Wavelets and multiscale methods in numerical analysis and simulation

• Research unit: EU RFP
• Project number: 97.0404

Key words (English)

Wavelets; boundary integral equations; multipole methods; numerical analysis

Alternative project number (English)

EU project number: FMRXCT980184

Research programs (English)

EU-programme: 4. Frame Research Programme - 10.1 Stimulation of training and mobility

Short description (English)

See abstract

Partners and International Organizations (English)

Coordinator: IAN-CNR (I)

Abstract (English)

In the period of 1.4.1999 - 30.3.2002, the Swiss team at ETH Zurich developed a wavelet/ multipole solver for strongly elliptic boundary integral equations on polyhedral domains.

The solver is based on the Galerkin formulation of the boundary integral equations and applied wavelet based matrix compression as well as fast multipole techniques. No stiffness matrix of the operator is generated. The solver has been realized as C++ class library CONCEPTS.

A class of new, kernel independent multipole methods has been developed and tested in CONCEPTS. It achieves performance similar to the wavelet code, but appears superior in test on complicated domains.

For weakly singular boundary integral equations of the first kind (single layer potential equations), a new preconditioner on unstructured surface triangulation has been developed which exhibits O(N) complexity and reduces the condition number of the matrices to
O(log N). It is based on an agglomeration technique and uses to a large extent the tree data structures already present in the in the multipole implementation in CONCEPTS.

In addition, during the project, a preconditioner for industrial eddy current problems which are formulated by boundary integral equations of the second kind at ABB Corp. Res. Center. has been developed and implemented.

Finally, during the project, a class of wavelet-based fast numerical schemes for the solution of parabolic equations has been developed and analyzed. Its implementation in MATLAB has shown superiority over standard numerical methods.

In joint work with the german team (TU Chemnitz), we obtained a completely new type of wavelet preconditioner for degerate elliptic problems which arise e.g. in mathematical finance or in spectral methods.

References in databases (English)

Swiss Database: Euro-DB of the
State Secretariat for Education and Research
Hallwylstrasse 4
CH-3003 Berne, Switzerland
Tel. [phone]
Swiss Project-Number: 97.0404