Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal--Oriented A Posteriori Error Estimation

In this article we consider the application of the generalization of the symmetric version of the interior penalty discontinuous Galerkin finite element method to the numerical approximation of the compressible Navier--Stokes equations. In particular, we consider the a posteriori error analysis and adaptive mesh design for the underlying discretization method. Indeed, by employing a duality argument (weighted) Type I a posteriori bounds are derived for the estimation of the error measured in terms of general target functionals of the solution; these error estimates involve the product of the finite element residuals with local weighting terms involving the solution of a certain dual problem that must be numerically approximated. This general approach leads to the design of economical finite element meshes specifically tailored to the computation of the target functional of interest, as well as providing efficient error estimation. Numerical experiments demonstrating the performance of the proposed approach will be presented.

Proof methods for structured corecursive programs

Corecursive programs produce values of greatest fixpoint types, in contrast to recursive programs, which consume values of least fixpoint types. There are a number of widely used methods for proving properties of corecursive programs, including fixpoint induction, the take lemma, and coinduction. However, these methods are all rather low level, in that they do not exploit the common structure that is often present in corecursive definitions. We argue for a more structured approach to proving properties of corecursive programs. In particular, we show that by writing corecursive programs using a simple operator that encapsulates a common pattern of corecursive definition, we can then use high-level algebraic properties of this operator to conduct proofs in a purely calculational style that avoids the use of inductive or coinductive methods.

Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems on Anisotropically Refined Meshes

In this paper we consider the a posteriori and a priori error analysis of discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution. Based on our a posteriori error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement. The theoretical results are illustrated by a series of numerical experiments.

Discontinuous Galerkin Methods on hp-Anisotropic Meshes II: A Posteriori Error Analysis and Adaptivity

We consider the a posteriori error analysis and hp-adaptation strategies for hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes with anisotropically enriched elemental polynomial degrees. In particular, we exploit duality based hp-error estimates for linear target functionals of the solution and design and implement the corresponding adaptive algorithms to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement and isotropic and anisotropic polynomial degree enrichment. The superiority of the proposed algorithm in comparison with standard hp-isotropic mesh refinement algorithms and an h-anisotropic/p-isotropic adaptive procedure is illustrated by a series of numerical experiments.

Calculation of sample size for stroke trials assessing functional outcome: comparison of binary and ordinal approaches

Background Many acute stroke trials have given neutral results. Sub-optimal statistical analyses may be failing to detect efficacy. Methods which take account of the ordinal nature of functional outcome data are more efficient. We compare sample size calculations for dichotomous and ordinal outcomes for use in stroke trials. Methods Data from stroke trials studying the effects of interventions known to positively or negatively alter functional outcome – Rankin Scale and Barthel Index – were assessed. Sample size was calculated using comparisons of proportions, means, medians (according to Payne), and ordinal data (according to Whitehead). The sample sizes gained from each method were compared using Friedman 2 way ANOVA. Results Fifty-five comparisons (54 173 patients) of active vs. control treatment were assessed. Estimated sample sizes differed significantly depending on the method of calculation (Po00001). The ordering of the methods showed that the ordinal method of Whitehead and comparison of means produced significantly lower sample sizes than the other methods. The ordinal data method on average reduced sample size by 28% (inter-quartile range 14–53%) compared with the comparison of proportions; however, a 22% increase in sample size was seen with the ordinal method for trials assessing thrombolysis. The comparison of medians method of Payne gave the largest sample sizes. Conclusions Choosing an ordinal rather than binary method of analysis allows most trials to be, on average, smaller by approximately 28% for a given statistical power. Smaller trial sample sizes may help by reducing time to completion, complexity, and financial expense. However, ordinal methods may not be optimal for interventions which both improve functional outcome

Can we improve the statistical analysis of stroke trials? Statistical re-analysis of functional outcomes in stroke trials

