Numerical reservoir simulation models are the basis for many decisions in regard to predicting, optimising, and improving production performance of oil and gas reservoirs. History matching is required to calibrate models to the dynamic behaviour of the reservoir, due to the existence of uncertainty in model parameters. Finally a set of history matched models are used for reservoir performance prediction and economic and risk assessment of different development scenarios. Various algorithms are employed to search and sample parameter space in history matching and uncertainty quantification problems. The algorithm choice and implementation, as done through a number of control parameters, have a significant impact on effectiveness and efficiency of the algorithm and thus, the quality of results and the speed of the process. This thesis is concerned with investigation, development, and implementation of improved and adaptive algorithms for reservoir history matching and uncertainty quantification problems. A set of evolutionary algorithms are considered and applied to history matching. The shared characteristic of applied algorithms is adaptation by balancing exploration and exploitation of the search space, which can lead to improved convergence and diversity. This includes the use of estimation of distribution algorithms, which implicitly adapt their search mechanism to the characteristics of the problem. Hybridising them with genetic algorithms, multiobjective sorting algorithms, and real-coded, multi-model and multivariate Gaussian-based models can help these algorithms to adapt even more and improve their performance. Finally diversity measures are used to develop an explicit, adaptive algorithm and control the algorithm’s performance, based on the structure of the problem. Uncertainty quantification in a Bayesian framework can be carried out by resampling of the search space using Markov chain Monte-Carlo sampling algorithms. Common critiques of these are low efficiency and their need for control parameter tuning. A Metropolis-Hastings sampling algorithm with an adaptive multivariate Gaussian proposal distribution and a K-nearest neighbour approximation has been developed and applied.

In 1984 Zilber conjectured that any strongly minimal structure is geometrically equivalent to one of the following types of strongly minimal structures, in the appropriate language: Pure sets, Vector Spaces over a fixed Division Ring and Algebraically Closed Fields. In 1993, in his article `A new strongly minimal set' Hrushovski produced a family of counterexamples to Zilber's conjecture. His method consists in two steps. Firstly he builds a `limit' structure from a suitable class of finite structures in a language consisting only of a ternary relational symbol. Secondly, in a step called the collapse, he defines a continuum of subclasses such that the corresponding `limit' structures are new strongly minimal structures. These new strongly minimal structures are non isomorphic but Hrushovski then asks if they are geometrically equivalent. We first analyze the pregeometries arising from different variations of the construction before the collapse. In particular we prove that if we repeat the construction starting with an n-ary relational symbol instead of a 3-ary relational symbol, then the pregeometries associated to the corresponding `limit' structures are not locally isomorphic when we vary the arity. Second we prove that these new strongly minimal structures are geometrically equivalent. In fact we prove that their geometries are isomorphic to the geometry of the `limit' structure obtained before the collapse.

This thesis develops the theory of Ordinal Time Turing Machines (OTTMs) and explores connections between this theory, inner model theory, α-recursion and Turing computation. We first provide a rigorous definition of an OTTM. We define how such machines may be taken to operate on sets, we prove that the class of OTTMs has a universal machine, we prove that the class 0 of OTTM computable sets is equal to L, we prove an analogue of the condensation lemma and we prove that the Generalized Continuum hypothesis & ◊ωl hold in L using lemmas concerning OTTMs. We also define several variants of computer limited to α time. We expose weaknesses in all bar one of the variants (uniform-α-computation) and then we use this remaining variant to develop a degree theory. Vie show this theory is isomorphic to the theory of α-recursion, we show that α-recursion is not equivalent to α-computation and we give a proof of a form of the SACKS SIMPSON theorem stated for the uniform-α-computer. We then prove results about halting computations, universal and metaversal programs ('metaversal program' is defined in this thesis) for the uniform-α-computer. We define a (B,α)-computer which is closely related to inner model theory. We prove an analogue of the Density theorem for the (B,α)-computer and find a metaversal program for the (B,α)-computer. Finally we compare the α-computers and OTTMs with Turing machines. The introduction consists entirely of pre-existing results and definitions which provide a necessary background for the rest of the thesis. The proof of SACKS-SIMPSON for uniform-a-computers is adapted from Benjamin Seyfferth's proof for non-uniform-a-computers. The Density theorem for α-recursion of Sacks is modified and adapted for uniform-(B,α)-computers. All other results are entirely my own work unless otherwise stated.

