Current research in CI combustion aims to reduce PM and NOx emissions by controlling mixture homogeneity. Low CN fuels are suitable due to their auto-ignition resistance, but the in-cylinder mixture stratification level must be carefully visualised and controlled. Numerous diagnostic techniques exist for imaging the in-cylinder hydrocarbon species concentration. Tomographic techniques based on spectroscopic modalities are minimally-intrusive and able to target species of interest even in multi-component fuel blends. The high-speed CST technique applied in this work is based on the NIRAT modality. A number of collimated LASER beams at 1700nm traverse the optically accessible engine combustion chamber and are spectroscopically absorbed by the first overtone of the C-H stretch bond. Non species-specific attenuation mechanisms are suppressed by a DWR scheme utilising a reference wavelength at 1651nm. Ratiometric data is used to tomographically reconstruct the spatially-varying fuel concentration. In this work the first application of NIRAT on a commercial CI engine is presented, using instrumentation capable of imaging 13 frames/CAD at 1200rpm using a 31-beam array. A novel method was developed to experimentally quantify the tomography system’s non-uniform spatial resolution. The method was applied in laboratory experiments involving free-space propane plumes and a map of the spatial resolution was created. The spatial resolution varies between 4mm and 14mm. The mean of 9mm is 72% better than previous estimates in the literature. Regions of poor performance correlated with non-uniformities in the sensitivity matrix, indicating that a regular beam array may contribute towards more accurate and objective reconstructions of unknown concentrations. The characterised tomography system was installed on an optically-accessible Volvo D5 CI engine. The optically-inaccessible CAD region achieved was ±18CAD, a reduction of ±12° from previous works. The vibration-tolerance of the optical access system was verified, concluding that the initial alignment of the beams is the dominant factor that determines beam integrity after prolonged engine operation. The behavior of individual beams was studied, finding strong cycle-to-cycle correlation of the anomalies present. This was exploited to develop a novel, robust analysis algorithm to process the engine data. The algorithm achieved a standard deviation of <10% of the maximum pk-pk magnitude of the transmission signal in the fuel vapour phase. The system was applied to qualitatively visualise the mixing of a 50/50% blend of iso-/n-dodecane in a motored, nitrogen-aspirated engine under a range of operating conditions. A study by simulation of the decomposition of n-dodecane concluded that only 0.492% of the quantity injected is pyrolytically converted during a compression stroke. Spray-phase imaging was not possible due to severe reduction of the optical throughput, lasting for 8-15 CAD for a lean mixture and for 15-30 CAD for a rich mixture. Vapour-phase reconstructions using the enhanced iterative Landweber algorithm were successful in resolving rich fuel pockets consistent with the injection pattern. It was shown that the degree of mixture homogeneity at TDC is dependent upon the initial intake temperature. PLIF was used to cross-validate the NIRAT reconstructions. Localisation of the features reconstructed with NIRAT was excellent, with a maximum angular deviation of ±10°. A swirl motion of the mixture by 1°/CAD was observed using both techniques, confirming the features previously observed in the NIRAT reconstructions. In conclusion, NIRAT has been, for the first time, successfully applied for in-cylinder fuel distribution imaging in a CI engine. The results, created using an original data analysis algorithm, were successfully cross-validated using PLIF. A novel spatial resolution quantification method was formulated and used to characterise the tomography system. The numerous findings and learning points from the individual stages of this work will be used to advance the field of combustion diagnostics as well as contribute towards the development of advanced in-cylinder tomographic imaging systems.

