GENERIC APPROXIMATE SPARSE INVERSE MATRIX TECHNIQUES

During the last decades explicit preconditioning methods have gained interest among the scientific community, due to their efficiency for solving large sparse linear systems in conjunction with Krylov subspace iterative methods. The effectiveness of explicit preconditioning schemes relies on the fact that they are close approximants to the inverse of the coefficient matrix. Herewith, we propose a Generic Approximate Sparse Inverse (GenASPI) matrix algorithm based on ILU(0) factorization. The proposed scheme applies to matrices of any structure or sparsity pattern unlike the previous dedicated implementations. The new scheme is based on the Generic Approximate Banded Inverse (GenAbI), which is a banded approximate inverse used in conjunction with Conjugate Gradient type methods for the solution of large sparse linear systems. The proposed GenASPI matrix algorithm, is based on Approximate Inverse Sparsity patterns, derived from powers of sparsified matrices and is computed with a modified procedure based on the GenAbI algorithm. Finally, applicability and implementation issues are discussed and numerical results along with comparative results are presented.

ON RULES FOR STOPPING THE CONJUGATE GRADIENT TYPE METHODS IN ILL‐POSED PROBLEMS

We consider stopping rules in conjugate gradient type iteration methods for solving linear ill‐posed problems with noisy data. The noise level may be known exactly or approximately or be unknown. We propose several new stopping rules, mostly for the case of unknown noise level. Numerical comparison with known rules (discrepancy principle, montone error rule, L‐curve rule, Hanke‐Raus rule) shows that the new rules are competitive.