Extending the applicability of the discrete dipole approximation for multi-scale features on surface

Many problems in science and engineering involve significant physical entities and processes that span a substantial range of dimensions. In the case of characterization of bacteria on growth media using light scattering the length scales of interest can be classified as micro-scale (single bacterium), macro-scale (bacterial colonies of more than 10¹² bacterium that have passed through the exponential growth phase and reached mm size), and the intermediate or meso-scale of several tens of hundreds of bacteria. Light scattering approaches, to be effective in determining physical properties such as morphology and material composition, must comprehend this spectrum of length scales. The discrete dipole approximation (DDA), a powerful modeling tool for rigorous 3-D vector scattering, has shown its capability to predict the light scattering from micro-scale objects. To be able to accommodate meso-scale objects, we need to extend the computational limits of the DDA method such that it could compute object sizes of 10λ-30λ characteristic dimension (i.e. volumes of 10³-10⁴ cubic wavelengths). To accomplish this, an analysis of the DDA method was performed for meso-scale cases of interest especially in biological applications. Based on this study, we propose new Sommerfeld integration paths and a revised iterative algorithm that combine to provide substantial improvements in the size of the computational domain that can be modeled for a given convergence criterion.     

By How Much Can Residual Minimization Accelerate the Convergence of Orthogonal Residual Methods?

We capitalize upon the known relationship between pairs of orthogonal and minimal residual methods (or, biorthogonal and quasi-minimal residual methods) in order to estimate how much smaller the residuals or quasi-residuals of the minimizing methods can be compared to those of the corresponding Galerkin or Petrov–Galerkin method. Examples of such pairs are the conjugate gradient (CG) and the conjugate residual (CR) methods, the full orthogonalization method (FOM) and the generalized minimal residual (GMRES) method, the CGNE and BiCG versions of applying CG to the normal equations, as well as the biconjugate gradient (BiCG) and the quasi-minimal residual (QMR) methods. Also the pairs consisting of the (bi)conjugate gradient squared (CGS) and the transpose-free QMR (TFQMR) methods can be added to this list if the residuals at half-steps are included, and further examples can be created easily.The analysis is more generally applicable to the minimal residual (MR) and quasi-minimal residual (QMR) smoothing processes, which are known to provide the transition from the results of the first method of such a pair to those of the second one. By an interpretation of these smoothing processes in coordinate space we deepen the understanding of some of the underlying relationships and introduce a unifying framework for minimal residual and quasi-minimal residual smoothing. This framework includes the general notion of QMR-type methods.     

Quantitative magnetic resonance (QMR) method for bone and whole-body-composition analysis

Objective: to evaluate the applicability, precision, and accuracy of the new EchoMRI quantitative magnetic resonance (QMR) method for in-vitro bovine bone analysis and in-vivo whole-body-composition analysis of conscious live mice. Research methods and procedures: bovine tibia bone samples were measured by QMR and dual-energy X-ray adsorptiometry (DEXA). Repeated measures of whole-body composition were made using live and dead mice with different levels of fat by QMR and DEXA and by classic chemical analysis of the mouse carcass. Results: bone-mineral density (BMD) and bone-mineral content (BMC) measured in bovine tibia by QMR and DEXA were highly correlated. Precision of fat and lean measurement in mice was found to be better for QMR than for DEXA. The coefficient of variation (CV) for fat was 0.34–0.71% for QMR compared with 3.06–12.60% for DEXA. Discussion: QMR offers more specific parameters of bone structure than does DEXA. QMR and DEXA did not differ in the total amount of fat detected in live mice but QMR had improved precision. QMR was superior to DEXA in measuring fat in very small mice. Conclusions: in bone tissue there is a strong correlation between hydrogen NMR signal and bone-mineral density as measured by X-ray. QMR provides a very precise, accurate, fast, and easy to use method for determining fat and lean mass of mice without the need for anesthesia. Its ability to detect differences and monitor changes in body composition in mice with great precision should be of great value in characterizing phenotypes and studying drugs affecting obesity.     

CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm

The Generalized Minimal Residual (GMRES) method and the Quasi-Minimal Residual (QMR) method are two Krylov methods for solving linear systems. The main difference between these methods is the generation of the basis vectors for the Krylov subspace. The GMRES method uses the Arnoldi process while QMR uses the Lanczos algorithm for constructing a basis of the Krylov subspace. In this paper we give a new method similar to QMR but based on the Hessenberg process instead of the Lanczos process. We call the new method the CMRH method. The CMRH method is less expensive and requires slightly less storage than GMRES. Numerical experiments suggest that it has behaviour similar to GMRES. This revised version was published online in June 2006 with corrections to the Cover Date.     

A QUASI-MINIMAL RESIDUAL VARIANT OF THE IOM (q) FOR LARGE UNSYMMETRIC LINEAR SYSTEMS

1　IntroductionThe solution of large N× N nonsingular unsymmetric( non-Hermitian) sparse sys-tems of linear equationsAx =b, ( 1 )is one of the most frequently encountered tasks in numerical computations.For example,such systems arise from finite difference or finite element approximations to partial differ-ential equationsA major class of methods for solving ( 1 ) is Krylov subspace or conjugate gradienttype methods.Most successful scheme of these methods are based on the orthogonal pro-jec…