Spectral Projected Gradient Method Research Articles

Transmission of visible light through oxidized copper films: feasibility of using a spectral projected gradient method.

Transmittance measurements at normal incidence were carried out over the visible spectral range for metallic thin films deposited by electron beam evaporation on thick glass substrates. The presence of an inhomogeneous thin layer of Cu2O covering the deposited Cu films is required for a satisfactory model of the measurements taken from various samples with increasing thickness. A spectral projected gradient method is used to invert the transmission spectra from which the wavelength dependence of the effective dielectric function of the oxidized coating layer is obtained. Then an effective medium model is used to estimate the volume fraction of internal voids randomly distributed through the surface layer.

Inexact spectral projected gradient methods on convex sets

A new method is introduced for large-scale convex constrained optimization. The general model algorithm involves, at each iteration, the approximate minimization of a convex quadratic on the feasible set of the original problem and global convergence is obtained by means of nonmonotone line searches. A specific algorithm, the Inexact Spectral Projected Gradient method (ISPG), is implemented using inexact projections computed by Dykstra's alternating projection method and generates interior iterates. The ISPG method is a generalization of the Spectral Projected Gradient method (SPG), but can be used when projections are difficult to compute. Numerical results for constrained least-squares rectangular matrix problems are presented.

Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method

The refractive index and the extinction coefficient of thin films deposited on absorbing substrates, as well as their thicknesses, are obtained from measurements of direct transmittance. The inversion approach is based on application of a spectral projected gradient method (SPGM). First, simulated transmittance spectra are used to compare the performance of the method with published results considering films with different degrees of absorption and thickness, and then inversions from real measurements taken over the solar spectral range are carried out. It is shown that the use of the SPGM, to minimize the difference between measured and computed transmittance spectra, becomes a powerful tool to solve efficiently the problem of obtaining the optical constants of thin films from spectroscopic measurements.

Visible spectral dependence of the scattering and absorption coefficients of pigmented coatings from inversion of diffuse reflectance spectra.

A spectral-projected gradient method and an extension of the Kubelka-Munk theory are applied to obtain the relevant parameters of the theory from measured diffuse reflectance spectra of pigmented samples illuminated with visible diffuse radiation. The initial estimate of the spectral dependence of the parameters, required by a recursive spectral-projected gradient method, was obtained by use of direct measurements and up-to-date theoretical estimates. We then tested the consistency of the Kubelka-Munk theory by repeating the procedure with samples of different thicknesses.

Nonmonotone Spectral Projected Gradient Methods on Convex Sets

Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo--Lampariello--Lucidi nonmonotone line search. In particular, the nonmonotone strategy is combined with the spectral gradient choice of steplength to accelerate the convergence process. In addition to the classical projected gradient nonlinear path, the feasible spectral projected gradient is used as a search direction to avoid additional trial projections during the one-dimensional search process. Convergence properties and extensive numerical results are presented.

Automatic differentiation and spectral projected gradient methods for optimal control problems

Automatic differentiation and nonmonotone spectral projected gradient techniques are used for solving optimal control problems. The original problem is reduced to a nonlinear programming one using general Runge–Kutta integration formulas. Canonical formulas which use a fast automatic differentiation strategy are given to compute derivatives of the objective function. On the basis of this approach, codes for solving optimal control problems are developed and some numerical results are presented.