Numerical simulation of the Riesz fractional diffusion equation with a nonlinear source term

In this paper, A Riesz fractional diffusion equation with a nonlinear source term (RFDE-NST) is considered. This equation is commonly used to model the growth and spreading of biological species. According to the equivalent of the Riemann-Liouville(R-L) and Gr¨unwald-Letnikov(GL) fractional derivative definitions, an implicit difference approximation (IFDA) for the RFDE-NST is derived. We prove the IFDA is unconditionally stable and convergent. In order to evaluate the efficiency of the IFDA, a comparison with a fractional method of lines (FMOL) is used. Finally, two numerical examples are presented to show that the numerical results are in good agreement with our theoretical analysis.

Elliptic curves, group law, and efficient computation

This thesis is about the derivation of the addition law on an arbitrary elliptic curve and efficiently adding points on this elliptic curve using the derived addition law. The outcomes of this research guarantee practical speedups in higher level operations which depend on point additions. In particular, the contributions immediately find applications in cryptology. Mastered by the 19th century mathematicians, the study of the theory of elliptic curves has been active for decades. Elliptic curves over finite fields made their way into public key cryptography in late 1980’s with independent proposals by Miller [Mil86] and Koblitz [Kob87]. Elliptic Curve Cryptography (ECC), following Miller’s and Koblitz’s proposals, employs the group of rational points on an elliptic curve in building discrete logarithm based public key cryptosystems. Starting from late 1990’s, the emergence of the ECC market has boosted the research in computational aspects of elliptic curves. This thesis falls into this same area of research where the main aim is to speed up the additions of rational points on an arbitrary elliptic curve (over a field of large characteristic). The outcomes of this work can be used to speed up applications which are based on elliptic curves, including cryptographic applications in ECC. The aforementioned goals of this thesis are achieved in five main steps. As the first step, this thesis brings together several algebraic tools in order to derive the unique group law of an elliptic curve. This step also includes an investigation of recent computer algebra packages relating to their capabilities. Although the group law is unique, its evaluation can be performed using abundant (in fact infinitely many) formulae. As the second step, this thesis progresses the finding of the best formulae for efficient addition of points. In the third step, the group law is stated explicitly by handling all possible summands. The fourth step presents the algorithms to be used for efficient point additions. In the fifth and final step, optimized software implementations of the proposed algorithms are presented in order to show that theoretical speedups of step four can be practically obtained. In each of the five steps, this thesis focuses on five forms of elliptic curves over finite fields of large characteristic. A list of these forms and their defining equations are given as follows: (a) Short Weierstrass form, y2 = x3 + ax + b, (b) Extended Jacobi quartic form, y2 = dx4 + 2ax2 + 1, (c) Twisted Hessian form, ax3 + y3 + 1 = dxy, (d) Twisted Edwards form, ax2 + y2 = 1 + dx2y2, (e) Twisted Jacobi intersection form, bs2 + c2 = 1, as2 + d2 = 1, These forms are the most promising candidates for efficient computations and thus considered in this work. Nevertheless, the methods employed in this thesis are capable of handling arbitrary elliptic curves. From a high level point of view, the following outcomes are achieved in this thesis. - Related literature results are brought together and further revisited. For most of the cases several missed formulae, algorithms, and efficient point representations are discovered. - Analogies are made among all studied forms. For instance, it is shown that two sets of affine addition formulae are sufficient to cover all possible affine inputs as long as the output is also an affine point in any of these forms. In the literature, many special cases, especially interactions with points at infinity were omitted from discussion. This thesis handles all of the possibilities. - Several new point doubling/addition formulae and algorithms are introduced, which are more efficient than the existing alternatives in the literature. Most notably, the speed of extended Jacobi quartic, twisted Edwards, and Jacobi intersection forms are improved. New unified addition formulae are proposed for short Weierstrass form. New coordinate systems are studied for the first time. - An optimized implementation is developed using a combination of generic x86-64 assembly instructions and the plain C language. The practical advantages of the proposed algorithms are supported by computer experiments. - All formulae, presented in the body of this thesis, are checked for correctness using computer algebra scripts together with details on register allocations.

Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.

Decoupled image-based visual servoing for cameras obeying the unified projection model

This paper proposes a generic decoupled imagebased control scheme for cameras obeying the unified projection model. The scheme is based on the spherical projection model. Invariants to rotational motion are computed from this projection and used to control the translational degrees of freedom. Importantly we form invariants which decrease the sensitivity of the interaction matrix to object depth variation. Finally, the proposed results are validated with experiments using a classical perspective camera as well as a fisheye camera mounted on a 6-DOF robotic platform.

Lie group symmetry analysis for granular media stress equations

The Airy stress function, although frequently employed in classical linear elasticity, does not receive similar usage for granular media problems. For plane strain quasi-static deformations of a cohesionless Coulomb–Mohr granular solid, a single nonlinear partial differential equation is formulated for the Airy stress function by combining the equilibrium equations with the yield condition. This has certain advantages from the usual approach, in which two stress invariants and a stress angle are introduced, and a system of two partial differential equations is needed to describe the flow. In the present study, the symmetry analysis of differential equations is utilised for our single partial differential equation, and by computing an optimal system of one-dimensional Lie algebras, a complete set of group-invariant solutions is derived. By this it is meant that any group-invariant solution of the governing partial differential equation (provided it can be derived via the classical symmetries method) may be obtained as a member of this set by a suitable group transformation. For general values of the parameters (angle of internal friction and gravity g) it is found there are three distinct classes of solutions which correspond to granular flows considered previously in the literature. For the two limiting cases of high angle of internal friction and zero gravity, the governing partial differential equation admit larger families of Lie point symmetries, and from these symmetries, further solutions are derived, many of which are new. Furthermore, the majority of these solutions are exact, which is rare for granular flow, especially in the case of gravity driven flows.

Generalised finite volume strategies for simulating transport in strongly orthotropic porous media

In this work two different finite volume computational strategies for solving a representative two-dimensional diffusion equation in an orthotropic medium are considered. When the diffusivity tensor is treated as linear, this problem admits an analytic solution used for analysing the accuracy of the proposed numerical methods. In the first method, the gradient approximation techniques discussed by Jayantha and Turner [Numerical Heat Transfer, Part B: Fundamentals, 40, pp.367–390, 2001] are applied directly to the

Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

In this paper, we consider the variable-order Galilei advection diffusion equation with a nonlinear source term. A numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the equation. The stability and convergence of the numerical scheme are analyzed. Besides, another numerical scheme for improving temporal accuracy is also developed. Finally, some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis. Keywords: The variable-order Galilei invariant advection diffusion equation with a nonlinear source term; The variable-order Riemann–Liouville fractional partial derivative; Stability; Convergence; Numerical scheme improving temporal accuracy

Position, rotation, and scale invariant recognition of images using higher-order spectra

A new approach to recognition of images using invariant features based on higher-order spectra is presented. Higher-order spectra are translation invariant because translation produces linear phase shifts which cancel. Scale and amplification invariance are satisfied by the phase of the integral of a higher-order spectrum along a radial line in higher-order frequency space because the contour of integration maps onto itself and both the real and imaginary parts are affected equally by the transformation. Rotation invariance is introduced by deriving invariants from the Radon transform of the image and using the cyclic-shift invariance property of the discrete Fourier transform magnitude. Results on synthetic and actual images show isolated, compact clusters in feature space and high classification accuracies

Digit recognition using trispectral features

Features derived from the trispectra of DFT magnitude slices are used for multi-font digit recognition. These features are insensitive to translation, rotation, or scaling of the input. They are also robust to noise. Classification accuracy tests were conducted on a common data base of 256× 256 pixel bilevel images of digits in 9 fonts. Randomly rotated and translated noisy versions were used for training and testing. The results indicate that the trispectral features are better than moment invariants and affine moment invariants. They achieve a classification accuracy of 95% compared to about 81% for Hu's (1962) moment invariants and 39% for the Flusser and Suk (1994) affine moment invariants on the same data in the presence of 1% impulse noise using a 1-NN classifier. For comparison, a multilayer perceptron with no normalization for rotations and translations yields 34% accuracy on 16× 16 pixel low-pass filtered and decimated versions of the same data.

