A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems

In this paper, we propose a two-step iterative algorithm based on projection technique for solving system of monotone nonlinear equations with convex constraints. The proposed two-step algorithm uses two search directions which are defined using the well-known Barzilai and Borwein (BB) spectral parameters.The BB spectral parameters can be viewed as the approximations of Jacobians with scalar multiple of identity matrices. If the Jacobians are close to symmetric matrices with clustered eigenvalues then the BB parameters are expected to behave nicely. We present a new line search technique for generating the separating hyperplane projection step of Solodov and Svaiter (1998) that generalizes the one used in most of the existing literature. We establish the convergence result of the algorithm under some suitable assumptions. Preliminary numerical experiments demonstrate the efficiency and computational advantage of the algorithm over some existing algorithms designed for solving similar problems. Finally, we apply the proposed algorithm to solve image deblurring problem.

An Enrichment Scheme for Polynomial Chaos Expansion Applied to Random Eigenvalue Problem

For a system with the parameters modeled as uncertain, polynomial approximations such as polynomial chaos expansion provide an effective way to estimate the statistical behavior of the eigenvalues and eigenvectors, provided the eigenvalues are widely spaced. For a system with a set of clustered eigenvalues, the corresponding eigenvalues and eigenvectors are very sensitive to perturbation of the system parameters. An enrichment scheme to the polynomial chaos expansion is proposed here in order to capture the behavior of such eigenvalues and eigenvectors. It is observed that for judiciously chosen enrichment functions, the enriched expansion provides better estimate of the statistical behavior of the eigenvalues and eigenvectors.