Data assimilation arises in a vast array of different topics: traditionally in meteorological and oceanographic modelling, wind tunnel or water tunnel experiments and recently from biomedical engineering. Data assimilation is a process for combine measured or observed data with a mathematical model, to obtain estimates of the expected data. The measured data usually contains inaccuracies and is given with low spatial and/or temporal resolution. In this thesis data assimilation for time dependent fluid flow is considered. The flow is assumed to satisfy a given partial differential equation, representing the mathematical model. The problem is to determine the initial state which leads to a flow field which satisfies the flow equation and is close to the given data. In the first part we consider one-dimensional flow governed by Burgers’ equation. We analyze two iterative methods for data assimilation problem for this equation. One of them so called adjoint optimization method, is based on minimization in L2-norm. We show that this minimization problem is ill-posed but the adjoint optimization iterative method is regularizing, and represents the well-known Landweber method in inverse problems. The second method is based on L2-minimization of the gradient. We prove that this problem always has a solution. We present numerical comparisons of these two methods. In the second part three-dimensional inviscid compressible flow represented by the Euler equations is considered. Adjoint technique is used to obtain an explicit formula for the gradient to the optimization problem. The gradient is used in combination with a quasi-Newton method to obtain a solution. The main focus regards the derivation of the adjoint equations with boundary conditions. An existing flow solver EDGE has been modified to solve the adjoint Euler equations and the gradient computations are validated numerically. The proposed iteration method are applied to a test problem where the initial pressure state is reconstructed, for exact data as well as when disturbances in data are present. The numerical convergence and the result are satisfying.

Ill-posed sets of linear equations typically arise when discretizing certain types of integral transforms. A well known example is image reconstruction, which can be modeled using the Radon transform. After expanding the solution into a finite series of basis functions a large, sparse and ill-conditioned linear system occurs. We consider the solution of such systems. In particular we study a new class of iteration methods named DROP (for Diagonal Relaxed Orthogonal Projections) constructed for solving both linear equations and linear inequalities. This class can also be viewed, when applied to linear equations, as a generalized Landweber iteration. The method is compared with other iteration methods using test data from a medical application and from electron microscopy. Our theoretical analysis include convergence proofs of the fully-simultaneous DROP algorithm for linear equations without consistency assumptions, and of block-iterative algorithms both for linear equations and linear inequalities, for the consistent case. When applying an iterative solver to an ill-posed set of linear equations the error usually initially decreases but after some iterations, depending on the amount of noise in the data, and the degree of ill-posedness, it starts to increase. This phenomenon is called semi-convergence. We study the semi-convergence performance of Landweber-type iteration, and propose new ways to specify the relaxation parameters. These are computed so as to control the propagated error. We also describe a class of stopping rules for Landweber-type iteration for solving linear inverse problems. The class includes the well known discrepancy principle, and the monotone error rule. We unify the error analysis of these two methods. The stopping rules depend critically on a certain parameter whose value needs to be specified. A training procedure is therefore introduced for securing robustness. The advantages of using trained rules are demonstrated on examples taken from image reconstruction from projections. Kaczmarz's method, also called ART (Algebraic Reconstruction Technique) is often used for solving the linear system which appears in image reconstruction. This is a fully sequential method. We examine and compare ART and its symmetric version. It is shown that the cycles of symmetric ART, unlike ART, converge to a weighted least squares solution if and only if the relaxation parameter lies between zero and two. Further we show that ART has faster asymptotic rate of convergence than symmetric ART. Also a stopping criterion is proposed and evaluated for symmetric ART. We further investigate a class of block-iterative methods used in image reconstruction. The cycles of the iterative sequences are characterized in terms of the original linear system. We define symmetric block-iteration and compare the behavior of symmetric and non-symmetric block-iteration. The results are illustrated using some well-known methods. A stopping criterion is offered and assessed for symmetric block-iteration.

