Numerical methods for strong solutions of stochastic differential equations : an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations

The pioneering work of Runge and Kutta a hundred years ago has ultimately led to suites of sophisticated numerical methods suitable for solving complex systems of deterministic ordinary differential equations. However, in many modelling situations, the appropriate representation is a stochastic differential equation and here numerical methods are much less sophisticated. In this paper a very general class of stochastic Runge-Kutta methods is presented and much more efficient classes of explicit methods than previous extant methods are constructed. In particular, a method of strong order 2 with a deterministic component based on the classical Runge-Kutta method is constructed and some numerical results are presented to demonstrate the efficacy of this approach.

High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula

In recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the difficulty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage [High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics 22 (1996) 81-101] were able to construct strong local order 1.5 stochastic Runge-Kutta methods for certain cases, it is known that all extant stochastic Runge-Kutta methods suffer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler-Maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques. An additional difficulty in solving SODEs arises even in the Linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper first the work of Magnus [On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (1954) 649-673] (applied to deterministic non-commutative Linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed - hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number (d) of Wiener processes it is shown that strong local order I Runge-Kutta methods with d + 1 stages can be constructed by evaluated a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the efficacy of these approaches. (C) 1999 Elsevier Science B.V. All rights reserved.

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if an s-stage SRK contains Stratonovich integrals up to order p then the strong order of the SRK cannot exceed min{(p + 1)/2, (s - 1)/2), p greater than or equal to 2, s greater than or equal to 3 or 1 if p = 1.

A strong quantitative trait locus for wing length on chromosome 2 in a wild population of great reed warblers

Wing length is a key character for essential behaviours related to bird flight such as migration and foraging. In the present study, we initiate the search for the genes underlying wing length in birds by studying a long-distance migrant, the great reed warbler (Acrocephalus arundinaceus). In this species wing length is an evolutionary interesting trait with pronounced latitudinal gradient and sex-specific selection regimes in local populations. We performed a quantitative trait locus (QTL) scan for wing length in great reed warblers using phenotypic, genotypic, pedigree and linkage map data from our long-term study population in Sweden. We applied the linkage analysis mapping method implemented in GRIDQTL (a new web-based software) and detected a genome-wide significant QTL for wing length on chromosome 2, to our knowledge, the first detected QTL in wild birds. The QTL extended over 25 cM and accounted for a substantial part (37%) of the phenotypic variance of the trait. A genome scan for tarsus length (a bodysize-related trait) did not show any signal, implying that the wing-length QTL on chromosome 2 was not associated with body size. Our results provide a first important step into understanding the genetic architecture of avian wing length, and give opportunities to study the evolutionary dynamics of wing length at the locus level. This journal is© 2010 The Royal Society.

Strong and bioactive tri-calcium phosphate scaffolds with tube-like macropores

Calcium phosphate ceramic scaffolds have been widely investigated for bone tissue engineering due to their excellent biocompatibility and biodegradation. Unfortunately, they have the shortcoming of low mechanical properties. In order to provide strong, bioactive, and biodegradable scaffolds, a new approach of infiltrating the macro-tube ABS (acrylontrile butadiene styrene) templates with a hydroxyapatite/bioactive glass mixed slurry was developed to fabricate porous Si-doped TCP (tri-calcium phosphate) scaffolds. The porous Si-doped TCP ceramics with a high porosity (~65%) and with interconnected macrotubes (~0.8mm in diameter) and micropores (5-100 m) had a high compressive strength (up to 14.68+0.2MPa), which was comparable to that of a trabecular bone and was much higher than those of pure TCP scaffolds. Additional cell attachment study and MTT cytotoxicity assay proved the bioactivity and biocompatibility of the new scaffolds. Thus a potential bioceramic material and a new approach to make the potential scaffolds were developed for bone tissue engineering.

Strong cryptography from weak secrets

Distributed-password public-key cryptography (DPwPKC) allows the members of a group of people, each one holding a small secret password only, to help a leader to perform the private operation, associated to a public-key cryptosystem. Abdalla et al. recently defined this tool [1], with a practical construction. Unfortunately, the latter applied to the ElGamal decryption only, and relied on the DDH assumption, excluding any recent pairing-based cryptosystems. In this paper, we extend their techniques to support, and exploit, pairing-based properties: we take advantage of pairing-friendly groups to obtain efficient (simulation-sound) zero-knowledge proofs, whose security relies on the Decisional Linear assumption. As a consequence, we provide efficient protocols, secure in the standard model, for ElGamal decryption as in [1], but also for Linear decryption, as well as extraction of several identity-based cryptosystems [6,4]. Furthermore, we strenghten their security model by suppressing the useless testPwd queries in the functionality.

Effect of strong acids on red mud structural and fluoride adsorption properties

The removal of fluoride using red mud has been improved by acidifying red mud with hydrochloric, nitric and sulphuric acid. This investigation shows that the removal of fluoride using red mud is significantly improved if red mud is initially acidified. The acidification of red mud causes sodalite and cancrinite phases to dissociate, confirmed by the release of sodium and aluminium into solution as well as the disappearance of sodalite bands and peaks in infrared and X-ray diffraction data. The dissolution of these mineral phases increases the amount of available iron and aluminium oxide/hydroxide sites that are accessible for the adsorption of fluoride. The removal of fluoride is dependent on the charge of iron and aluminium oxide/hydroxides on the surface of red mud. Acidifying red mud with hydrochloric, nitric and sulphuric acid resulted in surface sites of the form ≡ SOH2+ and ≡ SOH. Optimum removal is obtained when the majority of surface sites are in the form ≡ SOH2+ as the substitution of a fluoride ion doesn’t cause a significant increase in pH. This investigation shows the importance of having a low and consistent pH for the removal of fluoride from aqueous solutions using red mud.

