Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems

Computing (Vienna/New York)

Volumn 74, Issue 2, 2005, Pages 131-157

  Gavrilyuk, Ivan P ^a   Hackbusch, Wolfgang ^b   Khoromskij, Boris N ^b  

^a Staatliche Studienakademie ^*  (Germany)

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

Author keywords

Hierarchical matrices; High space dimensions; Kronecker tensor products; S i n c quadrature

Indexed keywords

BOUNDARY ELEMENT METHOD; COMPUTER SIMULATION; DATA MINING; DIFFERENTIAL EQUATIONS; FAST FOURIER TRANSFORMS; FINITE ELEMENT METHOD; INVERSE PROBLEMS; MATHEMATICAL OPERATORS; MATRIX ALGEBRA; TENSORS;

HIERARCHICAL MATRICES; HIGH SPACE DIMENSIONS; KRONECKER TENSOR PRODUCTS; SINC-QUADRATURE;

APPROXIMATION THEORY;

EID: 17144410769     PISSN: 0010485X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00607-004-0086-y     Document Type: Article

References (28)

1. Babenko, K. I.: Foundations of the numerical analysis. Moscow: Nauka 1986 (in Russian).
2. Beylkin, G., Mohlenkamp, M. J.: Numerical operator calculus in higher dimensions. University of Colorado, APPM preprint no. 476, August 2001.
3. Braess, D.: Nonlinear approximation theory. Springer 1986.
4. Dautray, R., Lions, J.-L.: Mathematical analysis and numerical methods for science and technology, vol. 5: Evolution problems I. Springer 1992.
5. Fujita, H., Saito, N., Suzuki, T.: Operator theory and numerical methods. Elsevier 2001.
6. Gavrilyuk, I.P.: Strongly P-positive operators and explicit representation of the solutions of initial value problems for second-order differential equations in Banach space. J. Math. Anal. Appl. 236, 327-349 (1999).
7. Gavrilyuk, I. P., Hackbusch, W., Khoromskij, B. N.: ℋ-matrix approximation for the operator exponential with applications. Numer. Math. 92, 83-111 (2002).
8. Gavrilyuk, I. P., Hackbusch, W.: Khoromskij, B. N.: Data-sparse approximation to operator-valued functions of elliptic operators. Math. Comp. 75(247), 1297-1324 (2004).
9. Gavrilyuk, I. P., Hackbusch, W., Khoromskij, B. N.: Data-sparse approximation to a class of operator-valued functions. Math. Comp. (forthcoming).
10. Grasedyck, L.: Existence and computation of a low Kronecker-rank approximation to the solution of a tensor system with tensor right-hand side. Computing 70, 295-334.
11. Grasedyck, L., Hackbusch, W.: Construction and arithmetics of ℋ-matrices. Computing 70, 295-334 (2003).
12. 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).
13. Griebel, M., Knapek, S.: Optimized tensor-product approximation spaces. Constr. Approx. 16, 303-332 (2000).
14. Hackbusch, W.: Elliptic differential equations: Theory and numerical treatment. Berlin: Springer 1992.
15. Hackbusch, W.: A sparse matrix arithmetic based on ℋ-matrices. Part I: Introduction to ℋ-matrices. Computing 62, 89-108 (1999).
16. Hackbusch, W., Khoromskij, B. N.: A sparse ℋ-matrix arithmetic. Part II: Application to multi-dimensional problems. Computing 64, 21-47 (2000).
17. Hackbusch, W., Khoromskij, B. N.: Towards ℋ-matrix approximation of the linear complexity. In: Operator theory: Advances and applications, vol. 121 (Elschner, J., Gohberg, I., Silbermann, B., eds.), pp 194-220. Basel: Birkhäuser 2001.
18. Hackbusch, W., Khoromskij, B. N.: Kronecker tensor-product approximation to certain matrix-valued functions in higher dimensions. Preprint 16, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig 2004.
19. 2- matrices. In: Lectures on Applied Mathematics (Bungartz, H.-J., Hoppe, R., Zenger, C., eds.), pp. 9-30. Berlin: Springer 2000.
20. Hackbusch, W., Khoromskij, B. N., Tyrtyshnikov: E.: Hierarchical Kronecker tensor-product approximation. Preprint 35, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig 2003.
21. Khoromskij, B. N.: ℋ-matrix approximation of integral operators. Lecture Notes No. 17, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig 2003.
22. Pazy, A.: Semigroups of linear operator and applications to partial differential equations. Springer 1983.
23. Sheen, D., Sloan, I. H., Thomée, V.: A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature. Math. Comp. 69, 177-195 (2000).
24. Stenger, F.: Numerical methods based on Sine and analytic functions. Springer 1993.
25. Temlyakov, V. N.: Approximation of functions with bounded mixed derivative. Proc. Steklov Inst. (1989) Math. 178(1), 275-293 (1989).
26. Tyrtyshnikov, E. E.: Tensor approximations of matrices generated by asymptotically smooth functions. (Russian) Mat. Sb. 194(6), 147-160 (2003) (translation in Sb. Math. 194, 941-954, 2003).
27. Tyrtyshnikov, E. E.: Tensor approximations of matrices generated by asymptotically smooth functions. (Russian) Mat. Sb. 194(6), 147-160 (2003) (translation in Sb. Math. 194, 941-954, 2003).
28. Yarvin, N., Rokhlin, V.: Generalized Gaussian quadratures and singular value decompositions of integral operators. SIAM J. Sci. Comput. 20, 699-718 (1998).