Conditioning and preconditioning of the variational data assimilation problem

Numerical weather prediction (NWP) centres use numerical models of the atmospheric flow to forecast future weather states from an estimate of the current state. Variational data assimilation (VAR) is used commonly to determine an optimal state estimate that miminizes the errors between observations of the dynamical system and model predictions of the flow. The rate of convergence of the VAR scheme and the sensitivity of the solution to errors in the data are dependent on the condition number of the Hessian of the variational least-squares objective function. The traditional formulation of VAR is ill-conditioned and hence leads to slow convergence and an inaccurate solution. In practice, operational NWP centres precondition the system via a control variable transform to reduce the condition number of the Hessian. In this paper we investigate the conditioning of VAR for a single, periodic, spatially-distributed state variable. We present theoretical bounds on the condition number of the original and preconditioned Hessians and hence demonstrate the improvement produced by the preconditioning. We also investigate theoretically the effect of observation position and error variance on the preconditioned system and show that the problem becomes more ill-conditioned with increasingly dense and accurate observations. Finally, we confirm the theoretical results in an operational setting by giving experimental results from the Met Office variational system.

Curriculum theory, curriculum policy and the problem of ill-disciplined thinking

This paper examines the implications of policy fracture and arms length governance within the decision making processes currently shaping curriculum design within the English education system. In particular it argues that an unresolved ‘ideological fracture’ at government level has been passed down to school leaders whose response to the dilemma is distorted by the target-driven agenda of arms length agencies. Drawing upon the findings of a large scale on-line survey of history teaching in English secondary schools, this paper illustrates the problems that occur when policy making is divorced from curriculum theory, and in particular from any consideration of the nature of knowledge. Drawing on the social realist theory of knowledge elaborated by Young (2008), we argue that the rapid spread of alternative curricular arrangements, implemented in the absence of an understanding of curriculum theory, undermines the value of disciplined thinking to the detriment of many young people, particularly those in areas of social and economic deprivation.

Simple solution for a complex problem: Proanthocyanidins, galloyl glucoses and ellagitannins fit on a single calibration curve in high performance-gel permeation chromatography

This study was undertaken to explore gel permeation chromatography (GPC) for estimating molecular weights of proanthocyanidin fractions isolated from sainfoin (Onobrychis viciifolia). The results were compared with data obtained by thiolytic degradation of the same fractions. Polystyrene, polyethylene glycol and polymethyl methacrylate standards were not suitable for estimating the molecular weights of underivatized proanthocyanidins. Therefore, a novel HPLC-GPC method was developed based on two serially connected PolarGel-L columns using DMF that contained 5% water, 1% acetic acid and 0.15 M LiBr at 0.7 ml/min and 50 degrees C. This yielded a single calibration curve for galloyl glucoses (trigalloyl glucose, pentagalloyl glucose), ellagitannins (pedunculagin, vescalagin, punicalagin, oenothein B, gemin A), proanthocyanidins (procyanidin B2, cinnamtannin B1), and several other polyphenols (catechin, epicatechin gallate, epicallocatechin gallate, amentoflavone). These GPC predicted molecular weights represented a considerable advance over previously reported HPLC-GPC methods for underivatized proanthocyanidins. (C) 2011 Elsevier B.V. All rights reserved.

The effects of landscape modifications on the long-term persistence of animal populations

Background: The effects of landscape modifications on the long-term persistence of wild animal populations is of crucial importance to wildlife managers and conservation biologists, but obtaining experimental evidence using real landscapes is usually impossible. To circumvent this problem we used individual-based models (IBMs) of interacting animals in experimental modifications of a real Danish landscape. The models incorporate as much as possible of the behaviour and ecology of four species with contrasting life-history characteristics: skylark (Alauda arvensis), vole (Microtus agrestis), a ground beetle (Bembidion lampros) and a linyphiid spider (Erigone atra). This allows us to quantify the population implications of experimental modifications of landscape configuration and composition. Methodology/Principal Findings: Starting with a real agricultural landscape, we progressively reduced landscape complexity by (i) homogenizing habitat patch shapes, (ii) randomizing the locations of the patches, and (iii) randomizing the size of the patches. The first two steps increased landscape fragmentation. We assessed the effects of these manipulations on the long-term persistence of animal populations by measuring equilibrium population sizes and time to recovery after disturbance. Patch rearrangement and the presence of corridors had a large effect on the population dynamics of species whose local success depends on the surrounding terrain. Landscape modifications that reduced population sizes increased recovery times in the short-dispersing species, making small populations vulnerable to increasing disturbance. The species that were most strongly affected by large disturbances fluctuated little in population sizes in years when no perturbations took place. Significance: Traditional approaches to the management and conservation of populations use either classical methods of population analysis, which fail to adequately account for the spatial configurations of landscapes, or landscape ecology, which accounts for landscape structure but has difficulty predicting the dynamics of populations living in them. Here we show how realistic and replicable individual-based models can bridge the gap between non-spatial population theory and non-dynamic landscape ecology. A major strength of the approach is its ability to identify population vulnerabilities not detected by standard population viability analyses.

Path dependence and Britain’s "coal wagon problem"

This article explores the nature and impact of path dependence in British rail coal haulage before 1939. It examines the factors which locked Britain's railways into a system of small coal wagons with highly fragmented ownership, the cost penalties of this system, and the reasons that attempts at modernization were unsuccessful. The analysis highlights the importance of decentralized ownership of a highly durable installed base of complementary infrastructure. Technical and institutional interrelatedness blocked incremental modernization, while the political requirement to compensate private wagon owners for the loss of their wagon stock made wholesale rationalization financially unattractive.

On computational algorithms for pole assignment

A number of computationally reliable direct methods for pole assignment by feedback have recently been developed. These direct procedures do not necessarily produce robust solutions to the problem, however, in the sense that the assigned poles are insensitive to perturbalions in the closed-loop system. This difficulty is illustrated here with results from a recent algorithm presented in this TRANSACTIONS and its causes are examined. A measure of robustness is described, and techniques for testing and improving robustness are indicated.

A technique for delivering individualised formative problems and examples

Active learning plays a strong role in mathematics and statistics, and formative problems are vital for developing key problem-solving skills. To keep students engaged and help them master the fundamentals before challenging themselves further, we have developed a system for delivering problems tailored to a student‟s current level of understanding. Specifically, by adapting simple methodology from clinical trials, a framework for delivering existing problems and other illustrative material has been developed, making use of macros in Excel. The problems are assigned a level of difficulty (a „dose‟), and problems are presented to the student in an order depending on their ability, i.e. based on their performance so far on other problems. We demonstrate and discuss the application of the approach with formative examples developed for a first year course on plane coordinate geometry, and also for problems centred on the topic of chi-square tests.