Copyright SIAM J. Matr. Anal. Appl. (2000)
_____________________________________________________________
       An Analysis of sparse approximate inverse
       preconditioners for boundary integral equations

       by Ke Chen   

       Department of Mathematical Sciences, 
       The University of Liverpool, 
       M \& O   Building,
       Peach  Street, Liverpool L69 7ZL, UK.   
       (Email : k.chen@liv.ac.uk, Web : http://www.liv.ac.uk/~cmchenke

       ABSTRACT

 Preconditioning techniques for dense linear systems arising from
singular boundary integral equations are described and analyzed.
A particular class of approximate inverse based preconditioners related
to the mesh neighbour methods is
known to be efficient. This paper shows that it is an operator
splitting preconditioner and clusters eigenvalues for the normal
equation matrix thus ensuring a fast convergence of the conjugate
gradient normal method (CGN).
Clustering of the eigenvalues of the preconditioned matrix and 
fast convergence of the generalized minimal residual method (GMRES)
are also observed.
For the type of problems considered, we demonstrate a crucial
connection between two essential features of eigenvalue clustering for a
sparse preconditioner --- approximate inversion for a small cluster radius
and operator splitting for a small cluster size.
Experimental results from several boundary integral equations
are presented.


Keywords:
        Preconditioning, Operator splitting, Approximate inversion,
        Singular boundary elements, Least squares solution,
        Conjugate gradients, CGN, GMRES (65F10, 65R20)
----------------------------------------------------------------------------