Approximate Gauss-Newton methods for nonlinear least squares problems

The Gauss–Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an “inner” direct or iterative process. In comparison with Newton’s method and its variants, the algorithm is attractive because it does not require the evaluation of second-order derivatives in the Hessian of the objective function. In practice the exact Gauss–Newton method is too expensive to apply operationally in meteorological forecasting, and various approximations are made in order to reduce computational costs and to solve the problems in real time. Here we investigate the effects on the convergence of the Gauss–Newton method of two types of approximation used commonly in data assimilation. First, we examine “truncated” Gauss–Newton methods where the inner linear least squares problem is not solved exactly, and second, we examine “perturbed” Gauss–Newton methods where the true linearized inner problem is approximated by a simplified, or perturbed, linear least squares problem. We give conditions ensuring that the truncated and perturbed Gauss–Newton methods converge and also derive rates of convergence for the iterations. The results are illustrated by a simple numerical example. A practical application to the problem of data assimilation in a typical meteorological system is presented.

Calcrete profile development in quaternary alluvial sequences, southeast Spain: Implications for using calcretes as a basis for landform chronologies

A detailed study of the morphology and micro-morphology of Quaternary alluvial calcrete profiles from the Sorbas Basin shows that calcretes may be morphologically simple or complex. The 'simple' profiles reflect pedogenesis occurring after alluvial terrace formation and consist of a single pedogenic horizon near the land surface. The 'complex' profiles reflect the occurrence of multiple calcrete events during terrace sediment aggradation and further periods of pedogenesis after terrace formation. These 'complex' calcrete profiles are consequently described as composite profiles. The exact morphology of the composite profiles depends upon: (1) the number of calcrete-forming events occurring during terrace sediment aggradation; (2) the amount of sediment accretion that occurs between each period of calcrete formation; and (3) the degree of pedogenesis after terrace formation. Simple calcrete profiles are most useful in establishing landform chronologies because they represent a single phase of pedogenesis after terrace formation. Composite profiles are more problematic. Pedogenic calcretes that form within them may inherit carbonate from calcrete horizons occurring lower down in the terrace sediments. In addition erosion may lead to the exhumation of older calcretes within the terrace sediment. Calcrete 'inheritance' may make pedogenic horizons appear more mature than they actually are and produce horizons containing carbonate embracing a range of ages. Calcrete exhumation exposes calcrete horizons whose morphology and radiometric ages are wholly unrelated to terrace surface age. Composite profiles are, therefore, only suitable for chronological studies if the pedogenic horizon capping the terrace sequence can be clearly distinguished from earlier calcrete-forming events. Thus, a detailed morphological/micro-morphological study is required before any chronological study is undertaken. This is the only way to establish whether particular calcrete profiles are suitable for dating purposes. Copyright (C) 2003 John Wiley Sons, Ltd.

Cusp observations during a sequence of fast IMF B-Z reversals

In recent years, a large number of papers have reported the response of the cusp to solar wind variations under conditions of northward or southward Interplanetary Magnetic Field (IMF) Z-component (BZ). These studies have shown the importance of both temporal and spatial factors in determining the extent and morphology of the cusp and the changes in its location, connected to variations in the reconnection geometry. Here we present a comparative study of the cusp, focusing on an interval characterised by a series of rapid reversals in the BZ-dominated IMF, based on observations from space-borne and ground-based instrumentation. During this interval, from 08:00 to 12:00 UT on 12 February 2003, the IMF BZ component underwent four reversals, remaining for around 30 min in each orientation. The Cluster spacecraft were, at the time, on an outbound trajectory through the Northern Hemisphere magnetosphere, whilst the mainland VHF and Svalbard (ESR) radars of the EISCAT facility were operating in support of the Cluster mission. Both Cluster and the EISCAT were, on occasion during the interval, observing the cusp region. The series of IMF reversal resulted in a sequence of poleward and equatorward motions of the cusp; consequently Cluster crossed the high altitude cusp twice before finally exiting the dayside magnetopause, both times under conditions of northward IMF BZ. The first magnetospheric cusp encounter, by all four Cluster spacecraft, showed reverse ion dispersion typical of lobe reconnection; subsequently, Cluster spacecraft 1 and 3 (only) crossed the cusp for a second time. We suggest that, during this second cusp crossing, these two spacecraft were likely to have been on newly closed field lines, which were first reconnected (opened) at low latitudes and later reconnected again (re-closed) poleward of the northern cusp.

