Efficient simulation of stochastic chemical kinetics with the Stochastic Bulirsch-Stoer extrapolation method

Background Biochemical systems with relatively low numbers of components must be simulated stochastically in order to capture their inherent noise. Although there has recently been considerable work on discrete stochastic solvers, there is still a need for numerical methods that are both fast and accurate. The Bulirsch-Stoer method is an established method for solving ordinary differential equations that possesses both of these qualities. Results In this paper, we present the Stochastic Bulirsch-Stoer method, a new numerical method for simulating discrete chemical reaction systems, inspired by its deterministic counterpart. It is able to achieve an excellent efficiency due to the fact that it is based on an approach with high deterministic order, allowing for larger stepsizes and leading to fast simulations. We compare it to the Euler τ-leap, as well as two more recent τ-leap methods, on a number of example problems, and find that as well as being very accurate, our method is the most robust, in terms of efficiency, of all the methods considered in this paper. The problems it is most suited for are those with increased populations that would be too slow to simulate using Gillespie’s stochastic simulation algorithm. For such problems, it is likely to achieve higher weak order in the moments. Conclusions The Stochastic Bulirsch-Stoer method is a novel stochastic solver that can be used for fast and accurate simulations. Crucially, compared to other similar methods, it better retains its high accuracy when the timesteps are increased. Thus the Stochastic Bulirsch-Stoer method is both computationally efficient and robust. These are key properties for any stochastic numerical method, as they must typically run many thousands of simulations.

Reconstructing 3D x-ray CT images of polymer gel dosimeters using the zero-scan method

In this study x-ray CT has been used to produce a 3D image of an irradiated PAGAT gel sample, with noise-reduction achieved using the ‘zero-scan’ method. The gel was repeatedly CT scanned and a linear fit to the varying Hounsfield unit of each pixel in the 3D volume was evaluated across the repeated scans, allowing a zero-scan extrapolation of the image to be obtained. To minimise heating of the CT scanner’s x-ray tube, this study used a large slice thickness (1 cm), to provide image slices across the irradiated region of the gel, and a relatively small number of CT scans (63), to extrapolate the zero-scan image. The resulting set of transverse images shows reduced noise compared to images from the initial CT scan of the gel, without being degraded by the additional radiation dose delivered to the gel during the repeated scanning. The full, 3D image of the gel has a low spatial resolution in the longitudinal direction, due to the selected scan parameters. Nonetheless, important features of the dose distribution are apparent in the 3D x-ray CT scan of the gel. The results of this study demonstrate that the zero-scan extrapolation method can be applied to the reconstruction of multiple x-ray CT slices, to provide useful 2D and 3D images of irradiated dosimetry gels.

A data mining driven risk profiling method for road asset management

Road surface skid resistance has been shown to have a strong relationship to road crash risk, however, applying the current method of using investigatory levels to identify crash prone roads is problematic as they may fail in identifying risky roads outside of the norm. The proposed method analyses a complex and formerly impenetrable volume of data from roads and crashes using data mining. This method rapidly identifies roads with elevated crash-rate, potentially due to skid resistance deficit, for investigation. A hypothetical skid resistance/crash risk curve is developed for each road segment, driven by the model deployed in a novel regression tree extrapolation method. The method potentially solves the problem of missing skid resistance values which occurs during network-wide crash analysis, and allows risk assessment of the major proportion of roads without skid resistance values.

Squared Extrapolation Methods (SQUAREM): A New Class of Simple and Efficient Numerical Schemes for Accelerating the Convergence of the EM Algorithm

We derive a new class of iterative schemes for accelerating the convergence of the EM algorithm, by exploiting the connection between fixed point iterations and extrapolation methods. First, we present a general formulation of one-step iterative schemes, which are obtained by cycling with the extrapolation methods. We, then square the one-step schemes to obtain the new class of methods, which we call SQUAREM. Squaring a one-step iterative scheme is simply applying it twice within each cycle of the extrapolation method. Here we focus on the first order or rank-one extrapolation methods for two reasons, (1) simplicity, and (2) computational efficiency. In particular, we study two first order extrapolation methods, the reduced rank extrapolation (RRE1) and minimal polynomial extrapolation (MPE1). The convergence of the new schemes, both one-step and squared, is non-monotonic with respect to the residual norm. The first order one-step and SQUAREM schemes are linearly convergent, like the EM algorithm but they have a faster rate of convergence. We demonstrate, through five different examples, the effectiveness of the first order SQUAREM schemes, SqRRE1 and SqMPE1, in accelerating the EM algorithm. The SQUAREM schemes are also shown to be vastly superior to their one-step counterparts, RRE1 and MPE1, in terms of computational efficiency. The proposed extrapolation schemes can fail due to the numerical problems of stagnation and near breakdown. We have developed a new hybrid iterative scheme that combines the RRE1 and MPE1 schemes in such a manner that it overcomes both stagnation and near breakdown. The squared first order hybrid scheme, SqHyb1, emerges as the iterative scheme of choice based on our numerical experiments. It combines the fast convergence of the SqMPE1, while avoiding near breakdowns, with the stability of SqRRE1, while avoiding stagnations. The SQUAREM methods can be incorporated very easily into an existing EM algorithm. They only require the basic EM step for their implementation and do not require any other auxiliary quantities such as the complete data log likelihood, and its gradient or hessian. They are an attractive option in problems with a very large number of parameters, and in problems where the statistical model is complex, the EM algorithm is slow and each EM step is computationally demanding.

Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Modelling water droplet movement on a leaf surface

The central aim for the research undertaken in this PhD thesis is the development of a model for simulating water droplet movement on a leaf surface and to compare the model behavior with experimental observations. A series of five papers has been presented to explain systematically the way in which this droplet modelling work has been realised. Knowing the path of the droplet on the leaf surface is important for understanding how a droplet of water, pesticide, or nutrient will be absorbed through the leaf surface. An important aspect of the research is the generation of a leaf surface representation that acts as the foundation of the droplet model. Initially a laser scanner is used to capture the surface characteristics for two types of leaves in the form of a large scattered data set. After the identification of the leaf surface boundary, a set of internal points is chosen over which a triangulation of the surface is constructed. We present a novel hybrid approach for leaf surface fitting on this triangulation that combines Clough-Tocher (CT) and radial basis function (RBF) methods to achieve a surface with a continuously turning normal. The accuracy of the hybrid technique is assessed using numerical experimentation. The hybrid CT-RBF method is shown to give good representations of Frangipani and Anthurium leaves. Such leaf models facilitate an understanding of plant development and permit the modelling of the interaction of plants with their environment. The motion of a droplet traversing this virtual leaf surface is affected by various forces including gravity, friction and resistance between the surface and the droplet. The innovation of our model is the use of thin-film theory in the context of droplet movement to determine the thickness of the droplet as it moves on the surface. Experimental verification shows that the droplet model captures reality quite well and produces realistic droplet motion on the leaf surface. Most importantly, we observed that the simulated droplet motion follows the contours of the surface and spreads as a thin film. In the future, the model may be applied to determine the path of a droplet of pesticide along a leaf surface before it falls from or comes to a standstill on the surface. It will also be used to study the paths of many droplets of water or pesticide moving and colliding on the surface.

Quantification of ebullitive and diffusive methane release to atmosphere from a water storage

Accurately quantifying total freshwater storage methane release to atmosphere requires the spatial–temporal measurement of both diffusive and ebullitive emissions. Existing floating chamber techniques provide localised assessment of methane flux, however, significant errors can arise when weighting and extrapolation to the entire storage, particularly when ebullition is significant. An improved technique has been developed that compliments traditional chamber based experiments to quantify the storage-scale release of methane gas to atmosphere through ebullition using the measurements from an Optical Methane Detector (OMD) and a robotic boat. This provides a conservative estimate of the methane emission rate from ebullition along with the bubble volume distribution. It also georeferences the area of ebullition activity across entire storages at short temporal scales. An assessment on Little Nerang Dam in Queensland, Australia, demonstrated whole storage methane release significantly differed spatially and throughout the day. Total methane emission estimates showed a potential 32-fold variation in whole-of-dam rates depending on the measurement and extrapolation method and time of day used. The combined chamber and OMD technique showed that 1.8–7.0% of the surface area of Little Nerang Dam is accounting for up to 97% of total methane release to atmosphere throughout the day. Additionally, over 95% of detectable ebullition occurred in depths less than 12 m during the day and 6 m at night. This difference in spatial and temporal methane release rate distribution highlights the need to monitor significant regions of, if not the entire, water storage in order to provide an accurate estimate of ebullition rates and their contribution to annual methane emissions.

Studies in the hydrolysis of metal ions -Part I.Copper

The hydrolysis of cupric ion has been studied at various ionic strengths (0·01, 0·05, 0·1 and 0·5 M). The results are analyzed employing 'core + links' theory, log-log plot, normalization plot, and extrapolation method for obtaining the pure mononuclear curve. The stability constants of Cu2(OH)2++, Cu3(OH)4++, Cu(OH)+ and Cu(OH)2 have been reported.

