Approximate Analytic Matrix Factorisations as Preconditioners for Newton's Method
نویسنده
چکیده
This paper treats direct methods of matrix factorisation as being approximated by analytic recursions. From this viewpoint, a new class of preconditioners is motivated, which is denoted by approximate analytic factorisation (AAF). Due to analyticity and exploitation of invariances, their computational and storage cost is usually rather cheap. AAF are derived for a variety of practically relevant matrices, and discussed for more general cases. It turns out that due to their ability to cope eeciently with both sparse and small-magnitude updates in the original matrix, AAF are very eecient as preconditioners for an iterative linear solver in Newton's Method.
منابع مشابه
Approximate Inverse Preconditioners for General Sparse Matrices
The standard Incomplete LU (ILU) preconditioners often fail for general sparse indeenite matrices because they give rise tòunstable' factors L and U. In such cases, it may be attractive to approximate the inverse of the matrix directly. This paper focuses on approximate inverse preconditioners based on minimizing kI?AMk F , where AM is the preconditioned matrix. An iterative descent-type method...
متن کاملAINVK: a Class of Approximate Inverse Preconditioners based on Krylov-subspace methods, for Large Indefinite Linear Systems
We propose a class of preconditioners for symmetric linear systems arising from numerical analysis and nonconvex optimization frameworks. Our preconditioners are specifically suited for large indefinite linear systems and may be obtained as by-product of Krylov-subspace solvers, as well as by applying L-BFGS updates. Moreover, our proposal is also suited for the solution of a sequence of linear...
متن کاملILU and IUL factorizations obtained from forward and backward factored approximate inverse algorithms
In this paper, an efficient dropping criterion has been used to compute the IUL factorization obtained from Backward Factored APproximate INVerse (BFAPINV) and ILU factorization obtained from Forward Factored APproximate INVerse (FFAPINV) algorithms. We use different drop tolerance parameters to compute the preconditioners. To study the effect of such a dropping on the quality of the ILU ...
متن کاملA Total Least Squares Methodfor Toeplitz
A Newton method to solve total least squares problems for Toeplitz systems of equations is considered. When coupled with a bisection scheme, which is based on an eecient algorithm for factoring Toeplitz matrices, global convergence can be guaranteed. Circulant and approximate factorization preconditioners are proposed to speed convergence when a conjugate gradient method is used to solve linear...
متن کاملBand preconditioners for the matrix-free truncated Newton method
This report is devoted to preconditioning techniques for the matrix-free truncated Newton method. After a review of basic known approaches, we propose new results concerning tridiagonal and pentadiagonal preconditioners based on the standard BFGS updates and on numerical differentiation. Furthermore, we present results of extensive numerical experiments serving for the careful comparison of sui...
متن کامل