Abstract
This work studies limited memory preconditioners for linear symmetric positive definite systems of equations. Connections are established between a partial Cholesky factorization from the literature and a variant of Quasi-Newton type preconditioners. Then, a strategy for enhancing the Quasi-Newton preconditioner via available information is proposed. Numerical experiments show the behaviour of the resulting preconditioner.
Highlights
The numerical solution of linear algebraic systems with symmetric positive definite (SPD) matrix is required in a broad range of applications, see e.g., [1,2,3,4,5,6]
Method or its variants [4] and we propose its use in combination with limited memory preconditioners
We show that the partial Cholesky factorization coincides with a Quasi-Newton preconditioner where the first-level preconditioner is diagonal and the low-dimensional subspace is constituted by a subset of columns of the identity matrix of dimension m
Summary
The numerical solution of linear algebraic systems with symmetric positive definite (SPD) matrix is required in a broad range of applications, see e.g., [1,2,3,4,5,6]. “partial” Cholesky factorization was proposed in [13,14] and used in the solution of compressed sensing, linear and quadratic programming, Lasso problems, maximum cut problems [3,14,15] This preconditioner is built by computing a trapezoidal partial Cholesky factorization limited to a prefixed and small number of columns and by approximating the resulting Schur complement via its diagonal. We show that the partial Cholesky factorization coincides with a Quasi-Newton preconditioner where the first-level preconditioner is diagonal and the low-dimensional subspace is constituted by a subset of columns of the identity matrix of dimension m.
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