Tensor-structured preconditioners and approximate inverse of elliptic operators in ℝd

Constructive Approximation

Volumn 30, Issue 3, 2009, Pages 599-620

  Khoromskij, Boris N ^a  

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

Author keywords

Boundary value problems; Elliptic resolvent; Green's kernels; High dimensions; Preconditioning; Spectral problems; Tensor approximation

Indexed keywords

EID: 70449726943     PISSN: 01764276     EISSN: 14320940     Source Type: Journal    
DOI: 10.1007/s00365-009-9068-9     Document Type: Article

References (30)

1. Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions. Dover, New York (1968).
2. Bertoglio, C., Khoromskij, B. N.: Low rank tensor-product approximation of the projected Green kernels via Sinc-quadratures. Preprint MPI MIS 79/2008, Leipzig, 2008 (submitted).
3. Beylkin, G., Mohlenkamp, M. J.: Numerical operator calculus in higher dimensions. Proc. Natl. Acad. Sci. USA 99, 10246-10251 (2002).
4. Beylkin, G., Mohlenkamp, M. J., Pérez, F.: Approximating a wavefunction as an unconstrained sum of Slater determinants. J. Math. Phys. 49, 032107 (2008).
5. Beylkin, G., Kurcz, Ch., Monzón, L.: Fast algorithms for Helmholtz Green's function. Proc. R. Soc., Ser. A 464, 3301-3326 (2008).
6. Beylkin, G., Kurcz, Ch., Monzón, L.: Fast Convolution with the Free Space Helmholtz Green's function. University of Colorado at Bolder, http://amath. colorado. edu/pub/wavelets/papers/BE-KU-MO-2008P. pdf.
7. De Lathauwer, L., De Moor, B., Vandewalle, J.: On the best rank-1 and rank-(R1,..., RN) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21, 1324-1342 (2000).
8. De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21, 1253-1278 (2000).
9. Gavrilyuk, I. P., Hackbusch, W., Khoromskij, B. N.: ℋ-matrix approximation for the operator exponential with applications. Numer. Math. 92, 83-111 (2002).
10. Gavrilyuk, I. P., Hackbusch, W., Khoromskij, B. N.: Tensor-product approximation to elliptic and parabolic solution operators in higher dimensions. Computing 74, 131-157 (2005).
11. Griebel, M., Hamaekers, J.: Sparse grids for the Schrödinger equation. ESAIM: M2AN 41, 215-247 (2007).
12. Hackbusch, W., Khoromskij, B. N.: Low-rank Kronecker product approximation to multi-dimensional nonlocal operators, part I: separable approximation of multi-variate functions. Computing 76, 177-202 (2006).
13. Hackbusch, W., Khoromskij, B. N., Sauter, S., Tyrtyshnikov, E.: Tensor formats in elliptic problems. Preprint MPI MIS 78/2008, Leipzig, 2008 (SINUM, submitted).
14. Hackbusch, W., Khoromskij, B. N., Tyrtyshnikov, E.: Hierarchical Kronecker tensor-product approximation. J. Numer. Math. 13, 119-156 (2005).
15. Hackbusch, W., Khoromskij, B. N., Tyrtyshnikov, E. E.: Approximate Iterations for Structured Matrices. Numer. Math. 109, 365-383 (2008).
16. Harrison, R. J., Fann, G. I., Yanai, T., Gan, Z., Beylkin, G.: Multiresolution quantum chemistry: basic theory and initial applications. J. Chem. Phys. 121(23), 11587-11598 (2004).
17. Khoromskij, B. N.: An Introduction to Structured Tensor-Product Representation of Discrete Nonlocal Operators, vol. 27. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig (2005).
18. Khoromskij, B. N.: Structured rank-(r1,..., rd) decomposition of function-related tensors in ℝd. Comput. Methods Appl. Math. 6(2), 194-220 (2006).
19. Khoromskij, B. N.: On tensor approximation of green iterations for Kohn-Sham equations. Comput. Vis. Sci. 11, 259-271 (2008). doi: 10. 1007/s00791-008-0097-x.
20. Khoromskij, B. N.: Fast and accurate tensor approximation of multivariate convolution with linear scaling in dimension. Preprint MPI MIS 36, Leipzig, 2008. J. Comput. Appl. Math. (accepted).
21. 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, 5749-5762 (2009).
22. Khoromskij, B. N., Khoromskaia, V.: Low rank tucker-type tensor approximation to classical potentials. Cent. Eur. J. Math. 5(3), 1-28 (2007).
23. Khoromskij, B. N., Khoromskaia, V.: Multigrid accelerated tensor approximation of function related multi-dimensional arrays. SIAM J. Sci. Comput. 31(4), 3002-3028 (2009).
24. Khoromskij, B. N., Schwab, Ch.: Tensor approximation of multi-parametric elliptic problems in stochastic PDEs. ETH Zurich, 2009 (in preparation).
25. Khoromskij, B. N., Wittum, G.: Numerical Solution of Elliptic Differential Equations by Reduction to the Interface. LNCSE, vol. 36. Springer, Berlin (2004).
26. Maz'ya, V., Schmidt, G.: Approximate Approximations. Math. Surveys and Monographs, vol. 141. AMS, Providence (2007).
27. Mohlenkamp, M., Young, M. J.: Convergence of Green's iterations for Schrödinger equations. Preprint 2007 (to appear).
28. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, I: Functional Analysis. Academic Press, New York (1972).
29. Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer, Berlin (1993).
30. Todor, R.-A., Schwab, Ch.: Convergence rate for sparse approximation of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27, 232-261 (2007).