From h to p efficiently: optimal implementation strategies for explicit time-dependent problems using the spectral/hp element method

We investigate the relative performance of a second-order Adams–Bashforth scheme and second-order and fourth-order Runge–Kutta schemes when time stepping a 2D linear advection problem discretised using a spectral/hp element technique for a range of different mesh sizes and polynomial orders. Numerical experiments explore the effects of short (two wavelengths) and long (32 wavelengths) time integration for sets of uniform and non-uniform meshes. The choice of time-integration scheme and discretisation together fixes a CFL limit that imposes a restriction on the maximum time step, which can be taken to ensure numerical stability. The number of steps, together with the order of the scheme, affects not only the runtime but also the accuracy of the solution. Through numerical experiments, we systematically highlight the relative effects of spatial resolution and choice of time integration on performance and provide general guidelines on how best to achieve the minimal execution time in order to obtain a prescribed solution accuracy. The significant role played by higher polynomial orders in reducing CPU time while preserving accuracy becomes more evident, especially for uniform meshes, compared with what has been typically considered when studying this type of problem.© 2014. The Authors. International Journal for Numerical Methods in Fluids published by John Wiley & Sons, Ltd

Optimising the performance of the spectral/hp element method with collective linear algebra operations

This is the final version of the article. Available from Elsevier via the DOI in this record.As computing hardware evolves, increasing core counts mean that memory bandwidth is becoming the deciding factor in attaining peak performance of numerical methods. High-order finite element methods, such as those implemented in the spectral/hp framework Nektar++, are particularly well-suited to this environment. Unlike low-order methods that typically utilise sparse storage, matrices representing high-order operators have greater density and richer structure. In this paper, we show how these qualities can be exploited to increase runtime performance on nodes that comprise a typical high-performance computing system, by amalgamating the action of key operators on multiple elements into a single, memory-efficient block. We investigate different strategies for achieving optimal performance across a range of polynomial orders and element types. As these strategies all depend on external factors such as BLAS implementation and the geometry of interest, we present a technique for automatically selecting the most efficient strategy at runtime.We thank D. Ekelschot and M. Turner for their assistance in generating the mesh and parameters for the simulation of Section 6. We also thank F. Witherden for initial discussions motivating this study. This work was funded in part by support from the libHPC II EPSRC project under grant EP/K038788/1. DM additionally acknowledges support under the Laminar Flow Control Centre funded by Airbus/EADS and EPSRC under grant EP/I037946. SJS acknowledges Royal Academy of Engineering support under their research chair scheme. We thank the Imperial College High Performance Computing Service for computing time used to calculate the results seen in Section 6. We additionally acknowledge access to ARCHER with support from the UK Turbulence Consortium under EPSRC grant EP/L000261/1

Reducing complexity and unidentifiability when modelling human atrial cells

Mathematical models of a cellular action potential in cardiac modelling have become increasingly complex, particularly in gating kinetics which control the opening and closing of individual ion channel currents. As cardiac models advance towards use in personalised medicine to inform clinical decision-making, it is critical to understand the uncertainty hidden in parameter estimates from their calibration to experimental data. This study applies approximate Bayesian computation to re-calibrate the gating kinetics of four ion channels in two existing human atrial cell models to their original datasets, providing a measure of uncertainty and indication of potential issues with selecting a single unique value given the available experimental data. Two approaches are investigated to reduce the uncertainty present: re-calibrating the models to a more complete dataset and using a less complex formulation with fewer parameters to constrain. The re-calibrated models are inserted back into the full cell model to study the overall effect on the action potential. The use of more complete datasets does not eliminate uncertainty present in parameter estimates. The less complex model, particularly for the fast sodium current, gave a better fit to experimental data alongside lower parameter uncertainty and improved computational speed

Left atrial enhancement correlates with myocardial conduction velocity in patients with persistent atrial fibrillation

Background: Conduction velocity (CV) heterogeneity and myocardial fibrosis both promote re-entry, but the relationship between fibrosis as determined by left atrial (LA) late-gadolinium enhanced cardiac magnetic resonance imaging (LGE-CMRI) and CV remains uncertain. Objective: Although average CV has been shown to correlate with regional LGE-CMRI in patients with persistent AF, we test the hypothesis that a localized relationship exists to underpin LGE-CMRI as a minimally invasive tool to map myocardial conduction properties for risk stratification and treatment guidance. Method: 3D LA electroanatomic maps during LA pacing were acquired from eight patients with persistent AF following electrical cardioversion. Local CVs were computed using triads of concurrently acquired electrograms and were co-registered to allow correlation with LA wall intensities obtained from LGE-CMRI, quantified using normalized intensity (NI) and image intensity ratio (IIR). Association was evaluated using multilevel linear regression. Results: An association between CV and LGE-CMRI intensity was observed at scales comparable to the size of a mapping electrode: −0.11 m/s per unit increase in NI (P < 0.001) and −0.96 m/s per unit increase in IIR (P < 0.001). The magnitude of this change decreased with larger measurement area. Reproducibility of the association was observed with NI, but not with IIR. Conclusion: At clinically relevant spatial scales, comparable to area of a mapping catheter electrode, LGE-CMRI correlates with CV. Measurement scale is important in accurately quantifying the association of CV and LGE-CMRI intensity. Importantly, NI, but not IIR, accounts for changes in the dynamic range of CMRI and enables quantitative reproducibility of the association

