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We consider the application of multilevel Monte Carlo methods to elliptic partial differential equations with random coefficients. Such equations arise, for example, in stochastic groundwater ow modelling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, lack uniform coercivity and boundedness with respect to the random parameter and have only limited spatial regularity. To give a rigorous bound on the cost of the multilevel Monte Carlo estimator to reach a desired accuracy, one needs to quantify the bias of the estimator. The bias, in this case, is the spatial discretisation error in the numerical solution of the partial differential equation. This thesis is concerned with establishing bounds on this discretisation error in the practically relevant and technically demanding case of coefficients which are not uniformly coercive or bounded with respect to the random parameter. Under mild assumptions on the regularity of the coefficient, we establish new results on the regularity of the solution for a variety of model problems. The most general case is that of a coefficient which is piecewise Hölder continuous with respect to a random partitioning of the domain. The established regularity of the solution is then combined with tools from classical discretisation error analysis to provide a full convergence analysis of the bias of the multilevel estimator for finite element and finite volume spatial discretisations. Our analysis covers as quantities of interest several spatial norms of the solution, as well as point evaluations of the solution and its gradient and any continuously Fréchet differentiable functional. Lastly, we extend the idea of multilevel Monte Carlo estimators to the framework of Markov chain Monte Carlo simulations. We develop a new multilevel version of a Metropolis Hastings algorithm, and provide a full convergence analysis.

Plank (tongue and groove) wood decking is a product that is commonly used in post and beam timber construction to transfer gravity loads on roofs and floors. In 2010, The National Building Code of Canada changed the application area of the specified concentrated roof live loads from 750 mm x 750 mm to 200 mm x 200 mm. The change was made to better reflect the area which a construction worker with equipment occupies. Preliminary analysis showed that the change in the application area of concentrated loads may have a significant impact on the design of decking systems. Little research or development has been done on plank decking since the 1950’s and 1960’s. An experimental program was undertaken at the University of Ottawa’s structural laboratory to better understand the behaviour of plank decking under uniformly distributed and concentrated loads. Non-destructive and destructive tests were conducted on plank decking systems to investigate their stiffness and failure mode characteristics under uniformly distributed as well as concentrated loads. The experimental test program was complimented with a detailed finite element model in order to predict the behaviour of a plank decking system, especially the force transfer between decks through the tongue and groove joint. The study showed that the published deflection coefficients for uniformly distributed loads can accurately predict the three types of decking layup patterns specified in the Canadian Design Standard (CSA O86, 2009). For unbalanced uniformly distributed loads on two-span continuous layup, it was found that the deflection coefficient of 0.42 was non-conservative. It was also found that under concentrated loads, the stiffness of the decking system increased significantly as more boards were added. A deflection coefficient of 0.40 is appropriate to calculate the deflection for the three types of decking layup patterns specified in the Canadian Design Standard (CSA O86, 2009) under concentrated load on an area of 200 mm by 200 mm. Significant load sharing was observed for plank decking under concentrated loads. An increase in capacity of about 1.5 to 2.5 times the capacity of the loaded boards was found. Furthermore, it was found that placing sheathing on top of a decking system had a significant effect in the case of concentrated load with an increase of over 50% in stiffness and over 100% in ultimate capacity.

Plank (tongue and groove) wood decking is a product that is commonly used in post and beam timber construction to transfer gravity loads on roofs and floors. In 2010, The National Building Code of Canada changed the application area of the specified concentrated roof live loads from 750 mm x 750 mm to 200 mm x 200 mm. The change was made to better reflect the area which a construction worker with equipment occupies. Preliminary analysis showed that the change in the application area of concentrated loads may have a significant impact on the design of decking systems. Little research or development has been done on plank decking since the 1950’s and 1960’s. An experimental program was undertaken at the University of Ottawa’s structural laboratory to better understand the behaviour of plank decking under uniformly distributed and concentrated loads. Non-destructive and destructive tests were conducted on plank decking systems to investigate their stiffness and failure mode characteristics under uniformly distributed as well as concentrated loads. The experimental test program was complimented with a detailed finite element model in order to predict the behaviour of a plank decking system, especially the force transfer between decks through the tongue and groove joint. The study showed that the published deflection coefficients for uniformly distributed loads can accurately predict the three types of decking layup patterns specified in the Canadian Design Standard (CSA O86, 2009). For unbalanced uniformly distributed loads on two-span continuous layup, it was found that the deflection coefficient of 0.42 was non-conservative. It was also found that under concentrated loads, the stiffness of the decking system increased significantly as more boards were added. A deflection coefficient of 0.40 is appropriate to calculate the deflection for the three types of decking layup patterns specified in the Canadian Design Standard (CSA O86, 2009) under concentrated load on an area of 200 mm by 200 mm. Significant load sharing was observed for plank decking under concentrated loads. An increase in capacity of about 1.5 to 2.5 times the capacity of the loaded boards was found. Furthermore, it was found that placing sheathing on top of a decking system had a significant effect in the case of concentrated load with an increase of over 50% in stiffness and over 100% in ultimate capacity.