Since the inverse problem in Diffusive Optical Tomography (DOT) is nonlinear and severely ill-posed, only low resolution reconstructions are feasible when noise is added to the data nowadays. The purpose of this thesis is to improve image reconstruction in DOT of the main optical properties of tissues with some novel mathematical methods. We have used the Landweber (L) method, the Landweber-Kaczmarz (LK) method and its improved Loping-Landweber-Kaczmarz (L-LK) method combined with sparsity or with total variation regularizations for single and simultaneous image reconstructions of the absorption and scattering coefficients. The sparsity method assumes the existence of a sparse solution which has a simple description and is superposed onto a known background. The sparsity method is solved using a smooth gradient and a soft thresholding operator. Moreover, we have proposed an improved sparsity method. For the total variation reconstruction imaging, we have used the split Bregman method and the lagged diffusivity method. For the total variation method, we also have implemented a memory-efficient method to minimise the storage of large Hessian matrices. In addition, an individual and simultaneous contrast value reconstructions are presented using the level set (LS) method. Besides, the shape derivative of DOT based on the RTE is derived using shape sensitivity analysis, and some reconstructions for the absorption coefficient are presented using this shape derivative via the LS method.\\Whereas most of the approaches for solving the nonlinear problem of DOT make use of the diffusion approximation (DA) to the radiative transfer equation (RTE) to model the propagation of the light in tissue, the accuracy of the DA is not satisfactory in situations where the medium is not scattering dominant, in particular close to the light sources and to the boundary, as well as inside low-scattering or non-scattering regions. Therefore, we have solved the inverse problem in DOT by the more accurate time-dependant RTE in two dimensions.

<p dir="ltr">In this thesis, we consider the inverse problem of reconstructing the shape, position, and size of an unknown scattering object. We will talk about different methods used for nondestructive testing in scattering theory. We will consider qualitative reconstruction methods to understand and determine important information about the support of unknown scattering objects. We will also discuss the material properties of the system and connect them to certain crucial aspects of the region of interest, as well as develop useful techniques to determine physical information using inverse scattering theory. </p><p><br></p><p dir="ltr">In the first part of the analysis, we consider the transmission eigenvalue (TE) problem associated with the scattering of a plane wave for an isotropic scatterer. In particular, we examine the transmission eigenvalue problem with two conductivity boundary parameters. In previous studies, this eigenvalue problem was analyzed with one conductive boundary parameter, whereas we will consider the case of two parameters. We will prove the existence and discreteness of the transmission eigenvalues. In addition, we will study the dependence of the TE's on the physical parameters and connect the first transmission eigenvalue to the physical parameters of the problem by a monotone-type argument. Lastly, we will consider the limiting procedure as the second boundary parameter vanishes at the boundary of the scattering region and provide numerical examples to validate the theory presented in Chapter 2. </p><p><br></p><p dir="ltr">The connection between transmission eigenvalues and the system's physical parameters provides a way to do testing in a nondestructive way. However, to understand the region of interest in terms of its shape, size, and position, one needs to use different techniques. As a result, we consider reconstructing extended scatterers using an analogous method to the Direct Sampling Method (DSM), a new sampling method based on the Landweber iteration. We will need a factorization of the far-field operator to analyze the corresponding imaging function for the new Landweber direct sampling method. Then, we use the factorization and the Funk--Hecke integral identity to prove that the new imaging function will accurately recover the scatterer. The method studied here falls under the category of qualitative reconstruction methods, where an imaging function is used to retrieve the scatterer. We prove the stability of our new imaging function as well as derive a discrepancy principle for recovering the regularization parameter. The theoretical results are verified with numerical examples to show how the reconstruction performs by the new Landweber direct sampling method.</p><p><br></p><p dir="ltr">Motivated by the work done with the transmission eigenvalue problem with two conductivity parameters, we also study the direct and inverse problem for isotropic scatterers with two conductive boundary conditions. In such a problem, one analyzes the behavior of the scattered field as one of the conductivity parameters vanishes at the boundary. Consequently, we prove the convergence of the scattered field dealing with two conductivity parameters to the scattered field dealing with only one conductivity parameter. We consider the uniqueness of recovering the coefficients from the known far-field data at a fixed incident direction for multiple frequencies. Then, we consider the inverse shape problem for recovering the scatterer for the measured far-field data at a fixed frequency. To this end, we study the direct sampling method for recovering the scatterer by studying the factorization for the far-field operator. The direct sampling method is stable concerning noisy data and valid in two dimensions for partial aperture data. The theoretical results are verified with numerical examples to analyze the performance using the direct sampling method. </p>