Effective hybrid recommendation combining users-searches correlations using tensors

Most recommendation methods employ item-item similarity measures or use ratings data to generate recommendations. These methods use traditional two dimensional models to find inter relationships between alike users and products. This paper proposes a novel recommendation method using the multi-dimensional model, tensor, to group similar users based on common search behaviour, and then finding associations within such groups for making effective inter group recommendations. Web log data is multi-dimensional data. Unlike vector based methods, tensors have the ability to highly correlate and find latent relationships between such similar instances, consisting of users and searches. Non redundant rules from such associations of user-searches are then used for making recommendations to the users.

Individual user behaviour modelling for effective web recommendation

With the growth of the Web, E-commerce activities are also becoming popular. Product recommendation is an effective way of marketing a product to potential customers. Based on a user’s previous searches, most recommendation methods employ two dimensional models to find relevant items. Such items are then recommended to a user. Further too many irrelevant recommendations worsen the information overload problem for a user. This happens because such models based on vectors and matrices are unable to find the latent relationships that exist between users and searches. Identifying user behaviour is a complex process, and usually involves comparing searches made by him. In most of the cases traditional vector and matrix based methods are used to find prominent features as searched by a user. In this research we employ tensors to find relevant features as searched by users. Such relevant features are then used for making recommendations. Evaluation on real datasets show the effectiveness of such recommendations over vector and matrix based methods.

Identifying interests of web users for effective recommendations

Search log data is multi dimensional data consisting of number of searches of multiple users with many searched parameters. This data can be used to identify a user’s interest in an item or object being searched. Identifying highest interests of a Web user from his search log data is a complex process. Based on a user’s previous searches, most recommendation methods employ two-dimensional models to find relevant items. Such items are then recommended to a user. Two-dimensional data models, when used to mine knowledge from such multi dimensional data may not be able to give good mappings of user and his searches. The major problem with such models is that they are unable to find the latent relationships that exist between different searched dimensions. In this research work, we utilize tensors to model the various searches made by a user. Such high dimensional data model is then used to extract the relationship between various dimensions, and find the prominent searched components. To achieve this, we have used popular tensor decomposition methods like PARAFAC, Tucker and HOSVD. All experiments and evaluation is done on real datasets, which clearly show the effectiveness of tensor models in finding prominent searched components in comparison to other widely used two-dimensional data models. Such top rated searched components are then given as recommendation to users.

User behaviour modelling in a multi-dimensional environment for personalization and recommendation