In the present research work, a three-dimensional Cellular Automata Finite Element (CAFE) multi-scale model was developed to simulate, ductile fracture, cleavage and the ductile-brittle transition in a structural steel. For the simulation of the ductile-brittle fracture, at least two Cellular Automata arrays are needed, one to represent the ductile material properties and the other one to account for the brittle fracture process. The cell sizes in both arrays are independent of each other and of the finite element size. The cell sizes in each Cellular Automata array are related to the microstructural process of each fracture mechanism. The finite elements size is chosen to represent the macro strain gradients accurately. The model was implemented through the user define material behavior subroutine VUMAT in the finite element program ABAQUS Explicit Version 5.6. In the CAFE model, the material information is moved from the structural response of finite elements and stored in the appropriated number of Cellular Automata (CA) arrays. In the present CAFE model, the Rousselier ductile damage model was applied to each ductile cell. The critical value of the maximum principal stress was used to assess the failure of each brittle cell. In the brittle CA arrays, four different cleavage fracture nucleation micromechanisms, found experimentally at te.st temperatures down to -196øC in a ferritic-pearlitic Grade A ship plate steel were included in the model. This was done in order to simulate the real microfeatures nucleating cleavage in ferritic steels. In this model, the physical damage parameters of the ductile and brittle parts were calibrated separately. After calibration the CAFE model simulated the experimentally measured distribution of brittle microcracks generated in the notch region of blunt four point double-notch bend tests performed at test temperatures from 25øC to -196øC. The ductile part of the CAFE model was calibrated with the simulation of tensile and impact Charpy tests performed at room temperature. Subsequently the model was applied to simulate the ductile-brittle transition of Grade A ship plate steel. When numerical against experimental data was obtained, the parameters were considered true material model parameters of the steel under analysis.

There has recently been a significant increase in academic interest in the field oftype-2 fuzzy sets and systems. Type-2 fuzzy systems offer the ability to model and reason with uncertain concepts. When faced with uncertainties type-2 fuzzy systems should, theoretically, give an increase in performance over type-l fuzzy systems. However, the computational complexity of generalised type-2 fuzzy systems is significantly higher than type-l systems. A direct consequence of this is that, prior to this thesis, generalised type-2 fuzzy logic has not yet been applied in a time critical domain, such as control. Control applications are the main application area of type-l fuzzy systems with the literature reporting many successes in this area. Clearly the computational complexity oftype-2 fuzzy logic is holding the field back. This restriction on the development oftype-2 fuzzy systems is tackled in this research. This thesis presents the novel approach ofdefining fuzzy sets as geometric objects - geometric fuzzy sets. The logical operations for geometric fuzzy sets are defined as geometric manipulations of these sets. This novel geometric approach is applied to type-I, type-2 interval and generalised type-2 fuzzy sets and systems. The major contribution of this research is the reduction in the computational complexity oftype-2 fuzzy logic that results from the application of the geometric approach. This reduction in computational complexity is so substantial that generalised type-2 fuzzy logic has, for the first time, been successfully applied to a control problem - mobile robot navigation. A detailed comparison between the performance of the generalised type-2 fuzzy controller and the performance of the type-l and type-2 interval controllers is given. The results indicate that the generalised type-2 fuzzy logic controller outperforms the other robot controllers. This outcome suggests that generalised type-2 fuzzy systems can offer an improved performance over type-l and type-2 interval systems.

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