A variable stepsize implementation for stochastic differential equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge-Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.

Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively. (C) 2000 Elsevier Science B.V. All rights reserved.

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.

General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems

In many modeling situations in which parameter values can only be estimated or are subject to noise, the appropriate mathematical representation is a stochastic ordinary differential equation (SODE). However, unlike the deterministic case in which there are suites of sophisticated numerical methods, numerical methods for SODEs are much less sophisticated. Until a recent paper by K. Burrage and P.M. Burrage (1996), the highest strong order of a stochastic Runge-Kutta method was one. But K. Burrage and P.M. Burrage (1996) showed that by including additional random variable terms representing approximations to the higher order Stratonovich (or Ito) integrals, higher order methods could be constructed. However, this analysis applied only to the one Wiener process case. In this paper, it will be shown that in the multiple Wiener process case all known stochastic Runge-Kutta methods can suffer a severe order reduction if there is non-commutativity between the functions associated with the Wiener processes. Importantly, however, it is also suggested how this order can be repaired if certain commutator operators are included in the Runge-Kutta formulation. (C) 1998 Elsevier Science B.V. and IMACS. 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.

Legal Profession Act 2004 (NSW) : costs agreement between law practice and client - whether binding on non-associated third party payer

The decision in the New South Wales Supreme Court in Boyce v McIntyre [2008] NSWSC 1218 involved determination of a number of issues relating to an assessment of costs under the Legal Profession Act 2004 (NSW). The issue of broad significance was whether a non-associated third party payer must pay the fixed fee that was agreed between the law practice and the client. The court found that the client agreement did not form the basis of assessing costs for the non-associated third party payer.

Costs - UCPR rr 681 and 684

In BHP Coal Pty Ltd v K Orenstein & Koppel AG (No 2) [2009] QSC 64 McMurdo J considered the circumstances in which the ordinary rule under r 681 of the Uniform Civil Procedure Rules 1999 (Qld) (UCPR) that costs should follow the event should be departed from in favour of a party who was unsuccessful overall, but who succeeded on particular questions. When the court is satisfied that a departure from the usual order under r 681 of the UCPR is justified, it appears increasingly willing to exercise the power in r 684(2) to declare what percentage of costs was applicable to a particular issue

Supreme Court Act 1995 (Qld) s 253 – whether refusal to stay execution of costs is an order “as to costs only”

In Virgtel Ltd v Zabusky [2009] QCA 92 the Queensland Court of Appeal considered the scope of an order “as to costs only” within the meaning of s 253 of the Supreme Court Act 1995 (Qld) (‘the Act”). The Court also declined to accept submissions from one of the parties after oral hearing, and made some useful comments which serve as a reminder to practitioners of their obligations in that regard.

Disclosure obligations on ex parte application for freezing order - duty of practitioners to the court - failure to comply with duty - serious dereliction of duty to court - supplemental order for costs against solicitors personally

The decision of Applegarth J in Heartwood Architectural & Joinery Pty Ltd v Redchip Lawyers [2009] QSC 195 (27 July 2009) involved a costs order against solicitors personally. This decision is but one of several recent decisions in which the court has been persuaded that the circumstances justified costs orders against legal practitioners on the indemnity basis. These decisions serve as a reminder to practitioners of their disclosure obligations when seeking any interlocutory relief in an ex parte application. These obligations are now clearly set out in r 14.4 of the Legal Profession (Solicitors) Rule 2007 and r 25 of 2007 Barristers Rule. Inexperience or ignorance will not excuse breaches of the duties owed to the court.

Supreme Court Act 1995 (Qld) s 253 - leave to appeal costs orders - appeal against substantive judgment insufficient - must be arguable case that discretion will be overturned on appeal

The recent decision of the Court of Appeal in AGL Sales (Qld) Pty Ltd v Dawson Sales Pty Ltd [2009] QCA 262 provides clear direction on the Court’s expectations of a party seeking leave to appeal a costs order.This decision is likely to impact upon common practice in relation to appeals against costs orders. It sends a clear message to trial judges that they should not give leave as of course when giving a judgment in relation to costs, and that parties seeking leave under s 253 of the Supreme Court Act 1995 (Qld) should make a separate application. The application should be supported by material presenting an arguable case that the trial judge made an error in the exercise of the discretion of the kind described in House v King (1936) 55 CLR 499. A different, and interesting, aspect of this appeal is that it was the first wholly electronic civil appeal. The court-provided technology had been adopted at trial, and the Court of Appeal dispensed with any requirement for hard copy appeal record books.