Handling information overload online, from the user's point of view is a big challenge, especially when the number of websites is growing rapidly due to growth in e-commerce and other related activities. Personalization based on user needs is the key to solving the problem of information overload. Personalization methods help in identifying relevant information, which may be liked by a user. User profile and object profile are the important elements of a personalization system. When creating user and object profiles, most of the existing methods adopt two-dimensional similarity methods based on vector or matrix models in order to find inter-user and inter-object similarity. Moreover, for recommending similar objects to users, personalization systems use the users-users, items-items and users-items similarity measures. In most cases similarity measures such as Euclidian, Manhattan, cosine and many others based on vector or matrix methods are used to find the similarities. Web logs are high-dimensional datasets, consisting of multiple users, multiple searches with many attributes to each. Two-dimensional data analysis methods may often overlook latent relationships that may exist between users and items. In contrast to other studies, this thesis utilises tensors, the high-dimensional data models, to build user and object profiles and to find the inter-relationships between users-users and users-items. To create an improved personalized Web system, this thesis proposes to build three types of profiles: individual user, group users and object profiles utilising decomposition factors of tensor data models. A hybrid recommendation approach utilising group profiles (forming the basis of a collaborative filtering method) and object profiles (forming the basis of a content-based method) in conjunction with individual user profiles (forming the basis of a model based approach) is proposed for making effective recommendations. A tensor-based clustering method is proposed that utilises the outcomes of popular tensor decomposition techniques such as PARAFAC, Tucker and HOSVD to group similar instances. An individual user profile, showing the user's highest interest, is represented by the top dimension values, extracted from the component matrix obtained after tensor decomposition. A group profile, showing similar users and their highest interest, is built by clustering similar users based on tensor decomposed values. A group profile is represented by the top association rules (containing various unique object combinations) that are derived from the searches made by the users of the cluster. An object profile is created to represent similar objects clustered on the basis of their similarity of features. Depending on the category of a user (known, anonymous or frequent visitor to the website), any of the profiles or their combinations is used for making personalized recommendations. A ranking algorithm is also proposed that utilizes the personalized information to order and rank the recommendations. The proposed methodology is evaluated on data collected from a real life car website. Empirical analysis confirms the effectiveness of recommendations made by the proposed approach over other collaborative filtering and content-based recommendation approaches based on two-dimensional data analysis methods.

Enriching XML documents clustering by using concise structure and content

With the growing number of XML documents on theWeb it becomes essential to effectively organise these XML documents in order to retrieve useful information from them. A possible solution is to apply clustering on the XML documents to discover knowledge that promotes effective data management, information retrieval and query processing. However, many issues arise in discovering knowledge from these types of semi-structured documents due to their heterogeneity and structural irregularity. Most of the existing research on clustering techniques focuses only on one feature of the XML documents, this being either their structure or their content due to scalability and complexity problems. The knowledge gained in the form of clusters based on the structure or the content is not suitable for reallife datasets. It therefore becomes essential to include both the structure and content of XML documents in order to improve the accuracy and meaning of the clustering solution. However, the inclusion of both these kinds of information in the clustering process results in a huge overhead for the underlying clustering algorithm because of the high dimensionality of the data. The overall objective of this thesis is to address these issues by: (1) proposing methods to utilise frequent pattern mining techniques to reduce the dimension; (2) developing models to effectively combine the structure and content of XML documents; and (3) utilising the proposed models in clustering. This research first determines the structural similarity in the form of frequent subtrees and then uses these frequent subtrees to represent the constrained content of the XML documents in order to determine the content similarity. A clustering framework with two types of models, implicit and explicit, is developed. The implicit model uses a Vector Space Model (VSM) to combine the structure and the content information. The explicit model uses a higher order model, namely a 3- order Tensor Space Model (TSM), to explicitly combine the structure and the content information. This thesis also proposes a novel incremental technique to decompose largesized tensor models to utilise the decomposed solution for clustering the XML documents. The proposed framework and its components were extensively evaluated on several real-life datasets exhibiting extreme characteristics to understand the usefulness of the proposed framework in real-life situations. Additionally, this research evaluates the outcome of the clustering process on the collection selection problem in the information retrieval on the Wikipedia dataset. The experimental results demonstrate that the proposed frequent pattern mining and clustering methods outperform the related state-of-the-art approaches. In particular, the proposed framework of utilising frequent structures for constraining the content shows an improvement in accuracy over content-only and structure-only clustering results. The scalability evaluation experiments conducted on large scaled datasets clearly show the strengths of the proposed methods over state-of-the-art methods. In particular, this thesis work contributes to effectively combining the structure and the content of XML documents for clustering, in order to improve the accuracy of the clustering solution. In addition, it also contributes by addressing the research gaps in frequent pattern mining to generate efficient and concise frequent subtrees with various node relationships that could be used in clustering.