Finite volume methods for simulating anomalous transport

In this thesis a new approach for solving a certain class of anomalous diffusion equations was developed. The theory and algorithms arising from this work will pave the way for more efficient and more accurate solutions of these equations, with applications to science, health and industry. The method of finite volumes was applied to discretise the spatial derivatives, and this was shown to outperform existing methods in several key respects. The stability and convergence of the new method were rigorously established.

Coverage algorithms for an under-actuated car-like vehicle in an uncertain environment

A coverage algorithm is an algorithm that deploys a strategy as to how to cover all points in terms of a given area using some set of sensors. In the past decades a lot of research has gone into development of coverage algorithms. Initially, the focus was coverage of structured and semi-structured indoor areas, but with time and development of better sensors and introduction of GPS, the focus has turned to outdoor coverage. Due to the unstructured nature of an outdoor environment, covering an outdoor area with all its obstacles and simultaneously performing reliable localization is a difficult task. In this paper, two path planning algorithms suitable for solving outdoor coverage tasks are introduced. The algorithms take into account the kinematic constraints of an under-actuated car-like vehicle, minimize trajectory curvatures, and dynamically avoid detected obstacles in the vicinity, all in real-time. We demonstrate the performance of the coverage algorithm in the field by achieving 95% coverage using an autonomous tractor mower without the aid of any absolute localization system or constraints on the physical boundaries of the area.

A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation

This paper aims to develop a meshless approach based on the Point Interpolation Method (PIM) for numerical simulation of a space fractional diffusion equation. Two fully-discrete schemes for the one-dimensional space fractional diffusion equation are obtained by using the PIM and the strong-forms of the space diffusion equation. Numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach.

Managing and learning with multiple models: Objectives and optimization algorithms

The quality of environmental decisions should be gauged according to managers' objectives. Management objectives generally seek to maximize quantifiable measures of system benefit, for instance population growth rate. Reaching these goals often requires a certain degree of learning about the system. Learning can occur by using management action in combination with a monitoring system. Furthermore, actions can be chosen strategically to obtain specific kinds of information. Formal decision making tools can choose actions to favor such learning in two ways: implicitly via the optimization algorithm that is used when there is a management objective (for instance, when using adaptive management), or explicitly by quantifying knowledge and using it as the fundamental project objective, an approach new to conservation.This paper outlines three conservation project objectives - a pure management objective, a pure learning objective, and an objective that is a weighted mixture of these two. We use eight optimization algorithms to choose actions that meet project objectives and illustrate them in a simulated conservation project. The algorithms provide a taxonomy of decision making tools in conservation management when there is uncertainty surrounding competing models of system function. The algorithms build upon each other such that their differences are highlighted and practitioners may see where their decision making tools can be improved. © 2010 Elsevier Ltd.

A semi-alternating direction method for a 2-D fractional FitzHugh–Nagumo monodomain model on an approximate irregular domain

A FitzHugh-Nagumo monodomain model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue. In this paper, we consider a two-dimensional fractional FitzHugh-Nagumo monodomain model on an irregular domain. The model consists of a coupled Riesz space fractional nonlinear reaction-diffusion model and an ordinary differential equation, describing the ionic fluxes as a function of the membrane potential. Secondly, we use a decoupling technique and focus on solving the Riesz space fractional nonlinear reaction-diffusion model. A novel spatially second-order accurate semi-implicit alternating direction method (SIADM) for this model on an approximate irregular domain is proposed. Thirdly, stability and convergence of the SIADM are proved. Finally, some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two-dimensional fractional Fitzhugh-Nagumo model on both an approximate circular and an approximate irregular domain.

