Approximation of Integral Operators by Variable-Order Interpolation

Summary.We employ a data-sparse, recursive matrix representation, so-called -matrices, for the efficient treatment of discretized integral operators. We obtain this format using local tensor product interpolants of the kernel function and replacing high-order approximations with piecewise lower-order ones. The scheme has optimal, i.e., linear, complexity in the memory requirement and time for the matrix-vector multiplication. We present an error analysis for integral operators of order zero. In particular, we show that the optimal convergence (h) is retained for the classical double layer potential discretized with piecewise constant functions.

[1]  G. Schmidlin,et al.  Fast solution algorithms for integral equations in R , 2003 .

[2]  Klaus Giebermann,et al.  Multilevel Approximation of Boundary Integral Operators , 2001, Computing.

[3]  Johannes Tausch,et al.  A variable order wavelet method for the sparse representation of layer potentials in the non-standard form , 2004, J. Num. Math..

[4]  Stefan A. Sauter.  Variable order panel clustering (extended version) , 1999 .

[5]  T. J. Rivlin.  The Chebyshev polynomials , 1974 .

[6]  Stefan A. Sauter,et al.  Variable Order Panel Clustering , 2000, Computing.

[7]  W Dahmen,et al.  Inverse Inequalities on Non � Quasiuniform Meshes and Application to the Mortar Element Method W � Dahmen , 2003 .

[8]  W. Hackbusch,et al.  On the fast matrix multiplication in the boundary element method by panel clustering , 1989 .

[9]  Wolfgang Hackbusch,et al.  Construction and Arithmetics of H-Matrices , 2003, Computing.

[10]  Wolfgang Hackbusch,et al.  A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices , 1999, Computing.

[11]  W. Hackbusch,et al.  H 2 -matrix approximation of integral operators by interpolation , 2002 .

[12]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[13]  A. Brandt,et al.  Multilevel matrix multiplication and fast solution of integral equations , 1990 .

[14]  W. Hackbusch,et al.  On H2-Matrices , 2000 .

[15]  Boris N. Khoromskij,et al.  A Sparse H-Matrix Arithmetic. Part II: Application to Multi-Dimensional Problems , 2000, Computing.

[16]  Christian Lage,et al.  Rapid solution of first kind boundary integral equations in R3 , 2003 .

[17]  HackbuschW..  A sparse matrix arithmetic based on H-matrices. Part I , 1999 .

[18]  Klaus Gibermann,et al.  Multilevel approximation of boundary integral operators , 2001 .

[19]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[20]  Lars Grasedyck,et al.  Theorie und Anwendungen Hierarchischer Matrizen , 2006 .

[21]  W. Hackbusch,et al.  Introduction to Hierarchical Matrices with Applications , 2003 .

[22]  S. Amini,et al.  Multi-level fast multipole solution of the scattering problem , 2003 .

[23]  E. Tyrtyshnikov.  Mosaic-Skeleton approximations , 1996 .

[24]  Wolfgang Dahmen,et al.  Wavelets on Manifolds I: Construction and Domain Decomposition , 1999, SIAM J. Math. Anal..

[25]  Stefan A. Sauter,et al.  Fast cluster techniques for BEM , 2003 .