Implementation and Evaluation of Vaidya’s Preconditioners

We show that a novel class of preconditioners, designed by Pravin Vaidya in 1991 but never before implemented, is remarkably robust and can outperform incomplete-Cholesky preconditioners. Our test suite includes problems arising from finitedifferences discretizations of elliptic PDEs in two and three dimensions. On 2D problems, Vaidya’s preconditioners often outperform drop-tolerance incomplete-Cholesky preconditioners with similar amounts of fill and sometimes outperform modified drop-tolerance incomplete preconditioners. Vaidya’s preconditioners do not appear to be effective on 3D problems. Vaidya’s preconditioners are robust in the sense that they are insensitive to the boundary conditions of the PDE or to the original ordering of the mesh.