A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2014. / A multi-dimensional data model provides a good conceptual view of the data in data warehousing and On-Line Analytical Processing (OLAP). A typical representation of such a data model is as a multi-dimensional array which is well suited when the array is dense. If the array is sparse, i.e., has a few number of non-zero elements relative to the product of the cardinalities of the dimensions, using a multi-dimensional array to represent the data set requires extremely large memory space while the actual data elements occupy a relatively small fraction of the space. Existing storage schemes for Multi-Dimensional Sparse Arrays (MDSAs) of higher dimensions k (k > 2), focus on optimizing the storage utilization, and offer little flexibility in data access efficiency. Most efficient storage schemes for sparse arrays are limited to matrices that are arrays in 2 dimensions. In this dissertation, we introduce four storage schemes for MDSAs that handle the sparsity of the array with two primary goals; reducing the storage overhead and maintaining efficient data element access. These schemes, including a well known method referred to as the Bit Encoded Sparse Storage (BESS), were evaluated and compared on four basic array operations, namely construction of a scheme, large scale random element access, sub-array retrieval and multi-dimensional aggregation. The four storage schemes being proposed, together with the evaluation results are: i.) The extended compressed row storage (xCRS) which extends CRS method for sparse matrix storage to sparse arrays of higher dimensions and achieves the best data element access efficiency among the methods compared; ii.) The bit encoded xCRS (BxCRS) which optimizes the storage utilization of xCRS by applying data compression methods with run length encoding, while maintaining its data access efficiency; iii.) A hybrid approach (Hybrid) that provides the best control of the balance between the storage utilization and data manipulation efficiency by combining xCRS and BESS. iv.) The PATRICIA trie compressed storage (PTCS) which uses PATRICIA trie to store the valid non-zero array elements. PTCS supports efficient data access, and has a unique property of supporting update operations conveniently. v.) BESS performs the best for the multi-dimensional aggregation, closely followed by the other schemes. We also addressed the problem of accelerating some selected array operations using General Purpose Computing on Graphics Processing Unit (GPGPU). The experimental results showed different levels of speed up, ranging from 2 to over 20 times, on large scale random element access and sub-array retrieval. In particular, we utilized GPUs on the computation of the cube operator, a special case of multi-dimensional aggregation, using BESS. This resulted in a 5 to 8 times of speed up compared with our CPU only implementation. The main contributions of this dissertation include the developments, implementations and evaluations of four efficient schemes to store multi-dimensional sparse arrays, as well as utilizing massive parallelism of GPUs for some data warehousing operations.

A research report submitted to the Faculty of Engineering and Built Environment, University of the Witwatersrand, Johannesburg, South Africa, in partial fulfilment of the requirements for the Degree of Master of Science in Engineering, November 2018 / The main aim of the study was to develop mathematical flow model of the shale gas at the Karoo Basin of South Africa (SA). The model development incorporates three systems (phases) to form a triple continuum flow model, the phases include matrix (m), natural (NF) and hydraulic fracture (HF). The model was developed from the continuity equation, and the general equations were formed. (0.05������ ���� = 3.90087 × 10−15 ��2���� ����2 + 3.90087 × 10−15 ��2���� ����2 − 1.95043 × 10−16(20 × 106 − ������), 0.01 �������� ���� = 2.00 × 10−15(20 × 106 − ������) − 2.00 × 10−9(20 × 106 − ������) + �� ���� [7.80 × 10−5 �������� ���� ] + �� ���� [7.80 × 10−5 �������� ���� ] �� ���� [0.1248269 �������� ���� ] + 0.1248269(20 × 106 − ������)− 4.98 × 10−4 = �������� ���� The model was solved using numerical method technique known as Finite Difference Method (FDM). For each phase a computer program MATLAB was used to plot the pressure gradient. Hydraulic pressure gradient fractures propagate between the distance of 100m and 500m. The model was verified using the data of Barnett Shale. Sensitivity analysis was also performed on the hydraulic permeability, drainage radius and the initial pressure of the reservoir. / XL2019

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2014. / Symmetries and conservation laws of partial di erential equations (pdes) have been instrumental in giving new approaches for reducing pdes. In this dissertation, we study the symmetries and conservation laws of the two-dimensional Schr odingertype equation and the Benney-Luke equation, we use these quantities in the Double Reduction method which is used as a way to reduce the equations into a workable pdes or even an ordinary di erential equations. The symmetries, conservation laws and multipliers will be determined though di erent approaches. Some of the reductions of the Schr odinger equation produced some famous di erential equations that have been dealt with in detail in many texts.

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We present an abstract framework for state space formulation and a generalized theorem on well-posedness which can be applied to a class of partial functional differential equations which arise in the modeling of viscoelastic and certain thermo-viscoelastic systems. Examples to which the theory applies include both second- and fourth-order equations with a variety of boundary conditions. The theory presented here allows for singular kernels as well as flexibility in the choice of state space. We discuss an approximation scheme using spline in the spatial variable and an averaging scheme in the delay variable. We compare a uniform mesh to a nonuniform mesh and give numerical results which indicate that the non-uniform mesh, which gives a better approximation of the kernel near the singularity, yields faster convergence. We give a proof of convergence of the simulation problem for singular kernels and of the control problem for bounded kernels. We use techniques of semigroup theory to establish the results on well-posedness and convergence. / Ph. D.

New numerical methods are proposed for computing with smooth scalar and vector valued functions of two variables defined on rectangular domains. Functions are approximated to essentially machine precision by an iterative variant of Gaussian elimination that constructs near-optimal low rank approximations. Operations such as integration, differentiation, and function evaluation are particularly efficient. Explicit convergence rates are shown for the singular values of differentiable and separately analytic functions, and examples are given to demonstrate some paradoxical features of low rank approximation theory. Analogues of QR, LU, and Cholesky factorizations are introduced for matrices that are continuous in one or both directions, deriving a continuous linear algebra. New notions of triangular structures are proposed and the convergence of the infinite series associated with these factorizations is proved under certain smoothness assumptions. A robust numerical bivariate rootfinder is developed for computing the common zeros of two smooth functions via a resultant method. Using several specialized techniques the algorithm can accurately find the simple common zeros of two functions with polynomial approximants of high degree (≥ 1,000). Lastly, low rank ideas are extended to linear partial differential equations (PDEs) with variable coefficients defined on rectangles. When these ideas are used in conjunction with a new one-dimensional spectral method the resulting solver is spectrally accurate and efficient, requiring O(n²) operations for rank $1$ partial differential operators, O(n³) for rank 2, and O(n⁴) for rank &geq,3 to compute an n x n matrix of bivariate Chebyshev expansion coefficients for the PDE solution. The algorithms in this thesis are realized in a software package called Chebfun2, which is an integrated two-dimensional component of Chebfun.

published_or_final_version / abstract / Mechanical Engineering / Doctoral / Doctor of Philosophy

In this thesis, we consider two problems in the Q-tensor theory of nematic liquid crystals. The first concerns eigenvalue constraints on the Q-tensor order parameter. In particular, by employing a singular potential constructed by Ball and Majumdar, we consider the existence, regularity and "strict physicality" of weak solutions to the Beris-Edwards equations of nemato-hydrodynamics. In the second part of the thesis, we consider a gradient flow of the well-studied Landau-de Gennes energy. We prove some rigorous results on the average long-time statistical behaviour of its solutions, which are in agreement with experimental observations in the condensed matter physics literature.

We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.