S. Bellavia, J. Gondzio and B. Morini
Abstract
We analyze and discuss matrix-free and limited-memory preconditioners
(LMP) for sparse symmetric positive definite systems and normal equations
of large and sparse least-squares problems. The preconditioners are
based on a partial Cholesky factorization and can be coupled with
a deflation strategy. The construction of the preconditioners requires
only matrix-vector products, is breakdown-free, and does not need
to form the matrix. The memory requirements of the preconditioners
are defined in advance and they do not depend on the number of nonzero
entries in the matrix. In case eigenvalue deflation is used,
the preconditioners turn out to be suitable for solving sequences
of slowly changing systems or linear systems with different right-hand
sides. Numerical results are provided, including the case of sequences
arising in constrained linear least-squares problems solved by interior
point methods.
Key words: sparse symmetric positive definite systems and least-squares problems, preconditioners, matrix-free, memory constraints, Cholesky factorization, deflation, interior point methods.