Sequence-selective assembly of tweezer-molecules on linear templates enables frameshift-reading of sequence information

Monomer-sequence information in synthetic copolyimides can be recognised by tweezer-type molecules binding to adjacent triplet-sequences on the polymer chains. In the present paper different tweezer-molecules are found to have different sequence-selectivities, as demonstrated in solution by 1H NMR spectroscopy and in the solid state by single crystal X-ray analyses of tweezer-complexes with linear and macrocyclic oligo-imides. This work provides clear-cut confirmation of polyimide chain-folding and adjacent-tweezer-binding. It also reveals a new and entirely unexpected mechanism for sequence-recognition which, by analogy with a related process in biomolecular information processing, may be termed "frameshift-reading". The ability of one particular tweezer-molecule to detect, with exceptionally high sensitivity, long-range sequence-information in chain-folding aromatic copolyimides, is readily explained by this novel process.

Sequence analysis of the cupin gene family in Synechocystis PCC6803

The recently described cupin superfamily of proteins includes the germin and germinlike proteins, of which the cereal oxalate oxidase is the best characterized. This superfamily also includes seed storage proteins, in addition to several microbial enzymes and proteins with unknown function. All these proteins are characterized by the conservation of two central motifs, usually containing two or three histidine residues presumed to be involved with metal binding in the catalytic active site. The present study on the coding regions of Synechocystis PCC6803 identifies a previously unknown group of 12 related cupins, each containing the characteristic two-motif signature. This group comprises 11 single-domain proteins, ranging in length from 104 to 289 residues, and includes two phosphomannose isomerases and two epimerases involved in cell wall synthesis, a member of the pirin group of nuclear proteins, a possible transcriptional regulator, and a close relative-of a cytochrome c551 from Rhodococcus. Additionally, there is a duplicated, two-domain protein that has close similarity to an oxalate decarboxylase from the fungus Collybia velutipes and that is a putative progenitor of the storage proteins of land plants.

Elusive relationships within order fabales: phylogenetic analyses using matK and rbcL sequence data

The order Fabales, including Leguminosae, Polygalaceae, Quillajaceae and Surianaceae, represents a novel hypothesis emerging from angiosperm molecular phylogenies. Despite good support for the order, molecular studies to date have suggested contradictory, poorly supported interfamilial relationships. Our reappraisal of relationships within Fabales addresses past taxon sampling deficiencies, and employs parsimony and Bayesian approaches using sequences from the plastid regions rbcL (166 spp.) and matK (78 spp.). Five alternative hypotheses for interfamilial relationships within Fabales were recovered. The Shimodaira-Hasegawa test found the likelihood of a resolved topology significantly higher than the one calculated for a polytomy, but did not favour any of the alternative hypotheses of relationship within Fabales. In the light of the morphological evidence available and the comparative behavior of rbcL and matK, the topology recovering Polygalaceae as sister to the rest of the order Fabales with Leguminosae more closely related to Quillajaceae + Surianaceae, is considered the most likely hypothesis of interfamilial relationships of the order. Dating of selected crown clades in the Fabales phylogeny using penalized likelihood suggests rapid radiation of the Leguminosae, Polygalaceae, and (Quillajaceae + Surianaceae) crown clades.

Simple sequence repeats reveal uneven distribution of genetic diversity in chloroplast genomes of Brassica oleracea L. and (n=9) wild relatives

Diversity in the chloroplast genome of 171 accessions representing the Brassica 'C' (n = 9) genome, including domesticated and wild B. oleracea and nine inter-fertile related wild species, was investigated using six chloroplast SSR (microsatellite) markers. The lack of diversity detected among 105 cultivated and wild accessions of B. oleracea contrasted starkly with that found within its wild relatives. The vast majority of B. oleracea accessions shared a single haplotype, whereas as many as six haplotypes were detected in two wild species, B. villosa Biv. and B. cretica Lam.. The SSRs proved to be highly polymorphic across haplotypes, with calculated genetic diversity values (H) of 0.23-0.87. In total, 23 different haplotypes were detected in C genome species, with an additional five haplotypes detected in B. rapa L. (A genome n = 10) and another in B. nigra L. (B genome, n = 8). The low chloroplast diversity of B. oleracea is not suggestive of multiple domestication events. The predominant B. oleracea haplotype was also common in B. incana Ten. and present in low frequencies in B. villosa, B. macrocarpa Guss, B. rupestris Raf. and B. cretica. The chloroplast SSRs reveal a wealth of diversity within wild Brassica species that will facilitate further evolutionary and phylogeographic studies of this important crop genus.