GMRES and the minimal polynomial

We present a qualitative model for the convergence behaviour of the Generalised Minimal Residual (GMRES) method for solving nonsingular systems of linear equationsAx =b in finite and infinite dimensional spaces. One application of our methods is the solution of discretised infinite dimensional problems, such as integral equations, where the constants in the asymptotic bounds are independent of the mesh size.Our model provides simple, general bounds that explain the convergence of GMRES as follows: If the eigenvalues ofA consist of a single cluster plus outliers then the convergence factor is bounded by the cluster radius, while the asymptotic error constant reflects the non-normality ofA and the distance of the outliers from the cluster. If the eigenvalues ofA consist of several close clusters, then GMRES treats the clusters as a single big cluster, and the convergence factor is the radius of this big cluster. We exhibit matrices for which these bounds are tight.Our bounds also lead to a simpler proof of existing r-superlinear convergence results in Hilbert space.This research was partially supported by National Science Foundation grants DMS-9122745, DMS-9423705, CCR-9102853, CCR-9400921, DMS-9321938, DMS-9020915, and DMS-9403224.     

Integrating 3D images using laboratory‐based micro X‐ray computed tomography and confocal X‐ray fluorescence techniques

The application of non‐destructive imaging to characterizing samples has become more important as the costs of samples increase. Imaging a sample via X‐ray techniques is preferable when altering or even touching the sample affects its properties, or when the sample is fielded after characterization. Two laboratory‐based X‐ray techniques used at Los Alamos include micro X‐ray computed tomography (MXCT) and confocal micro X‐ray fluorescence (confocal MXRF). Both methods create a 3D rendering of the sample non‐destructively. MXCT produces a high‐resolution (sub‐µm voxel) rendering of the sample based upon X‐ray absorption; the resulting model is a function of density and does not contain any elemental information. Confocal MXRF produces an elementally specific 3D rendering of the sample, but at a lower (30 × 30 × 65 µm) resolution. By combining data from these two techniques, scientists provided a more comprehensive method of analysis. We will describe a MATLAB routine written to render each of these data sets individually and/or within the same coordinate system. This approach is shown in the analysis of two samples: an integrated circuit surface mounted resistor and a machined piece of polystyrene foam. The samples chosen provide an opportunity to compare and contrast the two X‐ray techniques, identify their weaknesses and show how they are used in a complementary fashion. Copyright © 2010 John Wiley & Sons, Ltd.     

Precision Penning trap mass measurements of rare isotopes produced by projectile fragmentation

The low-energy beam and ion trap facility LEBIT at NSCL/MSU is at present the only facility where precision experiments are performed with stopped rare isotope beams produced by fast-beam fragmentation. LEBIT combines high-pressure-gas stopping with advanced ion manipulation techniques to provide brilliant low-energy beams. So far these beams have mainly been used for mass measurements on short-lived rare isotopes with a 9.4T Penning trap mass spectrometer. Recent examples include ^70m Br , located at the proton dripline, ³²Si and the iron isotopes ^63-65Fe . While the measurement of ³²Si helps to solve a long-standing dispute over the validity of the isobaric multiplet mass equation (IMME) for the A = 32 , T = 2 multiplet, the mass measurements of ^65m,g Fe marked the first time a nuclear isomeric state has been discovered by Penning trap mass spectrometry.     

Generalized phase diversity for wave-front sensing

Phase diversity is a phase-retrieval algorithm that uses a pair of intensity images taken symmetrically about the wave front to be determined. If these images are taken about the system input pupil this is equivalent to a curvature-sensing algorithm. Traditionally a defocus aberration kernel is used to produce the phase-diverse data. We present a generalization of this method to allow the use of other functions as the diversity kernel. We discuss the necessary and sufficient conditions that such a function must satisfy for use in a null wave-front sensor. Computer simulations were used to validate these results.     

Reduced occupancy of the deeply bound 0d(5/2) neutron state in 32Ar

The 9Be(32Ar, 31Ar)X reaction, leading to the 5/2+ ground state of a nucleus at the proton drip line, has a cross section of 10.4(13) mb at a beam energy of 65.1 MeV/nucleon. This translates into a spectroscopic factor that is only 24(3)% of that predicted by the many-body shell-model theory. We introduce refinements to the eikonal reaction theory used to extract the spectroscopic factor to clarify that this very strong reduction represents an effect of nuclear structure. We suggest that it reflects correlation effects linked to the high neutron separation energy (22.0 MeV) for this state.