Abstract
Multigrid methods are typically used to solve partial differential equations, i.e., they approximate the inverse of the corresponding partial differential operators. At least for elliptic PDEs, this inverse can be expressed in the form of an integral operator by Green’s theorem.
This implies that multigrid methods approximate certain integral operators, so it is straightforward to look for variants of multigrid methods that can be used to approximate more general integral operators.
ℋ2-matrices combine a multigrid-like structure with ideas from panel clustering algorithms in order to provide a very efficient method for discretizing and evaluating the integral operators found, e.g., in boundary element applications.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Börm, S., Grasedyck, L., Hackbusch, W: Introduction to hierarchical matrices with applications. Engineering Analysis with Boundary Elements 27, 405–422 (2003)
Börm, S., Hackbusch, W.: Data-sparse approximation by adaptive ℋ2-matrices. Computing 69, 1–35 (2002)
Börm, S., Hackbusch, W.: ℋ2-matrix approximation of integral operators by interpolation. Applied Numerical Mathematics 43, 129–143 (2002)
Börm, S., Hackbusch, W.: Approximation of boundary element operators by adaptive ℋ2-matrices. Foundations of Computational Mathematics 312, 58–75 (2004)
Börm, S., Löhndorf, M., Melenk, J.M.: Approximation of integral operators by variable-order interpolation. Tech. Rep. 82, Max Planck Institute for Mathematics in the Sciences, 2002. Submitted to Numerische Mathematik
Dahmen, W., Schneider, R.: Wavelets on manifolds I: Construction and domain decomposition. SIAM Journal of Mathematical Analysis 31, 184–230 (1999)
Giebermann, K.: Multilevel approximation of boundary integral operators. Computing 67, 183–207 (2001)
Grasedyck, L.: Theorie und Anwendungen Hierarchischer Matrizen. PhD thesis, Universität Kiel, 2001
Grasedyck, L., Hackbusch, W.: Construction and arithmetics of ℋ-matrices. Computing 70, 295–334 (2003)
Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. Journal of Computational Physics 73, 325–348 (1987)
Hackbusch, W.: A sparse matrix arithmetic based on ℋ-matrices. Part I: Introduction to ℋ-matrices. Computing 62, 89–108 (1999)
Hackbusch, W., Khoromskij, B.: A sparse ℋ-matrix arithmetic: General complexity estimates. J. Comp. Appl. Math. 125, 479–501 (2000)
Hackbusch, W., Khoromskij, B.: A sparse matrix arithmetic based on ℋ-matrices. Part II: Application to multi-dimensional problems. Computing 64, 21–47 (2000)
Hackbusch, W., Khoromskij, B., Sauter, S.: On ℋ2-matrices. In: Bungartz, H., Hoppe, R., Zenger, C. (eds.), Lectures on Applied Mathematics, Berlin: Springer-Verlag 2000, pp. 9–29
Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numerische Mathematik 54, 463–491 (1989)
Rokhlin, V.: Rapid solution of integral equations of classical potential theory. Journal of Computational Physics 60, 187–207 (1985)
Sauter, S.: Variable order panel clustering (extended version). Tech. Rep. 52, Max-Planck-Institut für Mathematik, Leipzig, Germany, 1999
Sauter, S.: Variable order panel clustering. Computing 64, 223–261 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by
G. Wittum
Rights and permissions
About this article
Cite this article
Börm, S. ℋ2-matrices – Multilevel methods for the approximation of integral operators. Comput. Visual Sci. 7, 173–181 (2004). https://doi.org/10.1007/s00791-004-0135-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00791-004-0135-2