Q-linear Convergence Research Articles

The iterative methods for monotone generalized variational inequalities

A new class of iterative methods are presented for monotone generalized variational inequality problems. These methods, which base on an equivalent formulation of the original problem, can be viewed as the extension of the symmetric projection rnethod for monotone variational inequalities. The global convergence of the methods is estab-lished under the monotonicity assumption on the functions associated the problem.Specialization of the proposed algorithms and related results to several special cases are also discussed. Moreover, two combination methods are presented for affine monotone problems. and their global and Q-linear convergence are also established

Asymptotic Convergence Analysis of the Forward-Backward Splitting Algorithm

The asymptotic convergence of the forward-backward splitting algorithm for solving equations of type 0 ∈ T(z) is analyzed, where T is a multivalued maximal monotone operator in the n-dimensional Euclidean space. When the problem has a nonempty solution set, and T is split in the form T = ℱ + h with ℱ being maximal monotone and h being co-coercive with modulus greater than ½, convergence rates are shown, under mild conditions, to be linear, superlinear or sublinear depending on how rapidly ℱ−1 and h−1 grow in the neighborhoods of certain specific points. As a special case, when both ℱ and h are polyhedral functions, we get R-linear convergence and 2-step Q-linear convergence without any further assumptions on the strict monotonicity on T or on the uniqueness of the solution. As another special case when h = 0, the splitting algorithm reduces to the proximal point algorithm, and we get new results, which complement R. T. Rockafellar's and F. J. Luque's earlier results on the proximal point algorithm.

Inexact Newton Methods for singular problems

In this paper we describe the effects of an inexact implementation of Newton's method on the behavior of the iteration for certain nonlinear equations in Banach space for which the Frechet derivative is singular at the solution. We give a termination criterion for the inner iteration that preserves not only the q-linear convergence of the Newton iterates but also the fine structure required for an acceleration method.

A fast two-grid method for matrix H-equations

Abstract We present a new two-grid method for solution of matrix H-equations that is a generalization of a method due to Atkinson. Our new algorithm provides mesh independent q-linear convergence and can be accelerated to a superlinearly convergent algorithm by means of a quasi-Newton method. In addition the algorithm requires far less storage than Newton's method. Such considerations are especially important for the large problems generated by discretizations of H-equations for matrix-valued functions. Such equations arise in radiative transfer, polarization, and multi-group neutron transport. As a numerical example we show measured convergence rates and discuss timings for a matrix H-equation arising in radiative transfer.

Least-Change Sparse Secant Update Methods with Inaccurate Secant Conditions

We investigate the role of the secant or quasi-Newton condition in the sparse Broyden or Schubert update method for solving systems of nonlinear equations whose Jacobians are either sparse, or can be approximated acceptably by conveniently sparse matrices. We develop a theory on perturbations to the secant equation that will still allow a proof of local q-linear convergence. To illustrate the theory, we show how to generalize the standard secant condition to the case when the function difference is contaminated by noise.