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.

Order conditions of stochastic Runge-Kutta methods by B-series

In this paper, general order conditions and a global convergence proof are given for stochastic Runge Kutta methods applied to stochastic ordinary differential equations ( SODEs) of Stratonovich type. This work generalizes the ideas of B-series as applied to deterministic ordinary differential equations (ODEs) to the stochastic case and allows a completely general formalism for constructing high order stochastic methods, either explicit or implicit. Some numerical results will be given to illustrate this theory.

Re-use of domain knowledge to provide confidence for adoption of off-site manufacturing for construction in Australia

Many construction industry decision-makers believe there is a lack of off-site manufacture (OSM) adoption for non-residential construction in Australia. Identification of construction business process was considered imperative in order to assist decision-makers to increase OSM utilisation. The premise that domain knowledge can be re-used to provide an intervention point in the construction process led a team of researchers to construct simple base-line process models for the complete construction process, segmented into six phases. Sixteen domain knowledge industry experts were asked to review the construction phase base-line models to answer the question “Where in the process illustrated by this base-line model phase is an OSM task?”. Through an iterative and generative process a number of off-site manufacture intervention points were identified and integrated into the process models. The re-use of industry expert domain knowledge provided suggestions for new ways to do basic tasks thus facilitating changes to current practice. It is expected that implementation of the new processes will lead to systemic industry change and thus a growth in productivity due to increased adoption of OSM.

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.

The use of hypnosis in labor and delivery : a preliminary study

Self-hypnosis was taught to 87 obstetric patients (HYP) and was not taught to 56 other patients (CNTRL), all delivered by the same family physician, in order to determine whether the use of self-hypnosis by low-risk obstetric patients leads to fewer technologic interventions during their deliveries or greater satisfaction of parturients with their delivery experience or both. The outcomes of the deliveries of these two groups were compared, and the HYP group was compared to 352 low-risk patients delivered by other family physicians at the same hospital (WCH). Questionnaires were mailed postpartum to 156 patients, all delivered by the same family physician, to determine satisfaction with delivery using the Labor and Delivery Satisfaction Index (LADSI). The hypnosis group showed a significant reduction in the number of epidurals (11.4% less than CNTRL and 17.9% less than WCH, p < 0.05) and the use of intravenous lines (18.5% less for both, p < 0.05). The number of episiotomies was significantly less in the HYP group compared to WCH (15.9%, p < 0.05) and 11.5% less when compared to CNTRL. The tear rate was not statistically different. Combined use of the intervention triad (epidural–forceps–episiotomy) was less for HYP than for CNTRL (15.8% less) and WCH (10.2% less, p < 0.05). More deliveries were done in the labor room with HYP than CNTRL (21%, p < 0.05). The second stage was shortened by 10 min (HYP vs CNTRL). Overall satisfaction of HYP and CNTRL patients was similar and generally favorable.

Collective sampling and analysis of high order tensors for chatroom communications

This work investigates the accuracy and efficiency tradeoffs between centralized and collective (distributed) algorithms for (i) sampling, and (ii) n-way data analysis techniques in multidimensional stream data, such as Internet chatroom communications. Its contributions are threefold. First, we use the Kolmogorov-Smirnov goodness-of-fit test to show that statistical differences between real data obtained by collective sampling in time dimension from multiple servers and that of obtained from a single server are insignificant. Second, we show using the real data that collective data analysis of 3-way data arrays (users x keywords x time) known as high order tensors is more efficient than centralized algorithms with respect to both space and computational cost. Furthermore, we show that this gain is obtained without loss of accuracy. Third, we examine the sensitivity of collective constructions and analysis of high order data tensors to the choice of server selection and sampling window size. We construct 4-way tensors (users x keywords x time x servers) and analyze them to show the impact of server and window size selections on the results.

