Preconditioner updates for solving sequences of linear systems in matrix-free environment

Abstract

SUMMARYWe present two new ways of preconditioning sequences of nonsymmetric linear systems in the special case where the implementation is matrix-free. Both approaches are fully algebraic, they are based on the general updates of incomplete LU decompositions recently introduced in [1], and they may be directly embedded into nonlinear algebraic solvers. The first of the approaches uses a new model of partial matrix estimation to compute the updates. The second approach exploits separability of function components to apply the updated factorized preconditioner via function evaluations with the discretized operator. Experiments with matrix-free implementations of test problems show that both new techniques offer useful, robust and black-box solution strategies. In addition, they show that the new techniques are often more efficient in matrix-free environment than either recomputing the preconditioner from scratch for every linear system of the sequence or than freezing the preconditioner throughout the whole sequence.

Jacobian-Free Implicit Inner-Iteration Preconditioner for Nonlinear Least Squares Problems

“…Some preconditioner construction techniques for sequence of matrices [3,28,29] and the incomplete LU factorization [12,28] may be used to determine the preconditioner in steps 3 and 11 in Algorithm 2. However, all these techniques do not work if we cannot form the full matrix or partial matrix of the Jacobian matrix explicitly in some Newton iterations.…”

Preconditioning Issues in the Numerical Solution of Nonlinear Equations and Nonlinear Least Squares

“…Therefore, in the optimization community there has been interest in updating strategy that can be implemented in a nearly matrix-free manner, that is close to true matrix-free settings. Specifically, nearly matrix-free preconditioning has the following properties: a few full matrices are formed; for preconditioning most systems of the sequence, matrices that are reduced in complexity with respect to the full A k 's are required; matrix-vector product approximations by finite differences can be used; see [25,32].…”

“…More precisely, let F be the function that, evaluated at v ∈ IR n , provides the product of A k times v. F is said separable if its evaluation can be easily separated in the evaluation of its function components, i.e. computing one component of F costs about an n-th part of the full function evaluation [25]. When F is the finite-differences operator, F is separable whenever the nonlinear function itself is separable.…”

“…The approaches proposed in [2,8,15,24,25] focus on updating the factorization of a preconditioner for a specific matrix A s of the sequence with the aim of building preconditioners for the subsequent matrices. These processes give rise to factorized preconditioners that are products of matrices easy to invert.…”

“…Approximate inverse preconditioners fall in this class [10,11]. Both quasi-Newton-type updates [12,13,34,35,37] and fully-algebraic procedures like those in [2,8,15,24,25] will be covered. Methods falling in the first class update the preconditioners by matrices of small rank, using quasi-Newton formula.…”