Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. Part I. Separable approximation of multi-variate functions

Computing (Vienna/New York)

Volumn 76, Issue 3-4, 2005, Pages 177-202

  Hackbusch, W ^a,b   Khoromskij, B N ^a,b  

^a Max Planck Institut für Mathematik in den Naturwissenschaften ^*

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

Author keywords

Approximation by exponential sums; Hierarchical matrices; Kronecker tensor product; Sinc interpolation; Sinc quadrature

Indexed keywords

APPROXIMATION BY EXPONENTIAL SUMS; HIERARCHICAL MATRICES; SINC INTERPOLATION; SINC QUADRATURE;

FINITE DIFFERENCE METHOD; FUNCTIONS; INTEGRAL EQUATIONS; INTERPOLATION; MATHEMATICAL OPERATORS; MATRIX ALGEBRA;

APPROXIMATION THEORY;

EID: 29144513721     PISSN: 0010485X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00607-005-0144-0     Document Type: Article

References (26)

1. Beylkin, G., Mohlenkamp, M. J.: Numerical operator calculus in higher dimensions. Proc. Natl. Acad. Sci. USA 99, 10246-10251 (2002).
2. Braess, D.: Nonlinear approximation theory. Berlin: Springer 1986.
3. Braess, D., Hackbusch, W.: Approximation of 1/x by exponential sums in [1, ∞). IMA J. Numer. Anal. 25, 685-697 (2005).
4. Gavrilyuk, I. P., Hackbusch, W., Khoromskij, B. N.: H-matrix approximation for the operator exponential with applications. Numer. Math. 92, 83-111 (2002).
5. Gavrilyuk, I. P., Hackbusch, W., Khoromskij, B. N.: Data-sparse approximation to operator-valued functions of elliptic operators. Math. Comp. 73, 1297-1324 (2004).
6. Gavrilyuk, I. P., Hackbusch, W., Khoromskij, B. N.: Data-sparse approximation to a class of operator-valued functions. Math. Comp. 74, 681-708 (2005).
7. Gavrilyuk, I. P., Hackbusch, W., Khoromskij, B. N.: esor-product approximation to elliptic and parabolic solution operators in higher dimensions. Computing 74, 131-157 (2005).
8. Gradshteyn, I. S., Ryzhik, I. M.: Table of integrals, series and products, 6th edition. San Diego: Academic Press 2000.
9. Grasedyck, L., Hackbusch, W.: Construction and arithmetics of H-matrices, Computing 70, 295-334 (2003).
10. Grasedyck, L., Hackbusch, W., Khoromskij, B. N.: Solution of large scale algebraic matrix Riccati equations by use of hierarchical matrices. Computing 70, 121-165 (2003).
11. Hackbusch, W: A sparse matrix arithmetic based on H-matrices. Part I: Introduction to H-matrices. Computing 62, 89-108 (1999).
12. Hackbusch, W., Khoromskij, B. N.: A sparse H-matrix arithmetic. Part II: Application to multi-dimensional problems. Computing 64, 21-47 (2000).
13. Hackbusch, W., Khoromskij, B. N.: Blended kernel approximation in the H-matrix techniques. Numer. Linear Algebra Appl. 9, 281-304 (2002).
14. Hackbusch, W., Khoromskij, B. N., Kriemann, R.: Hierarchical matrices based on a weak admissibility criterion. Computing 73, 207-243 (2004).
15. Hackbusch, W., Khoromskij, B. N., Tyrtyshnikov, E.: Hierarchical Kronecker tensor-product approximation. J. Numer. Math. 13, 119-156 (2005).
16. β by Sine functions. J. Appr. Theory 66, 44-52 (1991).
17. Khoromskij, B. N.: Structured data-sparse approximation to high-order tensors arising from the deterministic Boltzmann equation. Preprint 4, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig 2005 (to appear in Math. Comp.).
18. Stenger, F.: Numerical methods based on Sine and analytic functions. Springer, 1993.
19. Tyrtyshnikov, E. E.: Tensor approximations of matrices generated by asymptotically smooth functions, (in Russian) Mat. Sb. 194(6), 147-160 (2003);
20. translation in Sb. Math. 194, 941-954 (2003).
21. Yarkony, D. R.: Modern electronic structure theory. Part II. World Scientific 1995.
22. Yarvin, N., Rokhlin, V.: Generalized Gaussian quadratures and singular value decompositions of integral operators. SIAM J. Sci. Comput. 20, 699-718 (1998).
23. Zeidler, E. (ed.): Oxford Users' Guide to Mathematics. Oxford: Oxford University Press 2004.
24. Zhang, T., Golub, G. H.: Rank-one approximation to high-order tensors. SIAM J. Matrix Anal. Appl. 23, 534-550 (2001).
25. N) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21, 1324-1342 (2000).
26. De Lathauwer, L., De Moor, B., Vandewalle, J.: Computation of the canonical decomposition by means of a simultaneous generalized Schur decomposition. SIAM J. Matrix Anal. Appl. 26, 295-327 (2004).