An incomplete Cholesky preconditioner based on orthogonal approximations

We consider an incomplete Cholesky factorization preconditioner for the iterative solution of large sparse symmetric positive definite (SPD) systems of linear equations. The preconditioner exploits the numerical rank deficiency of some off-diagonal blocks of the Cholesky factor. As a distinctive feature, the approximations performed during the factorization procedure are orthogonal, and therefore the preconditioner falls within the framework introduced in [A. Napov, SIAM J. Matrix Anal. Appl. 34(2013), pp.1148–1173]. This implies that the incomplete factorization procedure isbreakdown-free, and that the resulting preconditioner is SPD. The aforementioned reference also gives some upper bounds on the spectral condition number of the preconditioned system based on the accuracy of individual approximations. The most accurate among these bounds is extended here to the considered preconditioner. On the practical side, we present and study an implementation of the preconditioner. It exploits the block sparsity structure as induced by nested dissection block partitioning, and identifies blocks with low numerical rank based on the sparsity pattern of the system matrix. The performance is assessed based on model PDE discretizations and, further, based on linear systems whose matrices correspond to large enough SPD matrices from the SuiteSparse matrix collection. The reported results are comparedwith those of some other solvers, including the SPD version of ILUPACK solver.