Combining Satellite And Gauge Precipitation Data With Co-Kriging Method For Jakarta Region

Jakarta is vulnerable to flooding mainly caused by prolonged and heavy rainfall and thus a robust hydrological modeling is called for. A good quality of spatial precipitation data is therefore desired so that a good hydrological model could be achieved. Two types of rainfall sources are available: satellite and gauge station observations. At-site rainfall is considered to be a reliable and accurate source of rainfall. However, the limited number of stations makes the spatial interpolation not very much appealing. On the other hand, the gridded rainfall nowadays has high spatial resolution and improved accuracy, but still, relatively less accurate than its counterpart. To achieve a better precipitation data set, the study proposes cokriging method, a blending algorithm, to yield the blended satellite-gauge gridded rainfall at approximately 10-km resolution. The Global Satellite Mapping of Precipitation (GSMaP, 0.1⁰×0.1⁰) and daily rainfall observations from gauge stations are used. The blended product is compared with satellite data by cross-validation method. The newly-yield blended product is then utilized to re-calibrate the hydrological model. Several scenarios are simulated by the hydrological models calibrated by gauge observations alone and blended product. The performance of two calibrated hydrological models is then assessed and compared based on simulated and observed runoff.

A Petrov-Galerkin method for a singularly perturbed ordinary differential equation with non-smooth data

In this paper, a singularly perturbed ordinary differential equation with non-smooth data is considered. The numerical method is generated by means of a Petrov-Galerkin finite element method with the piecewise-exponential test function and the piecewise-linear trial function. At the discontinuous point of the coefficient, a special technique is used. The method is shown to be first-order accurate and singular perturbation parameter uniform convergence. Finally, numerical results are presented, which are in agreement with theoretical results.

Constructive development of the solutions of linear equations in introductory ordinary differential equations

The solution of linear ordinary differential equations (ODEs) is commonly taught in first year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognising what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced studies in mathematics. In this study we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery based approach where students employ their existing skills as a framework for constructing the solutions of first and second order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first year undergraduate class. Finally we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar manner.

The intention to notice : the collection, the tour and ordinary landscapes

The Intention to Notice: the collection, the tour and ordinary landscapes is concerned with how ordinary landscapes and places are enabled and conserved through making itineraries that are framed around the ephemera encountered by chance, and the practices that make possible the endurance of these material traces. Through observing and then examining the material and temporal aspects of a variety of sites/places, the museum and the expanded garden are identified as spaces where the expression of contemporary political, ecological and social attitudes to cultural landscapes can be realised through a curatorial approach to design, to effect minimal intervention. Three notions are proposed to encourage investigation into contemporary cultural landscapes: To traverse slowly to allow space for speculations framed by the topographies and artefacts encountered; to [re]make/[re]write cultural landscapes as discursive landscapes that provoke the intention to notice; and to reveal and conserve the fabric of everyday places. A series of walking, recording and making projects undertaken across a variety of cultural landscapes in remote South Australia, Melbourne, Sydney, London, Los Angeles, Chandigarh, Padova and Istanbul, investigate how communities of practice are facilitated through the invitation to notice and intervene in ordinary landscapes, informed by the theory and practice of postproduction and the reticent auteur. This community of practice approach draws upon chance encounters and it seeks to encourage creative investigation into places. The Intention to Notice is a practice of facilitating that also leads to recording traces and events; large and small, material and immaterial, that encourages both conjecture and archive. Most importantly, there is an open-ended invitation to commit and exchange through design interaction.

Stable strong order 1.0 schemes for solving stochastic ordinary differential equations

The Balanced method was introduced as a class of quasi-implicit methods, based upon the Euler-Maruyama scheme, for solving stiff stochastic differential equations. We extend the Balanced method to introduce a class of stable strong order 1. 0 numerical schemes for solving stochastic ordinary differential equations. We derive convergence results for this class of numerical schemes. We illustrate the asymptotic stability of this class of schemes is illustrated and is compared with contemporary schemes of strong order 1. 0. We present some evidence on parametric selection with respect to minimising the error convergence terms. Furthermore we provide a convergence result for general Balanced style schemes of higher orders.

Numerical simulation of stochastic ordinary differential equations in biomathematical modelling

In this work we discuss the effects of white and coloured noise perturbations on the parameters of a mathematical model of bacteriophage infection introduced by Beretta and Kuang in [Math. Biosc. 149 (1998) 57]. We numerically simulate the strong solutions of the resulting systems of stochastic ordinary differential equations (SDEs), with respect to the global error, by means of numerical methods of both Euler-Taylor expansion and stochastic Runge-Kutta type.

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.

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.

Copula-based spatial modelling of geometallurgical variables

The most important aspect of modelling a geological variable, such as metal grade, is the spatial correlation. Spatial correlation describes the relationship between realisations of a geological variable sampled at different locations. Any method for spatially modelling such a variable should be capable of accurately estimating the true spatial correlation. Conventional kriged models are the most commonly used in mining for estimating grade or other variables at unsampled locations, and these models use the variogram or covariance function to model the spatial correlations in the process of estimation. However, this usage assumes the relationships of the observations of the variable of interest at nearby locations are only influenced by the vector distance between the locations. This means that these models assume linear spatial correlation of grade. In reality, the relationship with an observation of grade at a nearby location may be influenced by both distance between the locations and the value of the observations (ie non-linear spatial correlation, such as may exist for variables of interest in geometallurgy). Hence this may lead to inaccurate estimation of the ore reserve if a kriged model is used for estimating grade of unsampled locations when nonlinear spatial correlation is present. Copula-based methods, which are widely used in financial and actuarial modelling to quantify the non-linear dependence structures, may offer a solution. This method was introduced by Bárdossy and Li (2008) to geostatistical modelling to quantify the non-linear spatial dependence structure in a groundwater quality measurement network. Their copula-based spatial modelling is applied in this research paper to estimate the grade of 3D blocks. Furthermore, real-world mining data is used to validate this model. These copula-based grade estimates are compared with the results of conventional ordinary and lognormal kriging to present the reliability of this method.