Epidemic models, deterministic or stochastic, are usually based on the assumption that a single c... more Epidemic models, deterministic or stochastic, are usually based on the assumption that a single contact (one-hit) between an infective and a susceptible will result in the infection of the susceptible (see Bailey [1] and Daley and Gani [2]). In this paper, we consider the case where two consecutive contacts (two-hits) are necessary for the infection to result in symptoms of the disease. We compare the number of symptomatic infectives with those resulting from the single contact model. The infection may be thought of as the simple cold, where a one-hit contact may be resisted by a susceptible, but a two-hit contact causes cold symptoms.
We consider a Leslie-type matrix approach to an SIR epidemic in discrete time. We give examples o... more We consider a Leslie-type matrix approach to an SIR epidemic in discrete time. We give examples of the population of susceptibles, infectives and removals for different birthrates and two different infection rates. Finally, when the infection rate depends on the number of infectives, we derive conditions for a steady state.
this paper we propose two different algorithms for the solution of equation (3); Newton-Multigrid... more this paper we propose two different algorithms for the solution of equation (3); Newton-Multigrid method and the FAS scheme. Straightforward implementations of both of these methods either diverged or converged very slowly, depending upon the grid size and the shape of the target. We shall describe the modifications which were necessary to overcome these difficulties and compare the performance between the two methods.
1998 International Conference on Ion Implantation Technology. Proceedings (Cat. No.98EX144)
ABSTRACT The results of a particle-in-cell simulation for the plasma system in the plasma immersi... more ABSTRACT The results of a particle-in-cell simulation for the plasma system in the plasma immersion ion implantation context is presented. PIII of two-dimensional trenches is studied using this simulation. Modeling of the potential around the trenches and the development of the plasma sheath in dependence on the pulse time is demonstrated. The implanted concentration, the average impact ion angle and the ion energy are computed as function of the pulse time and discussed in dependence on the aspect ratios of trenches
The time-dependent evolution of the potential, the electrical field, and the particle movement su... more The time-dependent evolution of the potential, the electrical field, and the particle movement surrounding two-dimensional trenches during a high voltage pulse in the context of plasma immersion ion implantation is studied by a particle-in-cell simulation. The numerical procedure is based on the solution of Poisson`s equation on a grid and the determination of the movement of the particles on the grid. This simulation is combined with simulation codes for the calculation of depth profiles and sputtering yields. The retained ion dose and the depth resolved concentration distribution were determined in dependence on the rise time of the pulse between 0.1 and 2 mus, pulse durations between 1 and 10 mus and the ion mass (m=20-131, i.e., Ne,...,Xe) for trenches with two different aspect ratios (ε=3:1 and 3:2). The results are discussed on the basis of the temporal evolution of the energy of the ions and the impact angle of the ions during the pulse.
This article examines a continuous time Markov chain model for a plantation-nursery system in whi... more This article examines a continuous time Markov chain model for a plantation-nursery system in which diseased plantation trees are replaced at a daily rate by nursery seedlings. There is a random infection rate caused by insects, and the disease is also spread directly between the N plantation trees at the rate , starting with a diseased trees at time t ¼ 0; in addition, some replacement seedlings prove to be infected with probability 0 < p < 1. We find a formal solution to the system in terms of the Laplace transformsp p j , j ¼ 0;. .. ; N, of the probabilities p j ðtÞ of j infected plantation trees at time t. A very simple example for N ¼ 2, a ¼ 1 is used to illustrate the method. We then consider numerically the effect of the parameters , , and on the system, and for small t study the influence of the initial number a of infected trees on the expected number of such trees at time t 365. As t ! 1, stationarity is achieved, irrespective of the initial value a.
Three stochastic processes, the birth, death and birth-death processes, subject to immigration ca... more Three stochastic processes, the birth, death and birth-death processes, subject to immigration can be decomposed into the sum of each process in the absence of immigration and anindependent process. We examine these independent processes through their probability generating functions (pgfs) and derive their expectations.
