Partial di erential equations (PDEs) are important tools in modeling complex phenomena, and they arise in many physics and engineering applications. Due to the uncertainty in the input data, stochastic partial di erential equations (SPDEs) have become popular as a modelling tool in the last century. As the exact solutions are unknown, developing e cient numerical methods for simulating PDEs and SPDEs is a very important while challenging research topic. In this thesis we develop e cient numerical schemes for deterministic and stochastic porous media ows. More schemes are based on the computing of the matrix exponential functions of the non diagonal matrices, we use new e cient techniques: the real fast L eja points and the Krylov subspace techniques. For the deterministic ow and transport problem, we consider two deterministic exponential integrator schemes: the exponential time di erential stepping of order one (ETD1) and the exponential Euler midpoint (EEM) with nite volume method for discretization in space. We give the time and space convergence proof for the ETD1 scheme and illustrate with simulations in two and three dimensions that the exponential integrators are e - cient and accurate for advection dominated deterministic transport ow in heterogeneous anisotropic porous media compared to standard semi implicit and implicit schemes. For the stochastic ow and transport problem, we consider the general parabolic SPDEs in a Hilbert space, using the nite element method for discretization in space (although nite di erence or nite volume can be used as well). We use a linear functional of the noise and the standard Brownian increments to develop and give convergence proofs of three new e cient and accurate schemes for additive noise, one called the modi ed semi{ implicit Euler-Maruyama scheme and two stochastic exponential integrator schemes, and two stochastic exponential integrator schemes for multiplicative and additive noise. The schemes are applied to two dimensional ow and transport.

The aim of this thesis is to develop methodology for combining multiple endpoints within a single statistical analysis that compares the responses of patients treated with a novel treatment with those of control patients treated conventionally. The focus is on interval-censored bivariate survival data, and five real data sets from previous studies concerning multiple responses are used to illustrate the techniques developed. The background to survival analysis is introduced by a general description of survival data, and an overview of existing methods and underlying models is included. A review is given of two of the most popular survival analysis methods, namely the logrank test and Cox's proportional hazards model. The global score test methodology for combining multiple end points is described in detail, and application to real data demonstrates its benefits. The correlation between two score statistics arising from bivariate interval-censored survival data is the core of this research. The global score test methodology is extended to the case of bivariate interval-censored survival data and a complementary log-log link is applied to derive the covariance and the correlation between the two score statistics. A number of common scenarios are considered in this investigation and the accuracy of the estimator is evaluated by means of extensive simulations. An established method, namely the approach of Wei, Lin and Weissfeld, is examined and compared with the proposed method using both real and simulated data. It is concluded that our method is accurate, consistent and comparable to the competitor. This study marked the first successful development of the global score test methodology for bivariate survival data, employing a new approach to the derivation of the covariance between two score statistics on the basis of an interval-censored model. Additionally. the relationship between the jackknife technique and the Wei, Lin and Weissfeld method has been clarified.

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We are concerned in this thesis with the problem of how to extend standard methods of approximating analytic functions on an interval, say [0, 1], to representing with exponential accuracy the more general class of functions analytic on (0,1) and continuous on [0,1]. The particular approach we take is based upon using an exponential change of variables to transform (0, 1) to (-∞, ∞) thus in the process mapping any endpoint singularities to infinity. Under mild hypotheses, the resulting function is analytic on the real line and may therefore be approximated to exponential accuracy by a spectral interpolant. These ideas form the foundations of an established class of techniques referred to as sine methods. Vie begin by providing a thorough review of sine methods which takes the form of a description of a computational software package called Sincfun. Our experiences lead us to two key conclusions: (i) it is possible to use other bases instead of sine functions; and (ii) such methods in general seem to lead to highly inefficient representations, particularly for oscillatory functions. Regarding (i), we first set out a new convergence theory for variable transformation methods based on Chebyshev polynomials. We show that, 'when a function only has singularity at only one endpoint, mapping to a semi-infinite interval rather than an infinite one can lead to a much improved exponential rate of convergence, from C-√n to c-n2/3 We also perform a thorough analysis of related numerical methods based on so-called double-exponential transforms. Regarding (ii), we first quantify the inefficiency of exponential transform methods, showing that they are suboptimal in a certain precisely defined sense. We then proceed to derive a new' class of transforms which map conformally from an infinite strip to an infinite slit strip. The resulting numerical schemes are demonstrated to have highly desirable properties with regards to the resolution of oscillatory functions . Finally, we derive new quadrature rules, methods for indefinite integration, and spectral methods for application to singular boundary value problems. All of these techniques are based around our new approach of 'variable transformation used in conjunction with Chebyshev interpolation.

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In Chapter 2, I apply polyhedral theory to prove easily that the set of separable states is not a polytope; for the sake of completeness, I then review the role of polytopes in nonlocality. Next, I give a novel treatment of entanglement witnesses and define a new class of entanglement witnesses, which may prove to be useful beyond the examples given. In the last section, I briefly review the five basic convex body problems given in [1] (Groetschel et al., 1988), and their application to the quantum separability problem. In Chapter 3, I treat the separability problem as a computational decision problem and motivate its approximate formulations. After a review of basic complexity-theoretic notions, I discuss the computational complexity of the separability problems: I discuss the issue of NP-completeness, giving an alternative definition of the separability problem as an NP-hard problem in NP. I finish the chapter with a comprehensive survey of deterministic algorithm solutions to the separability problem, including one that follows from a second NP formulation. Chapters 1 and 3 motivate a new interior-point algorithm which, given the expected values of a subset of an orthogonal basis of observables of an otherwise unknown quantum state, searches for an entanglement witnesses in the span of the subset of observables. When all the expected values are known, the algorithm solves the separability problem. In Chapter 4, I give the motivation for the algorithm and show how it can be used in a particular physical scenario to detect entanglement (or decade separability) of an unknown quantum state using as few quantum resources as possible. I then explain the intuitive idea behind the algorithm and relate it to the standard algorithms of its kind. I end the chapter with a comparison of the complexities of the algorithms surveyed in Chapter 3. Finally, in Chapter 5, I present the details of the algorithm and discuss its performance relative to standard methods.

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