachlass: theoria interpolationis methodo nova tractata. In: Werke, vol. 3, pp. 265-330. Königliche Gesellschaft der Wissenschaften, Göttingem (1866).
10. Golub, G., van Loan, C.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996).
11. Goreinov, S., Oseledets, I., Savostyanov, D., Tyrtyshnikov, E., Zamarashkin, N.: How to find a good submatrix. In: Olshevsky, V., Tyrtyshnikov, E. (eds.) Matrix Methods: Theory, Algorithms, Applications, pp. 247-256. World Scientific, Hackensack (2010).
12. Goreinov, S. A., Oseledets, I. V., Savostyanov, D. V.: Wedderburn rank reduction and Krylov subspace method for tensor approximation. Part 1: Tucker case. SIAM J. Sci. Comput. 34(1), A1-A27 (2012). doi: 10. 1137/100792056.
13. Goreinov, S. A., Tyrtyshnikov, E. E.: The maximal-volume concept in approximation by low-rank matrices. Contemp. Math. 208, 47-51 (2001).
14. Grasedyck, L.: Polynomial approximation in hierarchical Tucker format by vector-tensorization (2010). DFG-SPP1324 Preprint 43. Philipps-University, Marburg. http://www. dfg-spp1324. de/download/preprints/preprint043. pdf.
15. Hackbusch, W.: Tensorisation of vectors and their efficient convolution. Numer. Math. 119(3), 465-488 (2011). doi: 10. 1007/s00211-011-0393-0.
16. Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer, Berlin (2012).
17. Hackbusch, W., Khoromskij, B. N.: Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. I. Separable approximation of multi-variate functions. Computing 76(3-4), 177-202 (2006). doi: 10. 1007/s00607-005-0144-0.
18. Hassanieh, H., Indyk, P., Katabi, D., Price, E.: Nearly optimal sparse Fourier transform (2012). Preprint. arXiv: 1201. 2501 [cs. DS].
19. rd annual ACM-SIAM symposium on discrete mathematics, pp. 1183-1194. SIAM, Philadelphia (2012).
20. Hegland, M.: A self-sorting in-place fast Fourier transform algorithm suitable for vector and parallel processing. Numer. Math. 68, 507-547 (1994).
21. Holtz, S., Rohwedder, T., Schneider, R.: The alternating linear scheme for tensor optimization in the tensor train format. SIAM J. Sci. Comput. 34(2), A683-A713 (2012). doi: 10. 1137/100818893.
22. Iwen, M. A.: AAFFT (Ann Arbor Fast Fourier Transform) (2008). Program Code. http://www. sourceforge. net/projects/aafftannarborfa/.
23. Iwen, M. A.: Combinatorial sublinear-time Fourier algorithms. Found. Comput. Math. 10, 303-338 (2010). doi: 10. 1007/s10208-009-9057-1.
24. Kazeev, V., Khoromskij, B. N.: Explicit low-rank QTT representation of Laplace operator and its inverse (2010). Preprint 75. MPI MIS, Leipzig. SIAM J. Matrix Anal. Appl. (2012, to appear). www. mis. mpg. de/preprints/2010/preprint2010_75. pdf.
25. Kazeev, V., Khoromskij, B. N., Tyrtyshnikov, E. E.: Multilevel Toeplitz matrices generated by tensor-structured vectors and convolution with logarithmic complexity (2011). Tech. Rep. 36, MPI MIS, Leipzig. http://www. mis. mpg. de/publications/preprints/2011/prepr2011-36. html.
26. Khoromskaia, V.: Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods. Ph. D. thesis, TU, Berlin (2010). http://opus. kobv. de/tuberlin/volltexte/2011/2948/.
27. Khoromskaia, V., Andrae, D., Khoromskij, B. N.: Fast and accurate tensor calculation of the Fock operator in a general basis (2012). Preprint 4. MPI MIS, Leipzig. www. mis. mpg. de/preprints/2012/preprint2012_4. pdf.
28. Khoromskaia, V., Khoromskij, B. N., Schneider, R.: QTT representation of the Hartree and exchange operators in electronic structure calculations. Comput. Methods Appl. Math. 11(3), 327-341 (2011).
29. Khoromskij, B. N.: On tensor approximation of Green iterations for Kohn-Sham equations. Comput. Vis. Sci. 11(4-6), 259-271 (2008). doi: 10. 1007/s00791-008-0097-x.
30. Khoromskij, B. N.: Fast and accurate tensor approximation of multivariate convolution with linear scaling in dimension. J. Comput. Appl. Math. 234(11), 3122-3139 (2010). doi: 10. 1016/j. cam. 2010. 02. 004.
31. Khoromskij, B. N.: O(d log n)-Quantics approximation of N-d tensors in high-dimensional numerical modeling. Constr. Approx. 34(2), 257-280 (2011). doi: 10. 1007/s00365-011-9131-1.