... Christian Wei , Hermann Hellwagner ... and set the restriction operator to be the transpose o... more ... Christian Wei , Hermann Hellwagner ... and set the restriction operator to be the transpose of the interpolation operator Rk-1 k = (Ik k-1)T . Additionally, the V-cycle will be used in a nested iteration in the form of a so-called FMV-scheme, see Brandt 2]. The interpolation operator eIk ...
. The numerical solution of partial differential equations leads to large,sparse systems of equat... more . The numerical solution of partial differential equations leads to large,sparse systems of equations with up a several millions of unknowns. Fast iterativealgorithms for the solution of these systems are typically based on the multilevel principle.Unfortunately, some of the commonly used programming techniques lead to a highoverhead on many advanced computer architectures. A fundamental problem arisesfrom hierarchical memory architectures with
BACKGROUND Effective teaching and learning require appropriate assessment tasks and prompt feedba... more BACKGROUND Effective teaching and learning require appropriate assessment tasks and prompt feedback. Feedback is critical to inform students if they are on track to achieve the course learning goals. According to Sadler (1989) useful feedback should provide evidence of learning that fills a gap between what is understood and what is aimed to be understood. In mathematics, the most effective form of assessment is weekly assignments to learn how to write mathematics. This is a difficult skill that requires a lot of practice and guidance. It is not sufficient to write down the correct answer or to give a list of calculations without adequate explanation. Students must employ precise use of words, formulae, symbols and punctuation. It is essential for students to receive feedback not only on the correctness of their answers, but also on their writing abilities in mathematics. AIMS The main aim of this project is to improve student learning of mathematics by promoting changes in mathemat...
Geoscience Australia and the Australian National University are developing a hydrodynamic inundat... more Geoscience Australia and the Australian National University are developing a hydrodynamic inundation modelling tool called anuga to help simulate the impact of natural inundation hazards such as riverine flooding, storm surges and tsunamis. The core of anuga is a Python implementation of a finite volume method, based on triangular meshes, for solving the conservative form of the Shallow Water Wave equation. We describe the parallelisation of the code using a domain decomposition strategy where we use the metis graph partitioning library to decompose the finite volume meshes. The parallel efficiency of our code is tested using a number of mesh partitions, and we verify that the metis graph partition is particularly efficient.
The thin plate spline method is often used to fit data in high dimensions. Standard thin plate sp... more The thin plate spline method is often used to fit data in high dimensions. Standard thin plate splines require the solution of a dense linear system of equations whose size increases with the number of data points and can be expensive when used on large data sets. In this paper we present a discrete thin plate spline method that uses polynomials with local support defined on finite element grids. The resulting system of equations is sparse and its size depends only on the number of nodes in the finite element grid so this method is efficient when dealing with large data sets. Theory is developed for general d-dimensional data sets and several example results are given for 3D models.
Traditional thin-plate splines use radial basis functions that produce dense linear system of equ... more Traditional thin-plate splines use radial basis functions that produce dense linear system of equations whose size increases with the number of data points. We present a discrete thin-plate spline method that uses polynomials with local support defined on finite-element grids. The resulting system of equations is sparse and its size depends only on the number of nodes in the finite element grid. Theory is developed for general d-dimensional data sets and model problems are presented in 3D to study the convergence behaviour.
The thin plate spline method is a widely used data fitting technique which has the ability to smo... more The thin plate spline method is a widely used data fitting technique which has the ability to smooth noisy data. We present some example applications of a new mixed finite element discretisation of the thin plate spline method. The new approach works with a pair of bases for the gradient and the Lagrange multiplier forming a biorthogonal system, thus ensuring that the scheme is numerically efficient and the formulation is stable. We overview of the theoretical foundations of the new approach and give numerical examples in both two and three dimensions.
Lecture Notes in Computational Science and Engineering, 2008
A commonly used method for fitting smooth functions to noisy data is the thin-plate spline method... more A commonly used method for fitting smooth functions to noisy data is the thin-plate spline method. Traditional thin-plate splines use radial basis functions and consequently require the solution of a dense linear system of equations whose dimension grows linearly with the number of data points. Here we discuss a method based on low order polynomial functions with locally supported basis functions. An advantage of such an approach is that the resulting system of equations is sparse and its dimension depends linearly on the number of nodes in the finite element grid instead of the number of data points. Another advantage is that an iterative solver, such as the conjugate gradient method, can be used. However it can be shown that the system of equations is similar to those arising from Tikhonov regularisation, and consequently the equations are ill-conditioned for certain choices of the parameters. To ensure that the method is robust an appropriate preconditioner must be used. In this paper we present the discrete thin-plate spline method and explore a set of preconditioners. We discuss some of the properties that are unique to our particular formulation and verify that the multiplicative Schwarz method is an effective preconditioner.