A turbulent boundary layer (TBL), generated in a water tunnel, extended to a highly turbulent and anisotropic “free stream” that consisted of a uniformly sheared flow (USF) with a mean shear that was in the opposite direction to that in the TBL. Extensive measurements of the fluctuating velocity were taken with the use of hot-film anemometry, laser Doppler velocimetry and particle image velocimetry. On either side of the TBL edge, defined as the location of maximum velocity, the turbulence relaxed to its canonical structures in TBL and USF, respectively, but, in the vicinity of the edge, the turbulence was multi-structure and exhibited strong departures from canonical behaviour. Of particular interest was the variation of the dissipation parameter, which, in contrast to its near-constancy in well-developed canonical flows, varied inversely proportionally to the turbulence Reynolds number. The entire flow contained horseshoe-shaped coherent structures, whose properties, however, varied from the TBL, across the multi-structure region and into the USF.

This thesis consists of five articles mainly devoted to problems in dynamical systems and ergodic theory. We consider non-uniformly hyperbolic two dimensional systems and limit distributions of point measures which are absolutely continuous with respect to the Lebesgue measure. Let $f_{a_0}(x)=a_0-x^2$ be a quadratic map where the parameter $a_0\in(1,2)$ is chosen such that the critical point $0$ is pre-periodic (but not periodic). In Papers A and B we study skew-products $(\th,x)\mapsto F(\th,x)=(g(\th),f_{a_0}(x)+\al s(\th))$, $(\th,x)\in S^1\times\real$. The functions $g:S^1\to S^1$ and $s:S^1\to[-1,1]$ are the base dynamics and the coupling functions, respectively, and $\al$ is a small, positive constant. Such quadratic skew-products are also called Viana maps. In Papers A and B we show for several choices of the base dynamics and the coupling function that the map $F$ has two positive Lyapunov exponents and for some cases we further show that $F$ admits also an absolutely continuous invariant probability measure. In Paper C we consider certain Bernoulli convolutions. By showing that a specific transversality property is satisfied, we deduce absolute continuity of the to these Bernoulli convolutions associated distributions. In Papers D and E we consider sequences of real numbers in the unit interval and study how they are distributed. The sequences in Paper D are given by the forward iterations of a point $x\in[0,1]$ under a piecewise expanding map $T_a:[0,1]\to[0,1]$ depending on a parameter $a$ contained in an interval $I$. Under the assumption that each $T_a$ admits a unique absolutely continuous invariant probability measure $\mu_a$ and that some technical conditions are satisfied, we show that the distribution of the forward orbit $T_a^j(x)$, $j\ge1$, is described by the distribution $\mu_a$ for Lebesgue almost every parameter $a\in I$. In Paper E we apply the ideas in Paper D to certain sequences which are equidistributed in the unit interval and give a geometrical proof of an old result by Koksma.

在行列設計(row-column design)的架構下，當行總和與列總和皆為總試驗處理數的倍數時，我們考慮行效果與列效果的相互比較之最適性。延續Shah和Sinha(1993)的結果，在給定行總和及列總和的情況下，我們導出達成齊一最適設計(uniformly optimal design)的充分條件。此外，當總實驗單位固定時，達成全域最適設計(universally optimal design)的充分條件亦被求出。我們同時列舉許多相關的設計排列法。 / We consider the problem of comparing row effects and column effects in the row-column design setup when the row sizes and column sizes are all multiples of the number of treatments. Following the work of Shah and Sinha (1993), we derive a sufficient condition for uniformly optimal designs for given values of the row sizes and column sizes. We also derive a sufficient condition for universally optimal designs when the total number of experimental units is fixed. Several examples of designs with high efficiencies are provided.