Dans cette thèse, j’exploite le cadre d’analyse de données fonctionnelles et développe l’analyse d’inférence et de prédiction, avec une application à des sujets sur les marchés financiers. Cette thèse est organisée en trois chapitres. Le premier chapitre est un article co-écrit avec Marine Carrasco. Dans ce chapitre, nous considérons un modèle de régression linéaire fonctionnelle avec une variable prédictive fonctionnelle et une réponse scalaire. Nous effectuons une comparaison théorique des techniques d’analyse des composantes principales fonctionnelles (FPCA) et des moindres carrés partiels fonctionnels (FPLS). Nous déterminons la vitesse de convergence de l’erreur quadratique moyen d’estimation (MSE) pour ces méthodes. Aussi, nous montrons cette vitesse est sharp. Nous découvrons également que le biais de régularisation de la méthode FPLS est plus petit que celui de FPCA, tandis que son erreur d’estimation a tendance à être plus grande que celle de FPCA. De plus, nous montrons que le FPLS surpasse le FPCA en termes de prédiction avec moins de composantes. Le deuxième chapitre considère un modèle autorégressif entièrement fonctionnel (FAR) pour prèvoir toute la courbe de rendement du S&P 500 a la prochaine journée. Je mène une analyse comparative de quatre techniques de Big Data, dont la méthode de Tikhonov fonctionnelle (FT), la technique de Landweber-Fridman fonctionnelle (FLF), la coupure spectrale fonctionnelle (FSC) et les moindres carrés partiels fonctionnels (FPLS). La vitesse de convergence, la distribution asymptotique et une stratégie de test statistique pour sélectionner le nombre de retard sont fournis. Les simulations et les données réelles montrent que les méthode FPLS performe mieux les autres en terme d’estimation du paramètre tandis que toutes ces méthodes affichent des performances similaires en termes de prédiction. Le troisième chapitre propose d’estimer la densité de neutralité au risque (RND) dans le contexte de la tarification des options, à l’aide d’un modèle fonctionnel. L’avantage de cette approche est qu’elle exploite la théorie d’absence d’arbitrage et qu’il est possible d’éviter toute sorte de paramétrisation. L’estimation conduit à un problème d’inversibilité et la technique fonctionnelle de Landweber-Fridman (FLF) est utilisée pour le surmonter. / In this thesis, I exploit the functional data analysis framework and develop inference, prediction and forecasting analysis, with an application to topics in the financial market. This thesis is organized in three chapters. The first chapter is a paper co-authored with Marine Carrasco. In this chapter, we consider a functional linear regression model with a functional predictor variable and a scalar response. We develop a theoretical comparison of the Functional Principal Component Analysis (FPCA) and Functional Partial Least Squares (FPLS) techniques. We derive the convergence rate of the Mean Squared Error (MSE) for these methods. We show that this rate of convergence is sharp. We also find that the regularization bias of the FPLS method is smaller than the one of FPCA, while its estimation error tends to be larger than that of FPCA. Additionally, we show that FPLS outperforms FPCA in terms of prediction accuracy with a fewer number of components. The second chapter considers a fully functional autoregressive model (FAR) to forecast the next day’s return curve of the S&P 500. In contrast to the standard AR(1) model where each observation is a scalar, in this research each daily return curve is a collection of 390 points and is considered as one observation. I conduct a comparative analysis of four big data techniques including Functional Tikhonov method (FT), Functional Landweber-Fridman technique (FLF), Functional spectral-cut off (FSC), and Functional Partial Least Squares (FPLS). The convergence rate, asymptotic distribution, and a test-based strategy to select the lag number are provided. Simulations and real data show that FPLS method tends to outperform the other in terms of estimation accuracy while all the considered methods display almost the same predictive performance. The third chapter proposes to estimate the risk neutral density (RND) for options pricing with a functional linear model. The benefit of this approach is that it exploits directly the fundamental arbitrage-free equation and it is possible to avoid any additional density parametrization. The estimation problem leads to an inverse problem and the functional Landweber-Fridman (FLF) technique is used to overcome this issue.