The thin plate spline method is a widely used data fitting technique as it has the ability to smo... more The thin plate spline method is a widely used data fitting technique as it has the ability to smooth noisy data. Here we consider a mixed finite element discretisation of the thin plate spline. By using mixed finite elements the formulation can be defined in-terms of relatively simple stencils, thus resulting in a system that is sparse and whose size only depends linearly on the number of finite element nodes. The mixed formulation is obtained by introducing the gradient of the corresponding function as an additional unknown. The novel approach taken in this paper is to work with a pair of bases for the gradient and the Lagrange multiplier forming a biorthogonal system thus ensuring that the scheme is numerically efficient, and the formulation is stable. Some numerical results are presented to demonstrate the performance of our approach. A preconditioned conjugate gradient method is an efficient solver for the arising linear system of equations.
Modeling, Simulation and Optimization of Complex Processes, 2008
... Boundary Conditions Linda Stals and Stephen Roberts Department of Mathematics, Australian Nat... more ... Boundary Conditions Linda Stals and Stephen Roberts Department of Mathematics, Australian National University, Canberra, ACT, 0200, Australia stals@maths.anu.edu.au ... JR Stat. Soc. Ser. B Stat. Methodol., 64(2):307319, 2002. 8. S. Roberts, M. Hegland, and I. Altas. ...
J. W. Larson, P. E. Strazdins, M. Hegland, B. Harding, S. Roberts , L. Stals , A. P. Rendell, Md.... more J. W. Larson, P. E. Strazdins, M. Hegland, B. Harding, S. Roberts , L. Stals , A. P. Rendell, Md. M. Ali , and J. Southern
Epidemic models, deterministic or stochastic, are usually based on the assumption that a single c... more Epidemic models, deterministic or stochastic, are usually based on the assumption that a single contact (one-hit) between an infective and a susceptible will result in the infection of the susceptible (see Bailey [1] and Daley and Gani [2]). In this paper, we consider the case where two consecutive contacts (two-hits) are necessary for the infection to result in symptoms of the disease. We compare the number of symptomatic infectives with those resulting from the single contact model. The infection may be thought of as the simple cold, where a one-hit contact may be resisted by a susceptible, but a two-hit contact causes cold symptoms.
We consider a Leslie-type matrix approach to an SIR epidemic in discrete time. We give examples o... more We consider a Leslie-type matrix approach to an SIR epidemic in discrete time. We give examples of the population of susceptibles, infectives and removals for different birthrates and two different infection rates. Finally, when the infection rate depends on the number of infectives, we derive conditions for a steady state.
this paper we propose two different algorithms for the solution of equation (3); Newton-Multigrid... more this paper we propose two different algorithms for the solution of equation (3); Newton-Multigrid method and the FAS scheme. Straightforward implementations of both of these methods either diverged or converged very slowly, depending upon the grid size and the shape of the target. We shall describe the modifications which were necessary to overcome these difficulties and compare the performance between the two methods.
1998 International Conference on Ion Implantation Technology. Proceedings (Cat. No.98EX144)
ABSTRACT The results of a particle-in-cell simulation for the plasma system in the plasma immersi... more ABSTRACT The results of a particle-in-cell simulation for the plasma system in the plasma immersion ion implantation context is presented. PIII of two-dimensional trenches is studied using this simulation. Modeling of the potential around the trenches and the development of the plasma sheath in dependence on the pulse time is demonstrated. The implanted concentration, the average impact ion angle and the ion energy are computed as function of the pulse time and discussed in dependence on the aspect ratios of trenches
The time-dependent evolution of the potential, the electrical field, and the particle movement su... more The time-dependent evolution of the potential, the electrical field, and the particle movement surrounding two-dimensional trenches during a high voltage pulse in the context of plasma immersion ion implantation is studied by a particle-in-cell simulation. The numerical procedure is based on the solution of Poisson`s equation on a grid and the determination of the movement of the particles on the grid. This simulation is combined with simulation codes for the calculation of depth profiles and sputtering yields. The retained ion dose and the depth resolved concentration distribution were determined in dependence on the rise time of the pulse between 0.1 and 2 mus, pulse durations between 1 and 10 mus and the ion mass (m=20-131, i.e., Ne,...,Xe) for trenches with two different aspect ratios (ε=3:1 and 3:2). The results are discussed on the basis of the temporal evolution of the energy of the ions and the impact angle of the ions during the pulse.