High-order propagation of jet noise on a tetrahedral mesh using large eddy simulation sources

Jet noise is an important area of research in commercial aviation due to its high contribution to the overall noise generated by an aircraft. Conventionally, CFD combined with surface integral methods is used to study jet noise because of its low cost. However, it is not always trivial to define integration surfaces around complex geometries. This study employs a different two-step approach that can handle complex geometries. It combines a large-eddy simulation (LES) to obtain the acoustic sources from the flow field, and an acoustic perturbation equations (APE) solver to propagate the sound to the far field. The LES is performed with an industrial 2nd-order finite volume solver. The APE code is a high-order discontinuous Galerkin (DG) spectral/hp solver of the Nektar+ + framework. The APE solver is validated on a canonical test case. A study on different polynomial expansion orders and meshes is further performed to estimate the mesh size for noise propagation in the high-order spectral/hp DG context. Finally, a three-dimensional jet noise case (Re = 10, 000 and Mach = 0.9) is simulated using unstructured tetrahedral mesh for the APE solver and improved noise results for high frequencies are obtained. The results demonstrate that the present approach is capable of predicting noise in complex geometry scenarios, such as installed jets under the aircraft wings

Measuring attitude towards Buddhism and Sikhism : internal consistency reliability for two new instruments

This paper describes and discusses the development and empirical properties of two new 24-item scales – one measuring attitude toward Buddhism and the other measuring attitude toward Sikhism. The scale is designed to facilitate inter-faith comparisons within the psychology of religion alongside the well-established Francis Scale of Attitude toward Christianity. Data were obtained from a multi-religious sample of 369 school pupils aged between 13 and 15 in London. Application of the two scales demonstrated that adolescents had a more positive attitude to Buddhism than Sikhism. The findings confirm the reliability of the scales and commend them for further use

A spectral element reduced basis method in parametric CFD

We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14,259 degrees of freedom. The steady-state snapshot solutions define a reduced order space, which allows to accurately evaluate the steady-state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation (Karniadakis and Sherwin, Spectral/hp element methods for computational fluid dynamics, 2nd edn. Oxford University Press, Oxford, 2005) in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios

Caveats of chronic exogenous corticosterone treatments in adolescent rats and effects on anxiety-like and depressive behavior and hypothalamic-pituitary-adrenal (HPA) axis function

<p>Abstract</p> <p>Background</p> <p>Administration of exogenous corticosterone is an effective preclinical model of depression, but its use has involved primarily adult rodents. Using two different procedures of administration drawn from the literature, we explored the possibility of exogenous corticosterone models in adolescence, a time of heightened risk for mood disorders in humans.</p> <p>Methods</p> <p>In experiment 1, rats were injected with 40 mg/kg corticosterone or vehicle from postnatal days 30 to 45 and compared with no injection controls on behavior in the elevated plus maze (EPM) and the forced swim test (FST). Experiment 2 consisted of three treatments administered to rats from postnatal days 30 to 45 or as adults (days 70 to 85): either corticosterone (400 μg/ml) administered in the drinking water along with 2.5% ethanol, 2.5% ethanol or water only. In addition to testing on EPM, blood samples after the FST were obtained to measure plasma corticosterone. Analysis of variance (ANOVA) and alpha level of <it>P </it>< 0.05 were used to determine statistical significance.</p> <p>Results</p> <p>In experiment 1, corticosterone treatment of adolescent rats increased anxiety in the EPM and decreased immobility in the FST compared to no injection control rats. However, vehicle injected rats were similar to corticosterone injected rats, suggesting that adolescent rats may be highly vulnerable to stress of injection. In experiment 2, the intake of treated water, and thus doses delivered, differed for adolescents and adults, but there were no effects of treatment on behavior in the EPM or FST. Rats that had ingested corticosterone had reduced corticosterone release after the FST. Ethanol vehicle also affected corticosterone release compared to those ingesting water only, but differently for adolescents than for adults.</p> <p>Conclusions</p> <p>The results indicate that several challenges must be overcome before the exogenous corticosterone model can be used effectively in adolescents.</p