Stochastic collocation on unstructured multivariate meshes

Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification (UQ) community. Techniques for least-squares regularization, compressive sampling recovery, and interpolatory reconstruction are becoming standard tools used in a variety of applications. Selection of a collocation mesh is frequently a challenge, but methods that construct geometrically "unstructured" collocation meshes have shown great potential due to attractive theoretical properties and direct, simple generation and implementation. We investigate properties of these meshes, presenting stability and accuracy results that can be used as guides for generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure

The Spin-Resolved Atomic Velocity Distribution and 21-cm Line Profile of Dark-Age Gas

The 21-cm hyperfine line of atomic hydrogen (HI) is a promising probe of the cosmic dark ages. In past treatments of 21-cm radiation it was assumed the hyperfine level populations of HI could be characterized by a velocity-independent ``spin temperature'' T_s determined by a competition between 21-cm radiative transitions, spin-changing collisions, and (at lower redshifts) Lyman-alpha scattering. However we show here that, if the collisional time is comparable to the radiative time, the spin temperature will depend on atomic velocity, T_s=T_s(v), and one must replace the usual hyperfine level rate equations with a Boltzmann equation describing the spin and velocity dependence of the HI distribution function. We construct here the Boltzmann equation relevant to the cosmic dark ages and solve it using a basis-function method. Accounting for the actual spin-resolved atomic velocity distribution results in up to a 2 per cent suppression of the 21-cm emissivity, and a redshift and angular-projection dependent suppression or enhancement of the linear power spectrum of 21-cm fluctuations of up to 5 per cent. The effect on the 21-cm line profile is more dramatic -- its full-width at half maximum (FWHM) can be enhanced by up to 60 per cent relative to the velocity-independent calculation. We discuss the implications for 21-cm tomography of the dark ages.Comment: 25 pages, 6 figures, submitted to Mon. Not. Roy. Astron. So

On Novel Approaches to Model-Based Structural Health Monitoring

Structural health monitoring (SHM) strategies have classically fallen into two main categories of approach: model-driven and data-driven methods. The former utilises physics-based models and inverse techniques as a method for inferring the health state of a structure from changes to updated parameters; hence defined as inverse model-driven approaches. The other frames SHM within a statistical pattern recognition paradigm. These methods require no physical modelling, instead inferring relationships between data and health states directly. Although successes with both approaches have been made, they both suffer from significant drawbacks, namely parameter estimation and interpretation difficulties within the inverse model-driven framework, and a lack of available full-system damage state data for data-driven techniques. Consequently, this thesis seeks to outline and develop a framework for an alternative category of approach; forward model-driven SHM. This class of strategies utilise calibrated physics-based models, in a forward manner, to generate health state data (i.e. the undamaged condition and damage states of interest) for training machine learning or pattern recognition technologies. As a result the framework seeks to provide potential solutions to these issues by removing the need for making health decisions from updated parameters and providing a mechanism for obtaining health state data. In light of this objective, a framework for forward model-driven SHM is established, highlighting key challenges and technologies that are required for realising this category of approach. The framework is constructed from two main components: generating physics-based models that accurately predict outputs under various damage scenarios, and machine learning methods used to infer decision bounds. This thesis deals with the former, developing technologies and strategies for producing statistically representative predictions from physics-based models. Specifically this work seeks to define validation within this context and propose a validation strategy, develop technologies that infer uncertainties from various sources, including model discrepancy, and offer a solution to the issue of validating full-system predictions when data is not available at this level. The first section defines validation within a forward model-driven context, offering a strategy of hypothesis testing, statistical distance metrics, visualisation tools, such as the witness function, and deterministic metrics. The statistical distances field is shown to provide a wealth of potential validation metrics that consider whole probability distributions. Additionally, existing validation metrics can be categorised within this fields terminology, providing greater insight. In the second part of this study emulator technologies, specifically Gaussian Process (GP) methods, are discussed. Practical implementation considerations are examined, including the establishment of validation and diagnostic techniques. Various GP extensions are outlined, with particular focus on technologies for dealing with large data sets and their applicability as emulators. Utilising these technologies two techniques for calibrating models, whilst accounting for and inferring model discrepancies, are demonstrated: Bayesian Calibration and Bias Correction (BCBC) and Bayesian History Matching (BHM). Both methods were applied to representative building structures in order to demonstrate their effectiveness within a forward model-driven SHM strategy. Sequential design heuristics were developed for BHM along with an importance sampling based technique for inferring the functional model discrepancy uncertainties. The third body of work proposes a multi-level uncertainty integration strategy by developing a subfunction discrepancy approach. This technique seeks to construct a methodology for producing valid full-system predictions through a combination of validated sub-system models where uncertainties and model discrepancy have been quantified. This procedure is demonstrated on a numerical shear structure where it is shown to be effective. Finally, conclusions about the aforementioned technologies are provided. In addition, a review of the future directions for forward model-driven SHM are outlined with the hope that this category receives wider investigation within the SHM community