This article examines a continuous time Markov chain model for a plantation-nursery system in whi... more This article examines a continuous time Markov chain model for a plantation-nursery system in which diseased plantation trees are replaced at a daily rate by nursery seedlings. There is a random infection rate caused by insects, and the disease is also spread directly between the N plantation trees at the rate , starting with a diseased trees at time t ¼ 0; in addition, some replacement seedlings prove to be infected with probability 0 < p < 1. We find a formal solution to the system in terms of the Laplace transformsp p j , j ¼ 0;. .. ; N, of the probabilities p j ðtÞ of j infected plantation trees at time t. A very simple example for N ¼ 2, a ¼ 1 is used to illustrate the method. We then consider numerically the effect of the parameters , , and on the system, and for small t study the influence of the initial number a of infected trees on the expected number of such trees at time t 365. As t ! 1, stationarity is achieved, irrespective of the initial value a.
Three stochastic processes, the birth, death and birth-death processes, subject to immigration ca... more Three stochastic processes, the birth, death and birth-death processes, subject to immigration can be decomposed into the sum of each process in the absence of immigration and anindependent process. We examine these independent processes through their probability generating functions (pgfs) and derive their expectations.
... Christian Wei , Hermann Hellwagner ... and set the restriction operator to be the transpose o... more ... Christian Wei , Hermann Hellwagner ... and set the restriction operator to be the transpose of the interpolation operator Rk-1 k = (Ik k-1)T . Additionally, the V-cycle will be used in a nested iteration in the form of a so-called FMV-scheme, see Brandt 2]. The interpolation operator eIk ...
. The numerical solution of partial differential equations leads to large,sparse systems of equat... more . The numerical solution of partial differential equations leads to large,sparse systems of equations with up a several millions of unknowns. Fast iterativealgorithms for the solution of these systems are typically based on the multilevel principle.Unfortunately, some of the commonly used programming techniques lead to a highoverhead on many advanced computer architectures. A fundamental problem arisesfrom hierarchical memory architectures with
BACKGROUND Effective teaching and learning require appropriate assessment tasks and prompt feedba... more BACKGROUND Effective teaching and learning require appropriate assessment tasks and prompt feedback. Feedback is critical to inform students if they are on track to achieve the course learning goals. According to Sadler (1989) useful feedback should provide evidence of learning that fills a gap between what is understood and what is aimed to be understood. In mathematics, the most effective form of assessment is weekly assignments to learn how to write mathematics. This is a difficult skill that requires a lot of practice and guidance. It is not sufficient to write down the correct answer or to give a list of calculations without adequate explanation. Students must employ precise use of words, formulae, symbols and punctuation. It is essential for students to receive feedback not only on the correctness of their answers, but also on their writing abilities in mathematics. AIMS The main aim of this project is to improve student learning of mathematics by promoting changes in mathemat...
Geoscience Australia and the Australian National University are developing a hydrodynamic inundat... more Geoscience Australia and the Australian National University are developing a hydrodynamic inundation modelling tool called anuga to help simulate the impact of natural inundation hazards such as riverine flooding, storm surges and tsunamis. The core of anuga is a Python implementation of a finite volume method, based on triangular meshes, for solving the conservative form of the Shallow Water Wave equation. We describe the parallelisation of the code using a domain decomposition strategy where we use the metis graph partitioning library to decompose the finite volume meshes. The parallel efficiency of our code is tested using a number of mesh partitions, and we verify that the metis graph partition is particularly efficient.