Multiplexed tandem polymerase chain reaction identifies strong expression of oestrogen receptor and Her-2 from single, formalin-fixed, paraffin-embedded breast cancer sections

Aim To establish the suitability of multiplex tandem polymerase chain reaction (MT-PCR) for rapid identification of oestrogen receptor (ER) and Her-2 status using a single, formalin-fixed, paraffin-embedded (FFPE) breast tumour section. Methods Tissue sections from 29 breast tumours were analysed by immunohistochemistry (IHC) and fluorescence in situ hybridisation (FISH). RNA extracted from 10μm FFPE breast tumour sections from 24 of 29 tumours (14 ER positive and 5 Her-2 positive) was analysed by MT-PCR. After establishing a correlation between IHC and/or FISH and MT-PCR results, the ER/Her-2 status of a further 32 randomly selected, archival breast tumour specimens was established by MT-PCR in a blinded fashion, and compared to IHC/FISH results. Results MT-PCR levels of ER and Her-2 showed good concordance with IHC and FISH results. Furthermore, among the ER positive tumours, MT-PCR provided a quantitative score with a high dynamic range. Threshold values obtained from this data set applied to 32 archival tumour specimens showed that tumours strongly positive for ER and/or Her-2 expression were easily identified by MT-PCR. Conclusion MT-PCR can provide rapid, sensitive and cost-effective analysis of FFPE material and may prove useful as triage to identify patients suited to endocrine or trastuzumab (Herceptin) treatment.

How thick SiO2 cap layer is needed to achieve strong visible photoluminescence from SiO2-buffered SiNx films?

The effect of a SiO2 nanolayer and annealing temperature on the UV/visible room-temperature photoluminescence (PL) from SiNx films synthesized by rf magnetron sputtering is studied. The PL intensity can be maximized when the SiO2 layer is 510 nm thick at 800 °C annealing temperature and only 2 nm at 1000 °C. A compositionstructureproperty analysis reveals that the PL intensity is directly related to both the surface chemical states and the content of the SiO and SiN bonds in the SiNx films. These results are relevant for the development of advanced optoelectronic and photonic emitters and sensors. © 2010 Elsevier B.V. All rights reserved.

Equilibrium and column studies of iron exchange with strong acid cation resin

The exchange of iron species from iron (III) chloride solutions with a strong acid cation resin has been investigated in relation to a variety of water and wastewater applications. A detailed equilibrium isotherm analysis was conducted wherein models such as Langmuir Vageler, Competitive Langmuir, Freundlich, Temkin, Dubinin Astakhov, Sips and Brouers-Sotolongo were applied to the experimental data. An important conclusion was that both the bottle-point method and solution normality used to generate the ion exchange equilibrium information influenced which sorption model fitted the isotherm profiles optimally. Invariably, the calculated value for the maximum loading of iron on strong acid cation resin was substantially higher than the value of 47.1 g/kg of resin which would occur if one Fe3+ ion exchanged for three “H+” sites on the resin surface. Consequently, it was suggested that above pH 1, various iron complexes sorbed to the resin in a manner which required less than 3 sites per iron moiety. Column trials suggested that the iron loading was 86.6 g/kg of resin when 1342 mg/L Fe (III) ions in water were flowed at 31.7 bed volumes per hour. Regeneration with 5 to 10 % HCl solutions reclaimed approximately 90 % of exchange sites.

An examination of isotherm generation : impact of bottle-point method upon potassium ion exchange with strong acid cation resin

This paper relates to the importance of impact of the chosen bottle-point method when conducting ion exchange equilibria experiments. As an illustration, potassium ion exchange with strong acid cation resin was investigated due to its relevance to the treatment of various industrial effluents and groundwater. The “constant mass” bottle-point method was shown to be problematic in that depending upon the resin mass used the equilibrium isotherm profiles were different. Indeed, application of common equilibrium isotherm models revealed that the optimal fit could be with either the Freundlich or Temkin equations, depending upon the conditions employed. It could be inferred that the resin surface was heterogeneous in character, but precise conclusions regarding the variation in the heat of sorption were not possible. Estimation of the maximum potassium loading was also inconsistent when employing the “constant mass” method. The “constant concentration” bottle-point method illustrated that the Freundlich model was a good representation of the exchange process. The isotherms recorded were relatively consistent when compared to the “constant mass” approach. Unification of all the equilibrium isotherm data acquired was achieved by use of the Langmuir Vageler expression. The maximum loading of potassium ions was predicted to be at least 116.5 g/kg resin.

Predictive motion planning for AUVs subject to strong time-varying currents and forecasting uncertainties

This paper presents a novel path planning method for minimizing the energy consumption of an autonomous underwater vehicle subjected to time varying ocean disturbances and forecast model uncertainty. The algorithm determines 4-Dimensional path candidates using Nonlinear Robust Model Predictive Control (NRMPC) and solutions optimised using A*-like algorithms. Vehicle performance limits are incorporated into the algorithm with disturbances represented as spatial and temporally varying ocean currents with a bounded uncertainty in their predictions. The proposed algorithm is demonstrated through simulations using a 4-Dimensional, spatially distributed time-series predictive ocean current model. Results show the combined NRMPC and A* approach is capable of generating energy-efficient paths which are resistant to both dynamic disturbances and ocean model uncertainty.