A general method to determine sampling windows for nonlinear mixed effects models with an application to population pharmacokinetic studies

Optimal design methods have been proposed to determine the best sampling times when sparse blood sampling is required in clinical pharmacokinetic studies. However, the optimal blood sampling time points may not be feasible in clinical practice. Sampling windows, a time interval for blood sample collection, have been proposed to provide flexibility in blood sampling times while preserving efficient parameter estimation. Because of the complexity of the population pharmacokinetic models, which are generally nonlinear mixed effects models, there is no analytical solution available to determine sampling windows. We propose a method for determination of sampling windows based on MCMC sampling techniques. The proposed method attains a stationary distribution rapidly and provides time-sensitive windows around the optimal design points. The proposed method is applicable to determine sampling windows for any nonlinear mixed effects model although our work focuses on an application to population pharmacokinetic models.

Nonlinear H_infinity control of UAVs for collision avoidance in gusty environments

This paper proposes a nonlinear H_infinity controller for stabilization of velocities, attitudes and angular rates of a fixed-wing unmanned aerial vehicle (UAV) in a windy environment. The suggested controller aims to achieve a steady-state flight condition in the presence of wind gusts such that the host UAV can be maneuvered to avoid collision with other UAVs during cruise flight with safety guarantees. This paper begins with building a proper model capturing flight aerodynamics of UAVs. Then a nonlinear controller is developed with gust attenuation and rapid response properties. Simulations are conducted for the Shadow UAV to verify performance of the proposed con- troller. Comparative studies with the proportional-integral-derivative (PID) controllers demonstrate that the proposed controller exhibits great performance improvement in a gusty environment, making it suitable for integration into the design of flight control systems for cruise flight of UAVs.

Mesopic rod and S-cone interactions revealed by modulation thresholds

We analyzed mesopic rod and S-cone interactions in terms of their contributions to the blue-yellow opponent pathway. Stimuli were generated using a 4-primary colorimeter. Mixed rod and S-cone modulation thresholds (constant L-, M-cone excitation) were measured as a function of their phase difference. Modulation amplitude was equated using threshold units and contrast ratios. This study identified three interaction types: (1) A linear and antagonistic rod:S-cone interaction, (2) probability summation (3) and a previously unidentified mutual nonlinear reinforcement. Linear rod:S-cone interactions occur within the blue-yellow opponent pathway. Probability summation involves signaling by different post-receptoral pathways. The origin of the nonlinear reinforcement is possibly at the photoreceptors.

Nonlinear pedagogy underpins intrinsic motivation in sports coaching

A key challenge for sports coaches is to provide performers with learning environments that result in sustainable motivation. In this paper, we will demonstrate that programmes based around the principles of Nonlinear Pedagogy can support the three basic psychological needs that underpin self-determined motivation. Coaches can therefore ensure that practice sessions provide for intrinsic motivation with its associated motivational and emotional benefits.

Comparison of robust criteria for D-optimal design

This study compared the performance of a local and three robust optimality criteria in terms of the standard error for a one-parameter and a two-parameter nonlinear model with uncertainty in the parameter values. The designs were also compared in conditions where there was misspecification in the prior parameter distribution. The impact of different correlation between parameters on the optimal design was examined in the two-parameter model. The designs and standard errors were solved analytically whenever possible and numerically otherwise.

Towards Bayesian experimental design for nonlinear models that require a large number of sampling times

The use of Bayesian methodologies for solving optimal experimental design problems has increased. Many of these methods have been found to be computationally intensive for design problems that require a large number of design points. A simulation-based approach that can be used to solve optimal design problems in which one is interested in finding a large number of (near) optimal design points for a small number of design variables is presented. The approach involves the use of lower dimensional parameterisations that consist of a few design variables, which generate multiple design points. Using this approach, one simply has to search over a few design variables, rather than searching over a large number of optimal design points, thus providing substantial computational savings. The methodologies are demonstrated on four applications, including the selection of sampling times for pharmacokinetic and heat transfer studies, and involve nonlinear models. Several Bayesian design criteria are also compared and contrasted, as well as several different lower dimensional parameterisation schemes for generating the many design points.

Fiber Bragg grating strain modulation based on nonlinear string transverse force amplifier

The only effective method of Fiber Bragg Grating (FBG) strain modulation has been by changing the distance between its two fixed ends. We demonstrate an alternative being more sensitive to force based on the nonlinear amplification relationship between a transverse force applied to a stretched string and its induced axial force. It may improve the sensitivity and size of an FBG force sensor, reduce the number of FBGs needed for multi-axial force monitoring, and control the resonant frequency of an FBG accelerometer.

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.

Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

A banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation

The method of lines is a standard method for advancing the solution of partial differential equations (PDEs) in time. In one sense, the method applies equally well to space-fractional PDEs as it does to integer-order PDEs. However, there is a significant challenge when solving space-fractional PDEs in this way, owing to the non-local nature of the fractional derivatives. Each equation in the resulting semi-discrete system involves contributions from every spatial node in the domain. This has important consequences for the efficiency of the numerical solver, especially when the system is large. First, the Jacobian matrix of the system is dense, and hence methods that avoid the need to form and factorise this matrix are preferred. Second, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. In this paper, we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two-sided, nonlinear space-fractional diffusion equation. A key contribution is to show, how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

Intrapreneurial bricolage: a contradiction in terms or useful construct?

This paper addresses the ambiguous relationship of internal, organizationa social capital and external social capital with corporate entrepreneurship performance. Drawing on social construction theory we argue that bricolage can mitigate some of the negative effects associated with social capital by recombining and redefining the purpose of available resources. We investigated our hypotheses through a random sample of 206 corporate entrepreneurship projects. We found that both internal and external social capital have no direct effect on performance of corporate entrepreneurship projects. The results indicate that bricolage mediates the relationship between social capital and performance of corporate entrepreneurship projects. Bricolage thrives in particularly when there is wide availability of social capital internal and external to the organization. The implications are that bricolage is a critical behavior in allowing corporate entrepreneur projects to benefit from resources available through their network of social relations inside and outside the company.

Lingodroids : learning terms for time

For humans and robots to communicate using natural language it is necessary for the robots to develop concepts and associated terms that correspond to the human use of words. Time and space are foundational concepts in human language, and to develop a set of words that correspond to human notions of time and space, it is necessary to take into account the way that they are used in natural human conversations, where terms and phrases such as `soon', `in a while', or `near' are often used. We present language learning robots called Lingodroids that can learn and use simple terms for time and space. In previous work, the Lingodroids were able to learn terms for space. In this work we extend their abilities by adding temporal variables which allow them to learn terms for time. The robots build their own maps of the world and interact socially to form a shared lexicon for location and duration terms. The robots successfully use the shared lexicons to communicate places and times to meet again.

Numerical approximation for a variable-order nonlinear reaction–subdiffusion equation

Fractional reaction–subdiffusion equations are widely used in recent years to simulate physical phenomena. In this paper, we consider a variable-order nonlinear reaction–subdiffusion equation. A numerical approximation method is proposed to solve the equation. Its convergence and stability are analyzed by Fourier analysis. By means of the technique for improving temporal accuracy, we also propose an improved numerical approximation. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.

Modelling sea water intrusion in coastal aquifers using heterogeneous computing

The objective of this PhD research program is to investigate numerical methods for simulating variably-saturated flow and sea water intrusion in coastal aquifers in a high-performance computing environment. The work is divided into three overlapping tasks: to develop an accurate and stable finite volume discretisation and numerical solution strategy for the variably-saturated flow and salt transport equations; to implement the chosen approach in a high performance computing environment that may have multiple GPUs or CPU cores; and to verify and test the implementation. The geological description of aquifers is often complex, with porous materials possessing highly variable properties, that are best described using unstructured meshes. The finite volume method is a popular method for the solution of the conservation laws that describe sea water intrusion, and is well-suited to unstructured meshes. In this work we apply a control volume-finite element (CV-FE) method to an extension of a recently proposed formulation (Kees and Miller, 2002) for variably saturated groundwater flow. The CV-FE method evaluates fluxes at points where material properties and gradients in pressure and concentration are consistently defined, making it both suitable for heterogeneous media and mass conservative. Using the method of lines, the CV-FE discretisation gives a set of differential algebraic equations (DAEs) amenable to solution using higher-order implicit solvers. Heterogeneous computer systems that use a combination of computational hardware such as CPUs and GPUs, are attractive for scientific computing due to the potential advantages offered by GPUs for accelerating data-parallel operations. We present a C++ library that implements data-parallel methods on both CPU and GPUs. The finite volume discretisation is expressed in terms of these data-parallel operations, which gives an efficient implementation of the nonlinear residual function. This makes the implicit solution of the DAE system possible on the GPU, because the inexact Newton-Krylov method used by the implicit time stepping scheme can approximate the action of a matrix on a vector using residual evaluations. We also propose preconditioning strategies that are amenable to GPU implementation, so that all computationally-intensive aspects of the implicit time stepping scheme are implemented on the GPU. Results are presented that demonstrate the efficiency and accuracy of the proposed numeric methods and formulation. The formulation offers excellent conservation of mass, and higher-order temporal integration increases both numeric efficiency and accuracy of the solutions. Flux limiting produces accurate, oscillation-free solutions on coarse meshes, where much finer meshes are required to obtain solutions with equivalent accuracy using upstream weighting. The computational efficiency of the software is investigated using CPUs and GPUs on a high-performance workstation. The GPU version offers considerable speedup over the CPU version, with one GPU giving speedup factor of 3 over the eight-core CPU implementation.