So far, there exist two kinds of fast wavelet compression method for solving boundary integral equations (BIEs), which we will refer to as the traditional WBEM and our WBEM.

    The main idea of  the wavelet compression method for BIEs is first proposed by the well-known paper [Beylkin G, Coifman R, Rokhlin V (1991) Fast wavelet transforms and numerical algorithms. Comm Pure Appl Math 37:141–183], or BCP-paper. Since then a large number of papers were published on the traditional method; representitive examples are those paper published by Dahmen, Schneider, Harbrecht, Schwab, Petersdorff, etc.. You can get their work from the web, or from me.

    The traditional method uses wavelet basis defined on interval to discretize the BIEs. Thus a boundary parameterization is necessary, which is the main drawback of the method. Since in many problems with complixated boundaries, paramerizations are usually not straightforward, or even can not be obtained. For more details, you can feel free to discuss with me.

    Our WBEM is first presented by Professor Johannes Tausch. It can construct wavelet basis on the usual boundary element partition, and thus can be used where the conventional BEM can be used. Our recent work shows that the method can even outperforms the FMM, at least at some cases.

    My experience on our WBEM indicates that the method is very competitive in handling large-scale problems, and thus worhty to study. Hope you could be interested in the method, and hope to discuss with you about any issue of the method.