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We investigate flows generated by bacteria near boundaries which are ubiquitous in biological systems. A bacterium such as \textit{Escherichia coli} is equipped with a number of rotary motors on its surface. Every motor has a helical flagellum attached to it and by rotating these motors the bacterium can move the fluid. We consider these flows in four different systems. Firstly, we explore a chiral flow which is always in the clockwise direction (when viewed from above) ahead of a dense suspension of bacteria on a moist surface. We quantitatively test a hypothesis that this flow is due to the action of cells stalled at the edge of a colony which extend their flagella outwards, moving fluid over a substrate. The model provides insight on the flagella orientations and their spatial distributions. Secondly, inspired by experiments which proposed to use confined bacteria in order to generate flows near surfaces, we develop a theoretical model of this fluid transport using a superposition of fundamental flow singularities. The rotation of a helical bacterial flagellum induces both a force and a torque on the surrounding fluid, both of which lead to a net flow along the surface. We investigate the optimal helical shapes to be used as micropumps near surfaces and show that bacterial flagella are nearly optimal. Thirdly, we build a theoretical model on a reorientation of peritrichous bacteria at the edge of a liquid drop on a Petri dish. Bacteria are more likely to turn clockwise because of the interaction between counterclockwise rotating flagella and boundaries which causes them to self-organise and circle clockwise (when viewed from above) around the outer edge of the colony. Finally, motivated by problems in biological physics occurring near corners, we derive the asymptotic behaviour for the Stokeslet (a flow due to a point force at low Reynolds number) both near and far from a corner geometry by using complex analysis on a known double integral solution for corner flows. We analyse flows in acute, obtuse and salient three-dimensional corners. We also use experiments on beads sedimenting in corn syrup to qualitatively test our results. The fundamental understanding of Stokes flows near boundaries is important for future developments in biophysics and bioengineering including applications to bacterial micropumps, steering microswimmers near corners, and preventing biofilm formation.

The central theme of this thesis is the design of optimal balanced black-box stopping criteria in iterative solvers of symmetric positive-definite, symmetric indefinite, and nonsymmetric linear systems arising from finite element approximation of stochastic (parametric) partial differential equations. For a given stochastic and spatial approximation, it is known that iteratively solving the corresponding linear(ized) system(s) of equations to too tight algebraic error tolerance results in a wastage of computational resources without decreasing the usually unknown approximation error. In order to stop optimally-by avoiding unnecessary computations and premature stopping-algebraic error and a posteriori approximation error estimate must be balanced at the optimal stopping iteration. Efficient and reliable a posteriori error estimators do exist for close estimation of the approximation error in a finite element setting. But the algebraic error is generally unknown since the exact algebraic solution is not usually available. Obtaining tractable upper and lower bounds on the algebraic error in terms of a readily computable and monotonically decreasing quantity (if any) of the chosen iterative solver is the distinctive feature of the designed optimal balanced stopping strategy. Moreover, this work states the exact constants, that is, there are no user-defined parameters in the optimal balanced stopping tests. Hence, an iterative solver incorporating the optimal balanced stopping methodology that is presented here will be a black-box iterative solver. Typically, employing such a stopping methodology would lead to huge computational savings and in any case would definitely rule out premature stopping. The constants in the devised optimal balanced black-box stopping tests in MINRES solver for solving symmetric positive-definite and symmetric indefinite linear systems can be estimated cheaply on-the- fly. The contribution of this thesis goes one step further for the nonsymmetric case in the sense that it not only provides an optimal balanced black-box stopping test in a memory-expensive Krylov solver like GMRES but it also presents an optimal balanced black-box stopping test in memory-inexpensive Krylov solvers such as BICGSTAB(L), TFQMR etc. Currently, little convergence theory exists for the memory-inexpensive Krylov solvers and hence devising stopping criteria for them is an active field of research. Also, an optimal balanced black-box stopping criterion is proposed for nonlinear (Picard or Newton) iterative method that is used for solving the finite dimensional Navier-Stokes equations. The optimal balanced black-box stopping methodology presented in this thesis can be generalized for any iterative solver of a linear(ized) system arising from numerical approximation of a partial differential equation. The only prerequisites for this purpose are the existence of a cheap and tight a posteriori error estimator for the approximation error along with cheap and tractable bounds on the algebraic error.