The thin plate spline method is often used to fit data in high dimensions. Standard thin plate sp... more The thin plate spline method is often used to fit data in high dimensions. Standard thin plate splines require the solution of a dense linear system of equations whose size increases with the number of data points and can be expensive when used on large data sets. In this paper we present a discrete thin plate spline method that uses polynomials with local support defined on finite element grids. The resulting system of equations is sparse and its size depends only on the number of nodes in the finite element grid so this method is efficient when dealing with large data sets. Theory is developed for general d-dimensional data sets and several example results are given for 3D models.
Traditional thin-plate splines use radial basis functions that produce dense linear system of equ... more Traditional thin-plate splines use radial basis functions that produce dense linear system of equations whose size increases with the number of data points. We present a discrete thin-plate spline method that uses polynomials with local support defined on finite-element grids. The resulting system of equations is sparse and its size depends only on the number of nodes in the finite element grid. Theory is developed for general d-dimensional data sets and model problems are presented in 3D to study the convergence behaviour.
The thin plate spline method is a widely used data fitting technique which has the ability to smo... more The thin plate spline method is a widely used data fitting technique which has the ability to smooth noisy data. We present some example applications of a new mixed finite element discretisation of the thin plate spline method. The new approach works with a pair of bases for the gradient and the Lagrange multiplier forming a biorthogonal system, thus ensuring that the scheme is numerically efficient and the formulation is stable. We overview of the theoretical foundations of the new approach and give numerical examples in both two and three dimensions.
Lecture Notes in Computational Science and Engineering, 2008
A commonly used method for fitting smooth functions to noisy data is the thin-plate spline method... more A commonly used method for fitting smooth functions to noisy data is the thin-plate spline method. Traditional thin-plate splines use radial basis functions and consequently require the solution of a dense linear system of equations whose dimension grows linearly with the number of data points. Here we discuss a method based on low order polynomial functions with locally supported basis functions. An advantage of such an approach is that the resulting system of equations is sparse and its dimension depends linearly on the number of nodes in the finite element grid instead of the number of data points. Another advantage is that an iterative solver, such as the conjugate gradient method, can be used. However it can be shown that the system of equations is similar to those arising from Tikhonov regularisation, and consequently the equations are ill-conditioned for certain choices of the parameters. To ensure that the method is robust an appropriate preconditioner must be used. In this paper we present the discrete thin-plate spline method and explore a set of preconditioners. We discuss some of the properties that are unique to our particular formulation and verify that the multiplicative Schwarz method is an effective preconditioner.
The thin plate spline method is a widely used data fitting technique as it has the ability to smo... more The thin plate spline method is a widely used data fitting technique as it has the ability to smooth noisy data. Here we consider a mixed finite element discretisation of the thin plate spline. By using mixed finite elements the formulation can be defined in-terms of relatively simple stencils, thus resulting in a system that is sparse and whose size only depends linearly on the number of finite element nodes. The mixed formulation is obtained by introducing the gradient of the corresponding function as an additional unknown. The novel approach taken in this paper is to work with a pair of bases for the gradient and the Lagrange multiplier forming a biorthogonal system thus ensuring that the scheme is numerically efficient, and the formulation is stable. Some numerical results are presented to demonstrate the performance of our approach. A preconditioned conjugate gradient method is an efficient solver for the arising linear system of equations.
Modeling, Simulation and Optimization of Complex Processes, 2008
... Boundary Conditions Linda Stals and Stephen Roberts Department of Mathematics, Australian Nat... more ... Boundary Conditions Linda Stals and Stephen Roberts Department of Mathematics, Australian National University, Canberra, ACT, 0200, Australia stals@maths.anu.edu.au ... JR Stat. Soc. Ser. B Stat. Methodol., 64(2):307319, 2002. 8. S. Roberts, M. Hegland, and I. Altas. ...
J. W. Larson, P. E. Strazdins, M. Hegland, B. Harding, S. Roberts , L. Stals , A. P. Rendell, Md.... more J. W. Larson, P. E. Strazdins, M. Hegland, B. Harding, S. Roberts , L. Stals , A. P. Rendell, Md. M. Ali , and J. Southern
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