Equivalent operator preconditioning for elliptic problems

The numerical solution of linear elliptic partial differential equations most often involves a finite element or finite difference discretization. To preserve sparsity, the arising system is normally solved using an iterative solution method, commonly a preconditioned conjugate gradient method. Preconditioning is a crucial part of such a solution process. In order to enable the solution of very large-scale systems, it is desirable that the total computational cost will be of optimal order, i.e. proportional to the degrees of freedom of the approximation used, which also induces mesh independent convergence of the iteration. This paper surveys the equivalent operator approach, which has proven to provide an efficient general framework to construct such preconditioners. Hereby one first approximates the given differential operator by some simpler differential operator, and then chooses as preconditioner the discretization of this operator for the same mesh. In this survey we give a uniform presentation of this approach, including theoretical foundation and several practically important applications for both symmetric and nonsymmetric equations and systems, and some nonlinear examples in the context of Newton linearization. Dedicated to the memory of Gene Golub for his friendly manner and for his broad interest and significant impact on numerical analysis.     

Validation of the extended Tanaka column characterization protocol by multivariate analysis of chromatographic retention of low‐molecular‐weight analytes on reversed phase columns using methanol and acetonitrile as organic modifiers

The validity of the extended Tanaka column characterization procedure against the retention behavior of 101 analytes of widely differing properties chromatographed on five differing stationary phase chemistries has been established using a chemometric technique called principal component analysis (PCA). It was concluded that the simple and conveniently determined column characterization parameters covered the same space in the PCA loading plot as the retention times for the 101 differing analytes. This confirms that the ten column characterization parameters of the extended Tanaka protocol encode the same information as the retention times of the 101 analytes. Significant selectivity differences were observed between stationary phases and the mobile‐phase modifiers – MeOH and MeCN. PCA contribution plots served as a convenient way to highlight specific selectivity differences between stationary phases. logD values exhibited a poor correlation with retention indicating that retention in RP‐LC is not solely dictated by the analyte's hydrophobicity. The use of MeOH was found to generate greater selectivity differences with the five stationary phases than when MeCN is used.     

One-nucleon removal cross-sections for 17, 19C and 8, 10B

One-neutron removal cross-sections ( ) of ^{17, 19}C in various targets at about 900 MeV/nucleon, and one-proton removal cross-sections ( ) of ^{8, 10}B at about 1400 MeV/nucleon were measured using the fragment separator FRS at GSI. A significant increase of for ¹⁹C compared to its neighbors was observed. The same behavior was found for for the neutron-deficient nucleus ⁸B compared to the stable isotope ¹⁰B. These results support a one-neutron halo structure in ¹⁹C and a one-proton halo in ⁸B. Received: 26 July 2000 / Accepted: 11 January 2001     

Real valued iterative methods for solving complex symmetric linear systems

Complex valued systems of equations with a matrix R + 1S where R and S are real valued arise in many applications. A preconditioned iterative solution method is presented when R and S are symmetric positive semi‐definite and at least one of R, S is positive definite. The condition number of the preconditioned matrix is bounded above by 2, so only very few iterations are required. Applications when solving matrix polynomial equation systems, linear systems of ordinary differential equations, and using time‐stepping integration schemes based on Padé approximation for parabolic and hyperbolic problems are also discussed. Numerical comparisons show that the proposed real valued method is much faster than the iterative complex symmetric QMR method. Copyright © 2000 John Wiley & Sons, Ltd.     

Quadratic estimates and functional calculi of perturbed Dirac operators

We prove quadratic estimates for complex perturbations of Dirac-type operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral projections of the Hodge–Dirac operator on compact manifolds depend analytically on L_∞ changes in the metric. We also recover a unified proof of many results in the Calderón program, including the Kato square root problem and the boundedness of the Cauchy operator on Lipschitz curves and surfaces.     

A survey of preconditioned iterative methods for linear systems of algebraic equations

We survey preconditioned iterative methods with the emphasis on solving large sparse systems such as arise by discretization of boundary value problems for partial differential equations.We discuss shortly various acceleration methods but the main emphasis is on efficient preconditioning techniques. Numerical simulations on practical problems have indicated that an efficient preconditioner is the most important part of an iterative algorithm. We report in particular on the state of the art of preconditioning methods for vectorizable and/or parallel computers.Dedicated to Carl-Erik Fröberg, a pioneer in Numerical Methods.     

A survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices

Eigenvalue and condition number estimates for preconditioned iteration matrices provide the information required to estimate the rate of convergence of iterative methods, such as preconditioned conjugate gradient methods. In recent years various estimates have been derived for (perturbed) modified (block) incomplete factorizations. We survey and extend some of these and derive new estimates. In particular we derive upper and lower estimates of individual eigenvalues and of condition number. This includes a discussion that the condition number of preconditioned second order elliptic difference matrices is O(h⁻¹). Some of the methods are applied to compute certain parameters involved in the computation of the preconditioner.     

Kähler geometry of Douady spaces

We consider the generalized Petersson–Weil metric on the moduli space of compact submanifolds of a Kähler manifold or a projective variety. It is extended as a positive current to the space of points corresponding to reduced fibers, and estimates are shown. For moduli of projective varieties the Petersson–Weil form is the curvature of a certain determinant line bundle equipped with a Quillen metric. We investigate its extension to the compactified moduli space.     

The potential of inductively coupled plasma mass spectrometry detection for high-performance liquid chromatography combined with accurate mass measurement of organic pharmaceutical compounds

Quantification of unknown components in pharmaceutical, metabolic and environmental samples is an important but difficult task. Most commonly used detectors (like UV, RI or MS) require standards of each analyte for accurate quantification. Even if the chemical structure or elemental composition is known, the response from these detectors is difficult to predict with any accuracy. In inductively coupled plasma mass spectrometry (ICP-MS) compounds are atomised and ionised irrespective of the chemical structure(s) incorporating the element of interest. Liquid chromatography coupled with inductively coupled plasma mass spectrometry (LC/ICP-MS) has been shown to provide a generic detection for structurally non-correlated compounds with common elements like phosphorus and iodine. Detection of selected elements gives a better quantification of tested 'unknowns' than UV and organic mass spectrometric detection. It was shown that the ultrasonic nebuliser did not introduce any measurable dead volume and preserves the separation efficiency of the system. ICP-MS can be used in combination with many different mobile phases ranging from 0-100% organic modifier. The dynamic range was found to exceed 2.5 orders of magnitude. The application of LC/ICP-MS to pharmaceutical drugs and formulations has shown that impurities can be quantified below the 0.1 mol-% level.