Three dimensional flow regimes are encountered in many types of industrial flow processes such as filtration, mixing, reaction engineering, polymerization and polymer forming as well as environmental systems. Thus, the analyses of phenomena involved fluid flow are of great importance and have been subject of numerous ongoing research projects. The analysis of these important phenomena can be conducted in laboratory through experiments or simply by using the emerging computational fluid dynamics (CFD) techniques. But when dealing with three dimensional fluid flow problems, the complexities encountered make the analysis via the traditional experimental techniques a daunting task. For this reason, researchers often prefer to use the CFD techniques which with some care taken, often produce accurate and stable results while maintaining cost as low as possible. Many CFD codes have been developed and tested in the past decades and the results have been successful and thus encouraging researchers to develop new codes and/or improve existing codes for the solutions of real world problems. In this present project, CFD techniques are used to simulate the fluid flow phenomena of interest by solving the flow governing equations numerically through the use of a personal computer. The aim of this present research is to develop a robust and reliable technique which includes a novel aspect for the solution of three dimensional generalized Newtonian fluids in domains including obstructions, and this must be done bearing in mind that both accuracy and cost efficiency have to be achieved. To this end, the finite element method (FEM) is chosen as the CFD computational method. There are many existing FEM techniques namely the streamline upwind Petrov-Galerkin methods, the streamline diffusion methods, the Taylor-Galerkin methods, among others. But after a thorough analysis of the physical conditions (geometries, governing equations, boundary conditions, assumptions …) of the fluid flow problems to be solve in this project, the appropriate scheme chosen is the UVWP family of the mixed finite element methods. It is scheme originally developed to solve two dimensional fluid flow problems but since the scheme produced accurate and stable results for two dimensional problems, then attempt is made in this present study to develop a new version of the UVWP scheme for the numerical analysis of three dimensional fluid flow problems. But, after some initial results obtained using the developed three dimensional scheme, investigations were made during the course of this study on how to speed up solutions' convergence without affecting the cost efficiency of the scheme. The outcomes of these investigations yield to the development of a novel scheme named the modified three dimensional UVWP scheme. Thus a computer model based on these two numerical schemes (UVWP and the Modified UVWP) is developed, tested, and validated through some benchmark problems, and then the model is used to solve some complicated tests problems in this study. Results obtained are accurate, and stable, moreover, the cost efficiency of the computer model must be mentioned because all the simulations carried out are done using a simple personal computer.

This dissertation consists of three main parts with each part focusing on numerical approximations of the stochastic Stokes and Navier-Stokes equations. Part One concerns the mixed finite element methods and Chorin projection methods for solving the stochastic Stokes equations with general multiplicative noise. We propose a modified mixed finite element method for solving the Stokes equations and show that the numerical solutions converge optimally to the PDE solutions. The convergence is under energy norms (strong convergence) for the velocity and in a time-averaged norm (weak convergence) for the pressure. In addition, after establishing the error estimates in second moment, high moment estimates are also established which serve as a bridge for deriving pathwise error estimates. The second portion of Part One focuses on Chorin projection methods for solving the stochastic Stokes equation with general multiplicative noise. Two types of the Chorin projection methods, the standard Chorin and modified Chorin projection methods, are considered. We establish sub-optimal order error estimates for the velocity and pressure approximations of the standard Chorin projection method, and optimal order estimates for the modified Chorin method Numerical experiments are also provided to verify the proved theoretical results. Part Two studies the mixed finite element methods for solving the stochastic Navier-Stokes equations with general additive noise. We establish the strong convergence for the velocity and pressure approximations. In addition, high moment error estimates are also established which serve as a bridge for deriving pathwise error estimates for both the velocity and pressure approximations. Part Three develops an iterative framework for parameter-dependent and random convection-diffusion problems which arise from many engineering and scientific applications. The main idea of the framework is to reformulate the underlying PDE problem into another problem with a parameter-independent convection-diffusion coefficient and a parameter-dependent (and solution-dependent) right-hand side, a fixed-point iteration is then employed to compute the solution of the reformulated problem. The computational saving is achieved at the solver level by using the LU direct method or block Krylov subspace methods as linear solvers so many computations can be reused. Numerical experiments are presented to validate the efficiency of the iterative method.

