Abstract:
The analysis of preconditioners has often lagged behind their practical application. Although quite powerful and general theories have been developed, questions about the expected performance of specific preconditioners remain hard to answer. This is particularly true for applications with irregular numerical or topological structure. For such problems, new algebraic analysis tools are required.
Over the past several years, just such a set of algebraic tools has been developed which are based upon the concept of "support number". This work descends from the techniques pioneered in the 80's by Beauwens, Notay, Vaidya and others. Although far from a panacea, methods from support theory have been applied to the analysis of existing preconditioners and to design new preconditioners with performance guarantees. This talk will introduce the ideas underlying "support theory", describe recent theoretical and practical progress, and mention some open problems.