Sampling theory studies the equivalence between continuous and discrete representations of information. This equivalence is ubiquitously used in communication engineering and signal processing. For example, it allows engineers to store continuous signals as discrete data on digital media. The classical sampling theorem, also known as the theorem of Whittaker-Shannon-Kotel'nikov, enables one to perfectly and stably reconstruct continuous signals with a constant bandwidth from their discrete samples at a constant Nyquist rate. The Nyquist rate depends on the bandwidth of the signals, namely, the frequency upper bound. Intuitively, a signal's `information density' and `effective bandwidth' should vary in time. Adjusting the sampling rate accordingly should improve the sampling efficiency and information storage. While this old idea has been pursued in numerous publications, fundamental problems have remained: How can a reliable concept of time-varying bandwidth been defined? How can samples taken at a time-varying Nyquist rate lead to perfect and stable reconstruction of the continuous signals? This thesis develops a new non-Fourier generalized sampling theory which takes samples only as often as necessary at a time-varying Nyquist rate and maintains the ability to perfectly reconstruct the signals. The resulting Nyquist rate is the critical sampling rate below which there is insufficient information to reconstruct the signal and above which there is redundancy in the stored samples. It is also optimal for the stability of reconstruction. To this end, following work by A. Kempf, the sampling points at a Nyquist rate are identified as the eigenvalues of self-adjoint extensions of a simple symmetric operator with deficiency indices (1,1). The thesis then develops and in a sense completes this theory. In particular, the thesis introduces and studies filtering, and yields key results on the stability and optimality of this new method. While these new results should greatly help in making time-variable sampling methods applicable in practice, the thesis also presents a range of new purely mathematical results. For example, the thesis presents new results that show how to explicitly calculate the eigenvalues of the complete set of self-adjoint extensions of such a symmetric operator in the Hilbert space. This result is of interest in the field of functional analysis where it advances von Neumann's theory of self-adjoint extensions.

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Computational Fluid Dynamics (CFD) is a main field that contributes to the development of high efficiency aircraft. CFD accuracy depends on the flow solver and the meshing of the geometry, and while it is doable to determine why a certain solver is more accurate than another, it is much more difficult to discern why two meshes produce differently accurate solutions. A framework is presented to evaluate the performance or ``goodness" of a mesh and to compare meshes. The framework is composed of quantifiable mesh parameters which define a mesh, and three performance measures: functional accuracy, their order of convergence, and their behaviour under the adjoint correction method. Although it seems that the relationships between parameters and results are not trivial, there are trends from which optimal mesh parameters are deduced. The H topology performs best, and the most important parameters are related to spacings and cell quality around the aerofoil leading edge.

Computational Fluid Dynamics (CFD) is a main field that contributes to the development of high efficiency aircraft. CFD accuracy depends on the flow solver and the meshing of the geometry, and while it is doable to determine why a certain solver is more accurate than another, it is much more difficult to discern why two meshes produce differently accurate solutions. A framework is presented to evaluate the performance or ``goodness" of a mesh and to compare meshes. The framework is composed of quantifiable mesh parameters which define a mesh, and three performance measures: functional accuracy, their order of convergence, and their behaviour under the adjoint correction method. Although it seems that the relationships between parameters and results are not trivial, there are trends from which optimal mesh parameters are deduced. The H topology performs best, and the most important parameters are related to spacings and cell quality around the aerofoil leading edge.

Trace gases and aerosols, or suspended liquid and solid material in the atmosphere, have significant climatological and societal impacts; consequently, accurate representation of their contribution to atmospheric composition is vital to predicting climate change and informing policy actions. Sensitivity analysis allows scientists and environmental decision makers alike to ascertain the role a specific component of the very complex system that is the atmosphere of the Earth. Anthropogenic and natural emissions of gases and aerosol are transported by winds and interact with sunlight, allowing significant transformation before these species reach the end of their atmospheric life on land or in water. The adjoint-based sensitivity method assesses the relative importance of each emissions source to selected results of interest, including aerosol and cloud droplet concentration. In this work, the adjoint of a comprehensive inorganic aerosol thermodynamic equilibrium model was produced to improve the representativeness of regional and global chemical transport modeling. Furthermore, a global chemical transport model adjoint equipped with the adjoint of a cloud droplet activation parameterization was used to explore the footprint of emissions contributing to current and potential future cloud droplet concentrations, which impact the radiative balance of the earth. In future work, these sensitivity relationships can be exploited in optimization frameworks for assimilation of observations of the system, such as satellite-based or in situ measurements of aerosol or precursor trace gas concentrations.

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