Imaging polarimetry has shown its advantage over other imaging techniques on various sensing tasks such as astronomical sensing, atmospheric sensing, target detection and biomedical diagnostics because it provides information of the imaged scene that are not obtainable to other imaging modalities such as shading and surface roughness of objects. Polarimeters are the devices designed to measure the polarization of light. In general imaging polarimeters can be divided into two broad classes, the wavefront division polarimeters and the modulated polarimeters. This thesis aims to contribute to the development of modulated polarimeters. The essential weakness of the modulated polarimeter is the resolution loss caused by the bandwidth reduction since this class of polarimeters divide the native bandwidth of the underlying detector to measure multiple pieces of polarization information “simultaneously”. In previous studies, researchers attempted to mitigate this resolution loss by designing more advanced modulation schemes that divide the native bandwidth in more economical fashions or combining multiple domains in modulation to expand the bandwidths of the polarimeters by trading off bandwidth of different domains. In recent years, our group and others have started looking at the latter approach, but it is still not particularly popular in the wider community. Many advanced multi-domain modulated polarimeter designs have been proposed but have not been verified with experiments. Therefore, the first study presented in this thesis demonstrates an experimental work where a previously proposed spatio-temporally modulated polarimeter is constructed and its advantages over single domain modulated polarimeters are verified. Secondly, this thesis examines the data reconstruction procedure of modulated polarimeters and proposes a novel machine learning based adaptive data reconstruction framework. This adaptive framework is demonstrated with a simulated classic spatially modulated polarimeter and compared with the conventional data reconstruction methods. The adaptive framework shows a promising improvement on the accuracy of the reconstructed data. Next, the machine learning based filtering framework is extended from single domain modulation schemes to multi-domain. The extended framework is then combined with the spatio-temporally modulated polarimeter from the experimental work to further improve the performance of the multi-domain modulated polarimeter. Finally, the appendix of the thesis presents a novel non-separable multi-domain modulation design that is able to suppress the systematic error introduced noises. This appendix is closely related to the other work presented in the thesis, but as it represents a different branch of research, it is presented separately to preserve the continuity of the document.

