Local results for the Gauss-Newton method on constrained rank-deficient nonlinear least squares

A nonlinear least squares problem with nonlinear constraints may be ill posed or even rank-deficient in two ways. Considering the problem formulated as subject to the constraints , the Jacobian and/or the Jacobian , , may be ill conditioned at the solution.

We analyze the important special case when and/or do not have full rank at the solution. In order to solve such a problem, we formulate a nonlinear least norm problem. Next we describe a truncated Gauss-Newton method. We show that the local convergence rate is determined by the maximum of three independent Rayleigh quotients related to three different spaces in .

Another way of solving an ill-posed nonlinear least squares problem is to regularize the problem with some parameter that is reduced as the iterates converge to the minimum. Our approach is a Tikhonov based local linear regularization that converges to a minimum norm problem. This approach may be used both for almost and rank-deficient Jacobians.

Finally we present computational tests on constructed problems verifying the local analysis.

     

Periodic oscillation within the chaotic region in a semiconductor laser subjected to external optical injection

We report on the experimental observation of periodic windows within the chaos of the output of a semiconductor laser that is subjected to external optical injection. Continuously sampled spectra of the slave-laser output as the injection level is increased reveal the general behavior of the system as well as features that suggest that the periodic oscillation forms a distinct island within the chaotic region in the parameter space.     

Synthesis and nonlinear light scattering of microemulsions and nanoparticle suspensions

Microemulsions composed of normal or inverse micellar solutions and aqueous suspensions of pristine (uncoated) or silica-coated iron oxide nanoparticles, mainly γ-Fe₂O₃, were synthesised and their optical limiting properties investigated. The microemulsions are colorless solutions with high transparency for visible wavelengths while the aqueous suspensions of iron oxide are of pale yellow colour. Optical limiting experiments performed in 2 mm cells using a f/5 optical system with a frequency doubled Nd:YAG laser delivering 5 ns pulses with 10 Hz repetition rate, showed clamping levels of ∼3 μJ for the suspensions of both pristine and silica-coated iron oxide nanoparticles. A strong photoinduced nonlinear light scattering was observed for the water-in-oil microemulsion and the aqueous suspensions of nanoparticles while oil-in-water microemulsions did not show a significant nonlinear effect. Measurements carried out using an integrating sphere further verified that the photoinduced nonlinear light scattering is the dominating nonlinear mechanism while the nonlinear absorption of iron oxide nanoparticles is negligible at 532 nm.     

JET quasistationary internal-transport-barrier operation with active control of the pressure profile

Quasistationary operation has been achieved on the Joint European Torus tokamak in internal-transport-barrier (ITB) scenarios, with the discharge time limited only by plant constraints. Full current drive was obtained over all the high performance phase by using lower hybrid current drive. For the first time feedback control on the total pressure and on the electron temperature profile was implemented by using, respectively, the neutral beams and the ion-cyclotron waves. Although impurity accumulation could be a problem in steady state ITBs, these experiments bring some elements to answer to it.     

The 10 MHz ultrasonic near-field: calculations, hydrophone and optical diffraction tomography measurements

The near-field from a 10 MHz ultrasound transducer has been studied using three different techniques; theoretical calculations, miniature hydrophone measurements and optical diffraction tomography (ODT). The motive was twofold; to evaluate the measurement possibilities of ODT for high-frequency ultrasound and to verify the calculated complexity of near-fields from real ultrasound transducers designed for use in blood perfusion measurements. Calculations of the field from an ideal piston transducer were done with surface integrals of the Kirchhoff function and the measured transducer was designed to approximate this ideal. Contour maps, from the three methods, of pressure amplitude from 1 to 30 mm along the beam are presented with beam profiles at 5 and 45 mm. The results imply that ODT has an increased spatial resolution compared with the 0.5 and 1.0 mm hydrophones. However, even better spatial resolution is needed to draw definite conclusions regarding the complexity of the near-field of the 10 MHz transducer.     

Mixing ratios in67Zn determined by gamma-gamma angular correlations

The gamma-gamma angular correlations of the 209.0–184.6, 494.3–393.6, 494.3–300.2 and 703.6–184.6 keV cascades in⁶⁷Zn have been measured. Mixing ratios of the 209.0, 300.2, 393.6 and 703.6 keV transitions are deduced. A spectroscopic study could not confirm the existence of a recently proposed level at 710 keV.     

Levels and transitions in136Ba

Excited states of¹³⁶Ba are studied in the beta decay of¹³⁶Cs. The gamma spectrum is measured with high-resolution and coincidence techniques. Several new transitions are suggested to be connected with two recently reported levels. Gamma-gamma directional correlations are performed in an attempt to determine the characteristics of these levels, and in order to determine mixing ratios. The lifetime of the 2207.5 keV state is remeasured, and the lifetime of the 2140.5 keV state is measured for the first time.