A general methodology for calculating mixed mode stress intensity factors and fracture toughness of solder joints with interfacial cracks

Solder joints in electronic packages undergo thermo-mechanical cycling, resulting in nucleation of micro-cracks, especially at the solder/bond-pad interface, which may lead to fracture of the joints. The fracture toughness of a solder joint depends on material properties, process conditions and service history, as well as strain rate and mode-mixity. This paper reports on a methodology for determining the mixed-mode fracture toughness of solder joints with an interfacial starter-crack, using a modified compact mixed mode (CMM) specimen containing an adhesive joint. Expressions for stress intensity factor (K) and strain energy release rate (G) are developed, using a combination of experiments and finite element (FE) analysis. In this methodology, crack length dependent geometry factors to convert for the modified CMM sample are first obtained via the crack-tip opening displacement (CTOD)-based linear extrapolation method to calculate the under far-field mode I and II conditions (f(1a) and f(2a)), (ii) generation of a master-plot to determine a(c), and (iii) computation of K and G to analyze the fracture behavior of joints. The developed methodology was verified using J-integral calculations, and was also used to calculate experimental fracture toughness values of a few lead-free solder-Cu joints. (C) 2014 Elsevier Ltd. All rights reserved.

Contrasting high-resolution characteristic shock-capturing methods in aeroacoustic test problems

Two shock-capturing methods are considered. One is based on a standard conservative Roe scheme with van Leer's MUSCL variable extrapolation method applied to characteristic variables and a Runge-Kutta time stepping scheme. The other is based on the novel CABARET space-time scheme, which uses two sets of staggered variables, one for the conservation step and the other for characteristic splitting into local Riemann invariants. The methods are compared in a range of 2-D inviscid compressible flow test cases. Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Microstructure and strain analysis of GaN epitaxial films using in-plane grazing incidence x-ray diffraction

This paper investigates the major structural parameters, such as crystal quality and strain state of (001)-oriented GaN thin films grown on sapphire substrates by metalorganic chemical vapour deposition, using an in-plane grazing incidence x-ray diffraction technique. The results are analysed and compared with a complementary out-of-plane x-ray diffraction technique. The twist of the GaN mosaic structure is determined through the direct grazing incidence measurement of (100) reflection which agrees well with the result obtained by extrapolation method. The method for directly determining the in-plane lattice parameters of the GaN layers is also presented. Combined with the biaxial strain model, it derives the lattice parameters corresponding to fully relaxed GaN films. The GaN epilayers show an increasing residual compressive stress with increasing layer thickness when the two dimensional growth stage is established, reaching to a maximum level of -0.89 GPa.

Study on binary system comprising a non-mesogen and a mesogen

Mixed liquid crystal formation has been studied in a new binary system comprising paranitroazobenzene derivatives, in which one component was a mesogen and the other was a non-mesogen. The mixtures were found to exhibit a monotropic nematic phase which was converted to an enantiotropic phase in specific ranges of temperature and concentration. The latent liquid crystal-isotropic transition temperature (LTP) of the non-mesogen was obtained by using the extrapolation method of the transition temperature-composition curve and the equal-G analysis method. The LTPs of the non-mesogen obtained by the above two methods showed good agreement with each other. The low-temperature transition of the mixtures detected by DSC was attributed to a change of the crystallite size.

Cure processing modeling and cure cycle simulation of epoxy-terminated poly(phenylene ether ketone) .1. DSC characterization of curing reaction

The curing reaction process of epoxy-terminated poly(phenylene ether ketone) (E-PEK) with 4,4'-diaminodiphenyl sulfone (DDS) and hexahydrophthalic acid anhydride (Nadic) as curing agents was investigated using isothermal differential scanning calorimetry (IDSC) and nonisothermal differential scanning calorimetry (DDSC) techniques. It was found that the curing reactions of E-PEK/DDS and E-PEK/Nadic are nth-order reactions but not autoaccelerating. The experimental results revealed that the curing reaction kinetics parameters measured from IDSC and DDSC are not equivalent. This means that, in the curing reaction kinetics model for our E-PEK system, both isothermal and nonisothermal reaction kinetics parameters are needed to describe isothermal and nonisothermal curing processes, The isothermal and nonisothermal curing processes were successfully simulated using this model. A new extrapolation method was suggested. On the basis of this method the maximum extent of the curing reaction (A(ult)) that is able to reach a certain temperature can be predicted. The A(ult) for the E-PEK system estimated by the new method agrees well with the results obtained from another procedure reported in the literature. (C) 1997 John Wiley & Sons, Inc.