The thermal convection of rock in Earth's mantle and associated plate tectonics are modeled by nonlinear incompressible Stokes and energy equations. This dissertation focuses on the development of advanced, scalable linear and nonlinear solvers for numerical simulations of realistic instantaneous mantle flow, where we must overcome several computational challenges. The most notable challenges are the severe nonlinearity, heterogeneity, and anisotropy due to the mantle's rheology as well as a wide range of spatial scales and highly localized features. Resolving the crucial small scale features efficiently necessitates adaptive methods, while computational results greatly benefit from a high accuracy per degree of freedom and local mass conservation. Consequently, the discretization of Earth's mantle is carried out by high-order finite elements on aggressively adaptively refined hexahedral meshes with a continuous, nodal velocity approximation and a discontinuous, modal pressure approximation. These velocity--pressure pairings yield optimal asymptotic convergence rates of the finite element approximation to the infinite-dimensional solution with decreasing mesh element size, are inf-sup stable on general, non-conforming hexahedral meshes with "hanging nodes,'' and have the advantage of preserving mass locally at the element level due to the discontinuous pressure. However, because of the difficulties cited above and the desired accuracy, the large implicit systems to be solved are extremely poorly conditioned and sophisticated linear and nonlinear solvers including powerful preconditioning techniques are required. The nonlinear Stokes system is solved using a grid continuation, inexact Newton--Krylov method. We measure the residual of the momentum equation in the H⁻¹-norm for backtracking line search to avoid overly conservative update steps that are significantly reduced from one. The Newton linearization is augmented by a perturbation of a highly nonlinear term in mantle's rheology, resulting in dramatically improved nonlinear convergence. We present a new Schur complement-based Stokes preconditioner, weighted BFBT, that exhibits robust fast convergence for Stokes problems with smooth but highly varying (up to 10 orders of magnitude) viscosities, optimal algorithmic scalability with respect to mesh refinement, and only a mild dependence on the polynomial order of high-order finite element discretizations. In addition, we derive theoretical eigenvalue bounds to prove spectral equivalence of our inverse Schur complement approximation. Finally, we present a parallel hybrid spectral--geometric--algebraic multigrid (HMG) to approximate the inverses of the Stokes system's viscous block and variable-coefficient pressure Poisson operators within weighted BFBT. Building on the parallel scalability of HMG, our Stokes solver demonstrates excellent parallel scalability to 1.6 million CPU cores without sacrificing algorithmic optimality.

Distributed Temperature Sensing (DTS) has been used in a variety of different applications. Its ability to detect temperature fluctuations along fiber optic lines that stretch for several kilometers has made it a popular topic in various fields of science, engineering, and technology. From pre-fire detection to ecological monitoring, DTS has taken a vital role in scientific research. DTS uses the principle of backscattering by three different spectral components, e.g., Rayleigh scattering, Brillouin scattering, and Raman scattering. Although there have been various improvements to DTS, its slow response time and poor spatial resolution have been hard to overcome. Its repetition rate is low because the pulse must travel the distance of the fiber optic line and return to the detector to record the temperature change along the fiber. A spatial resolution of 7.4 cm with a response time as low as 1 second and a temperature resolution of the 0.196 ℃ is achieved from the current Raman-based DTS system. This research proves that high-spatial resolution can be obtained with the use of a Silicon Avalanche Photodetector with a 1 GHz bandwidth.

Numerical solutions to fluid flow problems involve solving the linear systems arising from the discretization of the Stokes equation or a variant of it, which often have a saddle point structure and are difficult to solve. Geometric multigrid is a parallelizable method that can efficiently solve these linear systems especially for a large number of unknowns. We consider two approaches to solve these linear systems using geometric multigrid: First, we use a block preconditioner and apply geometric multigrid as in inner solver to the velocity block only. We develop deal.II tutorial step-56 to compare the use of geometric multigrid to other popular alternatives. This method is found to be competitive in serial computations in terms of performance and memory usage. Second, we design a special smoother to apply multigrid to the whole linear system. This smoother is analyzed as a Schwarz method using conforming and inf-sup stable discretization spaces. The resulting method is found to be competitive to a similar multigrid method using non-conforming finite elements that were studied by Kanschat and Mao. This approach has the potential to be superior to the first approach. Finally, expanding on the research done by Dannberg and Heister, we explore the analysis of a three-field Stokes formulation that is used to describe melt migration in the earth's mantle. Multiple discretizations were studied to find the best one to use in the ASPECT software package. We also explore improvements to ASPECT's linear solvers for this formulation utilizing block preconditioners and algebraic multigrid.