Preliminary evaluation of a primary care intervention for cry-fuss behaviours in the first 3-4 months of life (‘The Possums Approach’) : effects on cry-fuss behaviours and maternal mood

Problem crying in the first few months of life is both common and complex, arising out of multiple interacting and co-evolving factors. Parents whose babies cry and fuss a lot receive conflicting advice as they seek help from multiple health providers and emergency departments, and may be admitted into tertiary residential services. Conflicting advice is costly, and arises out of discipline-specific interpretations of evidence. An integrated, interdisciplinary primary care intervention (‘The Possums Approach’) for cry-fuss problems in the first months of life was developed from available peer-reviewed evidence. This study reports on preliminary evaluation of delivery of the intervention. A total of 20 mothers who had crying babies under 16 weeks of age (average age 6.15 weeks) completed questionnaires, including the Crying Patterns Questionnaire and the Edinburgh Postnatal Depression Scale, before and 3-4 weeks after their first consultation with trained primary care practitioners. Preliminary evaluation is promising. The Crying Patterns Questionnaire showed a significant decrease in crying and fussing duration, by 1 h in the evening (P = 0.001) and 30 min at night (P = 0.009). The median total amount of crying and fussing in a 24-h period was reduced from 6.12 to 3 h. The Edinburgh Postnatal Depression Scale showed a significant improvement in depressive symptoms, with the median score decreasing from 11 to 6 (P = 0.005). These findings are corroborated by an analysis of results for the subset of 16 participants whose babies were under 12 weeks of age (average age 4.71 weeks). These preliminary results demonstrate significantly decreased infant crying in the evening and during the night and improved maternal mood, validating an innovative interdisciplinary clinical intervention for cry-fuss problems in the first few months of life. This intervention, delivered by trained health professionals, has the potential to mitigate the costly problem of health professionals giving discipline-specific and conflicting advice post-birth.

Consumer acceptance of an SMS-assisted smoking cessation intervention : a multicountry study

This study assesses smokers' perceptions, motivations, and intentions towards using an SMS-assisted smoking cessation intervention in Australia, France, and Mexico through an extended technology acceptance model with mediating variables. Data was collected through online surveys. Results show that perceived usefulness and vicarious innovativeness predict use intentions for all three countries. Perceived ease of use is significant only for Mexico. Subjective norms are significant only for Mexico and Australia. Perceived monetary value and perceived annoyance are significant mediating variables for all three countries, whereas perceived enjoyment is significant only for Mexico and Australia. These results contribute to theory and practice.

Application of theory in developing peer-support based intervention to improve self-management for cardiac patients with diabetes

The biosafety of carbon nanomaterial needs to be critically evaluated with both experimental and theoretical validations before extensive biomedical applications. In this letter, we present an analysis of the binding ability of two dimensional monolayer carbon nanomaterial on actin by molecular simulation to understand their adhesive characteristics on F-actin cytoskeleton. The modelling results indicate that the positively charged carbon nanomaterial has higher binding stability on actin. Compared to crystalline graphene, graphene oxide shows higher binding influence on actin when carrying 11 positive surface charge. This theoretical investigation provides insights into the sensitivity of actin-related cellular activities on carbon nanomaterial.

A framework for a dense matching algorithm : Results using matching metrics and order statistic filters

An algorithm for computing dense correspondences between images of a stereo pair or image sequence is presented. The algorithm can make use of both standard matching metrics and the rank and census filters, two filters based on order statistics which have been applied to the image matching problem. Their advantages include robustness to radiometric distortion and amenability to hardware implementation. Results obtained using both real stereo pairs and a synthetic stereo pair with ground truth were compared. The rank and census filters were shown to significantly improve performance in the case of radiometric distortion. In all cases, the results obtained were comparable to, if not better than, those obtained using standard matching metrics. Furthermore, the rank and census have the additional advantage that their computational overhead is less than these metrics. For all techniques tested, the difference between the results obtained for the synthetic stereo pair, and the ground truth results was small.

Order statistic filters for image matching

The rank and census are two filters based on order statistics which have been applied to the image matching problem for stereo pairs. Advantages of these filters include their robustness to radiometric distortion and small amounts of random noise, and their amenability to hardware implementation. In this paper, a new matching algorithm is presented, which provides an overall framework for matching, and is used to compare the rank and census techniques with standard matching metrics. The algorithm was tested using both real stereo pairs and a synthetic pair with ground truth. The rank and census filters were shown to significantly improve performance in the case of radiometric distortion. In all cases, the results obtained were comparable to, if not better than, those obtained using standard matching metrics. Furthermore, the rank and census have the additional advantage that their computational overhead is less than these metrics. For all techniques tested, the difference between the results obtained for the synthetic stereo pair, and the ground truth results was small.