Background: Most large acute stroke trials have been neutral. Functional outcome is usually analysed using a yes or no answer, e.g. death or dependency vs. independence. We assessed which statistical approaches are most efficient in analysing outcomes from stroke trials. Methods: Individual patient data from acute, rehabilitation and stroke unit trials studying the effects of interventions which alter functional outcome were assessed. Outcomes included modified Rankin Scale, Barthel Index, and ‘3 questions’. Data were analysed using a variety of approaches which compare two treatment groups. The results for each statistical test for each trial were then compared. Results: Data from 55 datasets were obtained (47 trials, 54,173 patients). The test results differed substantially so that approaches which use the ordered nature of functional outcome data (ordinal logistic regression, t-test, robust ranks test, bootstrapping the difference in mean rank) were more efficient statistically than those which collapse the data into 2 groups (chi square) (ANOVA p<0.001). The findings were consistent across different types and sizes of trial and for the different measures of functional outcome. Conclusions: When analysing functional outcome from stroke trials, statistical tests which use the original ordered data are more efficient and more likely to yield reliable results. Suitable approaches included ordinal logistic regression, t-test, and robust ranks test.

Granulocyte-colony stimulating factor in experimental stroke and its effects on infarct size and functional outcome: a systematic review

Background Granulocyte-colony stimulating factor (G-CSF) shows promise as a treatment for stroke. This systematic review assesses G-CSF in experimental ischaemic stroke. Methods Relevant studies were identified with searches of Medline, Embase and PubMed. Data were extracted on stroke lesion size, neurological outcome and quality, and analysed using Cochrane Review Manager using random effects models; results are expressed as standardised mean difference (SMD) and odds ratio (OR). Results Data were included from 19 publications incorporating 666 animals. G-CSF reduced lesion size significantly in transient (SMD -1.63, p<0.00001) but not permanent (SMD -1.56, p=0.11) focal models of ischaemia. Lesion size was reduced at all doses and with treatment commenced within 4 hours of transient ischaemia. Neurological deficit (SMD -1.37, p=0.0004) and limb placement (SMD -1.88, p=0.003) improved with G-CSF; however, locomotor activity (>4 weeks post ischaemia) was not (SMD 0.76, p=0.35). Death (OR 0.27, p<0.0001) was reduced with G-CSF. Median study quality was 4 (range 0-7/8); Egger’s test suggested significant publication bias (p=0.001). Conclusions G-CSF significantly reduced lesion size in transient but not permanent models of ischaemic stroke. Motor impairment and death were also reduced. Further studies assessing dose-response, administration time, length of ischaemia and long-term functional recovery are needed.

ArrayMining: a modular web-application for microarray analysis combining ensemble and consensus methods with cross-study normalization

Background: Statistical analysis of DNA microarray data provides a valuable diagnostic tool for the investigation of genetic components of diseases. To take advantage of the multitude of available data sets and analysis methods, it is desirable to combine both different algorithms and data from different studies. Applying ensemble learning, consensus clustering and cross-study normalization methods for this purpose in an almost fully automated process and linking different analysis modules together under a single interface would simplify many microarray analysis tasks. Results: We present ArrayMining.net, a web-application for microarray analysis that provides easy access to a wide choice of feature selection, clustering, prediction, gene set analysis and cross-study normalization methods. In contrast to other microarray-related web-tools, multiple algorithms and data sets for an analysis task can be combined using ensemble feature selection, ensemble prediction, consensus clustering and cross-platform data integration. By interlinking different analysis tools in a modular fashion, new exploratory routes become available, e.g. ensemble sample classification using features obtained from a gene set analysis and data from multiple studies. The analysis is further simplified by automatic parameter selection mechanisms and linkage to web tools and databases for functional annotation and literature mining. Conclusion: ArrayMining.net is a free web-application for microarray analysis combining a broad choice of algorithms based on ensemble and consensus methods, using automatic parameter selection and integration with annotation databases.