Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
[1] S. Knapek. Hyperbolic cross approximation of integral operators with smooth kernel. submitted, 2000. also as Technical Report 665, SFB 256, Univ. Bonn.
bib | .ps.gz 1 ]
This paper is concerned with the construction and use of trigonometric approximation spaces for the approximate evaluation of integral operators with smooth kernels. The smoothness classes we consider are mixtures of classes of functions of dominating mixed derivative. We define a scale of nested approximation spaces for the approximation of the kernel that includes the standard full grid spaces as well as the spaces related to hyperbolic cross points. We present theoretical results on the approximation power of these spaces and show under which circumstances these approximation spaces lead to algorithms that break the curse of dimensionality. Blending schemes for these new approximation spaces allow the use of simple data structures and the direct application of fast hierarchical methods such as multilevel methods and fast Fourier transforms. Numerical examples illustrate the theoretical findings.