The transformed rejection method, a combination of the inversion and the rejection method, which is used to generate non-uniform random numbers from a variety of continuous distributions can be applied to discrete distributions as well. For the Poisson distribution a short and simple algorithm is obtained which is well suited for large values of the Poisson parameter $\mu$, even when $\mu$ may vary from call to call. The average number of uniform deviates required is lower than for any of the known uniformly fast algorithms. Timings for a C implementation show that the algorithm needs only half of the code but is - for $\mu$ not too small - at least as fast as the current state-of-the-art algorithms. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Synchrotron light source are accelerating research and development and fueling innovation in a wide range of research disciplines and industries worldwide. The third-generation synchrotron radiation facilities such as Advanced Photon Source (APS), produce ultra-brilliant x-rays using insertion devices consisting mainly of undulators, which provide exciting opportunities for advanced research into materials, earth science, life science, and medicine. Using high brightness x-ray radiation with high spatial coherence, unique coherence-based experiments are now becoming possible: coherence imaging techniques such as phase contrast imaging, holography, and tomography, are under intensive development, opening up a range of new areas of investigation. At the same time some useful optical elements used in the synchrotron radiation system have been created rapidly. Crucial to the development of all these fields is some knowledge of the spatial coherence of the light produced by these sources. In other words, the characterization of spatial coherence is a high priority. / The aim of this project is to develop a theoretical and experimental program to allow the measurement of the spatial coherence of synchrotron radiation. A technique to measure the spatial coherence of x-rays from undulators is presented. The essence of the coherence measurement technique is based on the interpretation of a complex diffraction pattern. We measure the spatial coherence function of a 7.9 keV x-ray beam from an undulator at a third-generation synchrotron (APS) using a sophisticated diffracting aperture known as a Uniformly Redundant Array (URA). The URA was also used to measure the spatial coherence function for soft x-rays at the APS. When a traditional Young’s double-slit experiment is used to test the degree of coherence, the separations of the two-slit have to be changed repeatedly to full map the spatial coherence function. The URA is a complex aperture consisting of many slits, (or, for a two-dimensional array, pinholes), organized such that all possible slit separations occur, and do so with exactly the same frequency. One might regard the URA as able to simultaneously perform many Young’s experiments a precisely equal number of times across the full range of slit separations permitted by the overall size of the URA. Therefore one experiment using a one-dimensional (1D) URA can perform the equivalent of multiple double-slit experiments. The diffraction theory developed in this thesis a convenient theoretical basis for interpreting this diffraction pattern.

Submitted by Automa??o e Estat?stica (sst@bczm.ufrn.br) on 2016-08-09T22:42:29Z No. of bitstreams: 1 FlaullesBooneBergamaschi_TESE.pdf: 610639 bytes, checksum: 58e4c11131b6241c88c80e6a1a2f579d (MD5) / Approved for entry into archive by Arlan Eloi Leite Silva (eloihistoriador@yahoo.com.br) on 2016-08-10T22:02:43Z (GMT) No. of bitstreams: 1 FlaullesBooneBergamaschi_TESE.pdf: 610639 bytes, checksum: 58e4c11131b6241c88c80e6a1a2f579d (MD5) / Made available in DSpace on 2016-08-10T22:02:43Z (GMT). No. of bitstreams: 1 FlaullesBooneBergamaschi_TESE.pdf: 610639 bytes, checksum: 58e4c11131b6241c88c80e6a1a2f579d (MD5) Previous issue date: 2015-12-11 / O principal objetivo deste trabalho ? propor a no??o de primalidade forte e uniformemente forte no ambiente fuzzy. Assim, ? apresentado um conceito de ideal uniformemente primo independente de alpha-cortes, mas de certa forma compat?vel. Tamb?m ? investigada a primalidade forte nos quantales, propondo uma defini??o de ideal primo e uniformemente primo coerente com o ambiente n?o-comutativo que serve de base para a investiga??o em fuzzy. Como ferramenta adicional, este trabalho introduz o conceito de t/m sistemas para o ambiente fuzzy sugerindo um caminho alternativo para lidar com primalidade. Por fim, ? desenvolvido uma vers?o fuzzy do teorema da correspond?ncia e no??es iniciais sobre o radical de um ideal fuzzy. / The main aim of this investigation is to propose the notion of uniform and strong primeness in fuzzy environment. First, it is proposed and investigated the concept of fuzzy strongly prime and fuzzy uniformly strongly prime ideal. As an additional tool, the concept of t/m systems for fuzzy environment gives an alternative way to deal with primeness in fuzzy. Second, a fuzzy version of correspondence theorem and the radical of a fuzzy ideal are proposed. Finally, it is proposed a new concept of prime ideal for Quantales which enable us to deal with primeness in a noncommutative setting.