Incomplete block LD^{-1} L^T factorization for constraint preconditioning techniques

We transform and partition the symmetric indefinite (saddle point) matrices into a block structure with blocks of orders 1 and 2 forming ‘a priori’ pivots. A sparse incomplete block $LD^{-1}L^T$ factorization of such a partitioned matrix is determined. We show that the reconstruction of a matrix from these incomplete factors forms a constraint preconditioner. The incomplete factorization depends on the existence of incomplete Schur complement reductions of symmetric positive definite matrices. Adding a semi-definite diagonal matrix to each of these incomplete Schur complement reductions addresses the existence and stability issues. Conjugate gradient method is applied to the preconditioned system and numerical results are presented for validation. Keywords: saddle point matrices, transformation, incomplete block factorization, precondition conjugate gradient, incomplete Schur complement reductio