A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

In this paper, we derive a new nonlinear two-sided space-fractional diffusion equation with variable coefficients from the fractional Fick’s law. A semi-implicit difference method (SIDM) for this equation is proposed. The stability and convergence of the SIDM are discussed. For the implementation, we develop a fast accurate iterative method for the SIDM by decomposing the dense coefficient matrix into a combination of Toeplitz-like matrices. This fast iterative method significantly reduces the storage requirement of O(n2)O(n2) and computational cost of O(n3)O(n3) down to n and O(nlogn)O(nlogn), where n is the number of grid points. The method retains the same accuracy as the underlying SIDM solved with Gaussian elimination. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

A Crank--Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction-diffusion equation

In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank--Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order $2$ in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh--Nagumo model. Numerical results are provided to verify the theoretical analysis.

Maximum principle and numerical method for the multi-term time–space Riesz–Caputo fractional differential equations

The maximum principle for the space and time–space fractional partial differential equations is still an open problem. In this paper, we consider a multi-term time–space Riesz–Caputo fractional differential equations over an open bounded domain. A maximum principle for the equation is proved. The uniqueness and continuous dependence of the solution are derived. Using a fractional predictor–corrector method combining the L1 and L2 discrete schemes, we present a numerical method for the specified equation. Two examples are given to illustrate the obtained results.

Efficiency of parallelisation of genetic algorithms in the data analysis context

Most real-life data analysis problems are difficult to solve using exact methods, due to the size of the datasets and the nature of the underlying mechanisms of the system under investigation. As datasets grow even larger, finding the balance between the quality of the approximation and the computing time of the heuristic becomes non-trivial. One solution is to consider parallel methods, and to use the increased computational power to perform a deeper exploration of the solution space in a similar time. It is, however, difficult to estimate a priori whether parallelisation will provide the expected improvement. In this paper we consider a well-known method, genetic algorithms, and evaluate on two distinct problem types the behaviour of the classic and parallel implementations.

Optimisation algorithms for microarray biclustering

In providing simultaneous information on expression profiles for thousands of genes, microarray technologies have, in recent years, been largely used to investigate mechanisms of gene expression. Clustering and classification of such data can, indeed, highlight patterns and provide insight on biological processes. A common approach is to consider genes and samples of microarray datasets as nodes in a bipartite graphs, where edges are weighted e.g. based on the expression levels. In this paper, using a previously-evaluated weighting scheme, we focus on search algorithms and evaluate, in the context of biclustering, several variations of Genetic Algorithms. We also introduce a new heuristic “Propagate”, which consists in recursively evaluating neighbour solutions with one more or one less active conditions. The results obtained on three well-known datasets show that, for a given weighting scheme,optimal or near-optimal solutions can be identified.

Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains

Subdiffusion equations with distributed-order fractional derivatives describe some important physical phenomena. In this paper, we consider the time distributed-order and Riesz space fractional diffusions on bounded domains with Dirichlet boundary conditions. Here, the time derivative is defined as the distributed-order fractional derivative in the Caputo sense, and the space derivative is defined as the Riesz fractional derivative. First, we discretize the integral term in the time distributed-order and Riesz space fractional diffusions using numerical approximation. Then the given equation can be written as a multi-term time–space fractional diffusion. Secondly, we propose an implicit difference method for the multi-term time–space fractional diffusion. Thirdly, using mathematical induction, we prove the implicit difference method is unconditionally stable and convergent. Also, the solvability for our method is discussed. Finally, two numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

Numerical simulation of anomalous infiltration in porous media

Nonlinear time-fractional diffusion equations have been used to describe the liquid infiltration for both subdiffusion and superdiffusion in porous media. In this paper, some problems of anomalous infiltration with a variable-order timefractional derivative in porous media are considered. The time-fractional Boussinesq equation is also considered. Two computationally efficient implicit numerical schemes for the diffusion and wave-diffusion equations are proposed. Numerical examples are provided to show that the numerical methods are computationally efficient.

Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation

In this paper, we consider a two-sided space-fractional diffusion equation with variable coefficients on a finite domain. Firstly, based on the nodal basis functions, we present a new fractional finite volume method for the two-sided space-fractional diffusion equation and derive the implicit scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the implicit fractional finite volume method and conclude that the method is unconditionally stable and convergent. Finally, some numerical examples are given to show the effectiveness of the new numerical method, and the results are in excellent agreement with theoretical analysis.