Estimating health related costs and savings from balcony acoustic design for road traffic noise

A multi-faceted study is conducted with the objective of estimating the potential fiscal savings in annoyance and sleep disturbance related health costs due to providing improved building acoustic design standards. This study uses balcony acoustic treatments in response to road traffic noise as an example. The study area is the State of Queensland in Australia, where regional road traffic noise mapping data is used in conjunction with standard dose–response curves to estimate the population exposure levels. The background and the importance of using the selected road traffic noise indicators are discussed. In order to achieve the objective, correlations between the mapping indicator (LA10 (18 hour)) and the dose response curve indicators (Lden and Lnight) are established via analysis on a large database of road traffic noise measurement data. The existing noise exposure of the study area is used to estimate the fiscal reductions in health related costs through the application of simple estimations of costs per person per year per degree of annoyance or sleep disturbance. The results demonstrate that balcony acoustic treatments may provide a significant benefit towards reducing the health related costs of road traffic noise in a community.

Improved estimates of percolation and anisotropic permeability from 3-D X-ray microtomography using stochastic analyses and visualization

X-ray microtomography (micro-CT) with micron resolution enables new ways of characterizing microstructures and opens pathways for forward calculations of multiscale rock properties. A quantitative characterization of the microstructure is the first step in this challenge. We developed a new approach to extract scale-dependent characteristics of porosity, percolation, and anisotropic permeability from 3-D microstructural models of rocks. The Hoshen-Kopelman algorithm of percolation theory is employed for a standard percolation analysis. The anisotropy of permeability is calculated by means of the star volume distribution approach. The local porosity distribution and local percolation probability are obtained by using the local porosity theory. Additionally, the local anisotropy distribution is defined and analyzed through two empirical probability density functions, the isotropy index and the elongation index. For such a high-resolution data set, the typical data sizes of the CT images are on the order of gigabytes to tens of gigabytes; thus an extremely large number of calculations are required. To resolve this large memory problem parallelization in OpenMP was used to optimally harness the shared memory infrastructure on cache coherent Non-Uniform Memory Access architecture machines such as the iVEC SGI Altix 3700Bx2 Supercomputer. We see adequate visualization of the results as an important element in this first pioneering study.

Insights into special costs orders : WA court examines compliance and retrospectivity

In Walter v Buckeridge [No.5] [2012] WASC 495 Le Miere J considered an application by the defendants for special costs orders under the applicable legislation in Western Australia. Aspects of the decision may be of persuasive value in dealing with similar issues under Queensland legislation.

Surgical treatment approaches and reimbursement costs of surgical site infections post hip arthroplasty in Australia : a retrospective analysis

BACKGROUND: The treatment for deep surgical site infection (SSI) following primary total hip arthroplasty (THA) varies internationally and it is at present unclear which treatment approaches are used in Australia. The aim of this study is to identify current treatment approaches in Queensland, Australia, show success rates and quantify the costs of different treatments. METHODS: Data for patients undergoing primary THA and treatment for infection between January 2006 and December 2009 in Queensland hospitals were extracted from routinely used hospital databases. Records were linked with pathology information to confirm positive organisms. Diagnosis and treatment of infection was determined using ICD-10-AM and ACHI codes, respectively. Treatment costs were estimated based on AR-DRG cost accounting codes assigned to each patient hospital episode. RESULTS: A total of n=114 patients with deep surgical site infection were identified. The majority of patients (74%) were first treated with debridement, antibiotics and implant retention (DAIR), which was successful in eradicating the infection in 60.3% of patients with an average cost of $13,187. The remaining first treatments were 1-stage revision, successful in 89.7% with average costs of $27,006, and 2-stage revisions, successful in 92.9% of cases with average costs of $42,772. Multiple treatments following 'failed DAIR' cost on average $29,560, for failed 1-stage revision were $24,357, for failed 2-stage revision were $70,381 and were $23,805 for excision arthroplasty. CONCLUSIONS: As treatment costs in Australia are high primary prevention is important and the economics of competing treatment choices should be carefully considered. These currently vary greatly across international settings.

Distinguishing between mean-field, moment dynamics and stochastic descriptions of birth-death-movement processes

Mathematical descriptions of birth–death–movement processes are often calibrated to measurements from cell biology experiments to quantify tissue growth rates. Here we describe and analyze a discrete model of a birth–death-movement process applied to a typical two–dimensional cell biology experiment. We present three different descriptions of the system: (i) a standard mean–field description which neglects correlation effects and clustering; (ii) a moment dynamics description which approximately incorporates correlation and clustering effects, and; (iii) averaged data from repeated discrete simulations which directly incorporates correlation and clustering effects. Comparing these three descriptions indicates that the mean–field and moment dynamics approaches are valid only for certain parameter regimes, and that both these descriptions fail to make accurate predictions of the system for sufficiently fast birth and death rates where the effects of spatial correlations and clustering are sufficiently strong. Without any method to distinguish between the parameter regimes where these three descriptions are valid, it is possible that either the mean–field or moment dynamics model could be calibrated to experimental data under inappropriate conditions, leading to errors in parameter estimation. In this work we demonstrate that a simple measurement of agent clustering and correlation, based on coordination number data, provides an indirect measure of agent correlation and clustering effects, and can therefore be used to make a distinction between the validity of the different descriptions of the birth–death–movement process.