A scheme for giving steady-state solutions of the Navier-Stokes equations on unstructured grids in two-dimensions is presented. The method, called fluctuation splitting, is designed to be independent of the underlying grid geometry and distributes the residual in the cell to the vertices in a manner based upon upwinding ideas. The development starts with a discussion of schemes for the advection equation on triangles which is then extended to include the diffusive terms in the advection-diffusion equation. This scheme is then evaluated on the Smith and Hutton problem, which involves the simultaneous advection and diffusion of a scalar quantity at high and low Peclet numbers around a bend. By integrating continuity on a structured grid, the Blasius boundary layer equations are solved as a scalar advection-diffusion problem. The extension to systems of equations begins with the Euler equations, which involves the development of wave-models to model the local solution within cells of the domain. The vertices of the cells are updated based upon the passage of these simple waves. This is then extended to the Navier-Stokes equations by discretizing the viscous fluxes in a manner similar to that done in the scalar case for advection-diffusion. Results are shown for the Blasius boundary layer and a NACA0012 airfoil. Finally, the parallel aspects of the scheme, which uses a very compact stencil for each vertex, are discussed. It is shown that the algorithm is readily parallelizable and that the performance of the parallel algorithm on an IBM SP/2 is excellent.

A Cartesian, cell-based scheme for solving the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal cut cells are created. The geometry of the cut cells is computed using polygon-clipping algorithms. The grid is stored in a binary-tree data structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded, with a limited linear reconstruction of the primitive variables used to provide input states to an approximate Riemann solver for computing the fluxes between neighboring cells. A multi-stage time-stepping scheme is used to reach a steady-state solution. Validation of the Euler solver with benchmark numerical and exact solutions is presented. An assessment of the accuracy of the approach is made by uniform and adaptive grid refinements for a steady, transonic, exact solution to the Euler equations. The error of the approach is directly compared to a structured solver formulation. A non-smooth flow is also assessed for grid convergence, comparing uniform and adaptively refined results. Several formulations of the viscous terms are assessed analytically, both for accuracy and positivity. The two best formulations are used to compute adaptively refined solutions of the Navier-Stokes equations. These solutions are compared to each other, to experimental results and/or theory for a series of low and moderate Reynolds numbers flow fields. The most suitable viscous discretization is demonstrated for geometrically-complicated internal flows. For flows at high Reynolds numbers, both an altered grid-generation procedure and a different formulation of the viscous terms are shown to be necessary. A hybrid Cartesian/body-fitted grid generation approach is demonstrated. In addition, a grid-generation procedure based on body-aligned cell cutting coupled with a viscous stencil-construction procedure based on quadratic programming is presented.

A convergence acceleration technique for the Euler and Navier-Stokes equations is presented, based on local preconditioning of these systems of equations. The key to the success of local Euler preconditioning is equalizing the characteristic wave speeds of the Euler equations as much as possible. By equalizing the wave speeds, the efficiency of wave propagation is improved, resulting in convergence acceleration of the time-marching scheme and other benefits such as clustering of numerical eigenvalues and accuracy improvement in the low-speed limit. A large family of Euler preconditioners is studied with regard to various design criteria formulated for enhancing the performance of Euler time-marching schemes in the areas of efficiency, accuracy, and robustness. A major result is the derivation of a class of preconditioners that can successfully compute stagnating flow, which has previously been problematic for preconditioners developed before this work. The Navier-Stokes preconditioning also accelerates the convergence and clusters eigenvalues. It is developed based on a Fourier analysis of the discretized equations and a dispersion analysis of the differential equations. The principle of the Navier-Stokes preconditioning is to remove the dependence of the physical time scales on both the Mach number and the cell-Reynolds number. The discrete Fourier analysis suggests a hybrid Navier-Stokes preconditioner, combining an optimal Euler preconditioner with the Jacobi block for the discretized viscous/conductive terms. The dispersion analysis produces an analytical form of the preconditioner, which can equalize, in absolute value, the complex wave speeds of the Navier-Stokes equations; these include both effects of viscous damping and wave propagation. The Jacobi and analytical techniques can also be combined in a single preconditioner, for greater robustness and efficiency. Numerical tests confirm that both types of preconditioners can increase the efficiency of the convergence.