32. Khoromskij, B. N.: Tensor-structured numerical methods in scientific computing: survey on recent advances. Chemom. Intell. Lab. Syst. 110(1), 1-19 (2012). doi: 10. 1016/j. chemolab. 2011. 09. 001.
33. Khoromskij, B. N., Khoromskaia, V.: Multigrid accelerated tensor approximation of function related multidimensional arrays. SIAM J. Sci. Comput. 31(4), 3002-3026 (2009). doi: 10. 1137/080730408.
34. Khoromskij, B. N., Khoromskaia, V., Chinnamsetty, S. R., Flad, H.-J.: Tensor decomposition in electronic structure calculations on 3D Cartesian grids. J. Comput. Phys. 228(16), 5749-5762 (2009). doi: 10. 1016/j. jcp. 2009. 04. 043.
35. Khoromskij, B. N., Khoromskaia, V., Flad, H.-J.: Numerical solution of the Hartree-Fock equation in multilevel tensor-structured format. SIAM J. Sci. Comput. 33(1), 45-65 (2011). doi: 10. 1137/090777372.
36. Khoromskij, B. N., Oseledets, I. V.: DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator (2010). Preprint 69. MPI MIS, Leipzig. www. mis. mpg. de/preprints/2010/preprint2010_69. pdf.
37. Khoromskij, B. N., Oseledets, I. V.: QTT-approximation of elliptic solution operators in high dimensions. Russ. J. Numer. Anal. Math. Model. 26(3), 303-322 (2011). doi: 10. 1515/rjnamm. 2011. 017.
38. Kolda, T. G., Bader, B. W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455-500 (2009). doi: 10. 1137/07070111X.
39. Oseledets, I. V.: A new tensor decomposition. Dokl. Math. 80(1), 495-496 (2009). doi: 10. 1134/S1064562409040115.
40. d matrices using tensor decomposition. SIAM J. Matrix Anal. Appl. 31(4), 2130-2145 (2010). doi: 10. 1137/090757861.
41. Oseledets, I. V.: Constructive representation of functions in tensor formats (2010). Preprint 2010-04. INM RAS, Moscow. http://pub. inm. ras. ru.
42. Oseledets, I. V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295-2317 (2011). doi: 10. 1137/090752286.
43. Oseledets, I. V., Savostyanov, D. V., Tyrtyshnikov, E. E.: Cross approximation in tensor electron density computations. Numer. Linear Algebra Appl. 17(6), 935-952 (2010). doi: 10. 1002/nla. 682.
44. Oseledets, I. V., Tyrtyshnikov, E. E.: Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput. 31(5), 3744-3759 (2009). doi: 10. 1137/090748330.
45. Oseledets, I. V., Tyrtyshnikov, E. E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432(1), 70-88 (2010). doi: 10. 1016/j. laa. 2009. 07. 024.
46. Oseledets, I. V., Tyrtyshnikov, E. E.: Algebraic wavelet transform via quantics tensor train decomposition. SIAM J. Sci. Comput. 33(3), 1315-1328 (2011). doi: 10. 1137/100811647.
47. Savostyanov, D. V.: Fast revealing of mode ranks of tensor in canonical format. Numer. Math. Theory Method Appl. 2(4), 439-444 (2009). doi: 10. 4208/nmtma. 2009. m9006s.
48. Savostyanov, D. V.: QTT-rank-one vectors with QTT-rank-one and full-rank Fourier images. Linear Algebra Appl. 436(9), 3215-3224 (2012). doi: 10. 1016/j. laa. 2011. 11. 008.
49. Savostyanov, D. V., Oseledets, I. V.: Fast adaptive interpolation of multi-dimensional arrays in tensor train format. In: Proceedings of nDS-2011 Conference. IEEE Press, New York (2011). doi: 10. 1109/nDS. 2011. 6076873.
50. Savostyanov, D. V., Tyrtyshnikov, E. E.: Approximate multiplication of tensor matrices based on the individual filtering of factors. J. Comput. Math. Math. Phys. 49(10), 1662-1677 (2009). doi: 10. 1134/s0965542509100029.
51. Savostyanov, D. V., Tyrtyshnikov, E. E., Zamarashkin, N. L.: Fast truncation of mode ranks for bilinear tensor operations. Numer. Linear Algebra Appl. 19(1), 103-111 (2012). doi: 10. 1002/nla. 765.
52. Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princenton (1971).
53. Strang, G.: The discrete cosine transform. SIAM Rev. 41(1), 135-147 (1999).
54. White, S. R.: Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69(19), 2863-2866 (1992). doi: 10. 1103/PhysRevLett. 69. 2863.
55. Zou, J., Gilbert, A., Strauss, M., Daubechies, I.: Theoretical and experimental analysis of a randomized algorithm for Sparse Fourier transform analysis. J. Comput. Phys. 211, 572-595 (2006). doi: 10. 1016/j. jcp. 2005. 06. 005.