Numerical study of transition in the Rayleigh-B&#039 / enard problem of thermal convection between rigid plates heated from below under the influence of gravity with and without rotation is presented. The first numerical approach uses spectral element method with Fourier expansion for horizontal extent and Legendre polynomal for vertical extent for the purpose of generating a database for the subsequent analysis by using Karhunen-Lo&#039 / eve (KL) decomposition. KL decompositions is a statistical tool to decompose the dynamics underlying a database representing a physical phenomena to its basic components in the form of an orthogonal KL basis. The KL basis satisfies all the spatial constraints such as the boundary conditions and the solenoidal (divergence-free) character of the underlying flow field as much as carried by the flow database. The optimally representative character of the orthogonal basis is used to investigate the convective flow for different parameters, such as Rayleigh and Prandtl numbers. The second numerical approach uses divergence free basis functions that by construction satisfy the continuity equation and the boundary conditions in an expansion of the velocity flow field. The expansion bases for the thermal field are constructed to satisfy the boundary conditions. Both bases are based on the Legendre polynomials in the vertical direction in order to simplify the Galerkin projection procedure, while Fourier representation is used in the horizontal directions due to the horizontal extent of the computational domain taken as periodic. Dual bases are employed to reduce the governing Boussinesq equations to a dynamical system for the time dependent expansion coefficients. The dual bases are selected so that the pressure term is eliminated in the projection procedure. The resulting dynamical system is used to study the transitional regimes numerically. The main difference between the two approaches is the accuracy with which the solenoidal character of the flow is satisfied. The first approach needs a numerically or experimentally generated database for the generation of the divergence-free KL basis. The degree of the accuracy for the KL basis in satisfying the solenoidal character of the flow is limited to that of the database and in turn to the numerical technique used. This is a major challenge in most numerical simulation techniques for incompressible flow in literature. It is also dependent on the parameter values at which the underlying flow field is generated. However the second approach is parameter independent and it is based on analytically solenoidal basis that produces an almost exactly divergence-free flow field. This level of accuracy is especially important for the transition studies that explores the regions sensitive to parameter and flow perturbations.

We considered a Bayesian approach to nonlinear inverse problems in which the unknown quantity is a high dimension spatial field. The Bayesian approach contains a natural mechanism for regularization in the form of prior information, can incorporate information from heterogeneous sources and provides a quantitative assessment of uncertainty in the inverse solution. The Bayesian setting casts the inverse solution as a posterior probability distribution over the model parameters. Karhunen-Lo'eve expansion and Discrete Cosine transform were used for dimension reduction of the random spatial field. Furthermore, we used a hierarchical Bayes model to inject multiscale data in the modeling framework. In this Bayesian framework, we have shown that this inverse problem is well-posed by proving that the posterior measure is Lipschitz continuous with respect to the data in total variation norm. The need for multiple evaluations of the forward model on a high dimension spatial field (e.g. in the context of MCMC) together with the high dimensionality of the posterior, results in many computation challenges. We developed two-stage reversible jump MCMC method which has the ability to screen the bad proposals in the first inexpensive stage. Channelized spatial fields were represented by facies boundaries and variogram-based spatial fields within each facies. Using level-set based approach, the shape of the channel boundaries was updated with dynamic data using a Bayesian hierarchical model where the number of points representing the channel boundaries is assumed to be unknown. Statistical emulators on a large scale spatial field were introduced to avoid the expensive likelihood calculation, which contains the forward simulator, at each iteration of the MCMC step. To build the emulator, the original spatial field was represented by a low dimensional parameterization using Discrete Cosine Transform (DCT), then the Bayesian approach to multivariate adaptive regression spline (BMARS) was used to emulate the simulator. Various numerical results were presented by analyzing simulated as well as real data.

Detection of a human emotion from human speech is always a challenging task. Factors like intonation, pitch, and loudness of signal vary from different human voice. So, it's important to know the exact pitch, intonation and loudness of a speech for making it a challenging task for detection. Some voices exhibit high background noise which will affect the amplitude or pitch of the signal. So, knowing the detailed properties of a speech to detect emotion is mandatory. Detection of emotion in humans from speech signals is a recent research field. One of the scenarios where this field has been applied is in situations where the human integrity and security are at risk In this project we are proposing a set of features based on the decomposition signals from discrete wavelet transform to characterize different types of negative emotions such as anger, happy, sad, and desperation. The features are measured in three different conditions: (1) the original speech signals, (2) the signals that are contaminated with noise or are affected by the presence of a phone channel, and (3) the signals that are obtained after processing using an algorithm for Speech Enhancement Transform. According to the results, when the speech enhancement is applied, the detection of emotion in speech is increased and compared to results obtained when the speech signal is highly contaminated with noise. Our objective is to use Artificial neural network because the brain is the most efficient and best machine to recognize speech. The brain is built with some neural network. At the same time, Artificial neural networks are clearly advanced with respect to several features, such as their nonlinearity and high classification capability. If we use Artificial neural networks to evolve the machine or computer that it can detect the emotion. Here we are using feedforward neural network which is suitable for classification process and using sigmoid function as activation function. The detection of human emotion from speech is achieved by training the neural network with features extracted from the speech. To achieve this, we need proper features from the speech. So, we must remove background noise in the speech. We can remove background noise by using filters. wavelet transform is the filtering technique used to remove the background noise and enhance the required features in the speech.

Although generally accurate, the quality of SeaWinds on QuikSCAT scatterometer ocean vector winds is compromised by certain natural phenomena and retrieval algorithm limitations. This dissertation addresses three main contributers to scatterometer estimate error: poor ambiguity selection, estimate uncertainty at low wind speeds, and rain corruption. A quality assurance (QA) analysis performed on SeaWinds data suggests that about 5% of SeaWinds data contain ambiguity selection errors and that scatterometer estimation error is correlated with low wind speeds and rain events. Ambiguity selection errors are partly due to the "nudging" step (initialization from outside data). A sophisticated new non-nudging ambiguity selection approach produces generally more consistent wind than the nudging method in moderate wind conditions. The non-nudging method selects 93% of the same ambiguities as the nudged data, validating both techniques, and indicating that ambiguity selection can be accomplished without nudging. Variability at low wind speeds is analyzed using tower-mounted scatterometer data. According to theory, below a threshold wind speed, the wind fails to generate the surface roughness necessary for wind measurement. A simple analysis suggests the existence of the threshold in much of the tower-mounted scatterometer data. However, the backscatter does not "go to zero" beneath the threshold in an uncontrolled environment as theory suggests, but rather has a mean drop and higher variability below the threshold. Rain is the largest weather-related contributer to scatterometer error, affecting approximately 4% to 10% of SeaWinds data. A simple model formed via comparison of co-located TRMM PR and SeaWinds measurements characterizes the average effect of rain on SeaWinds backscatter. The model is generally accurate to within 3 dB over the tropics. The rain/wind backscatter model is used to simultaneously retrieve wind and rain from SeaWinds measurements. The simultaneous wind/rain (SWR) estimation procedure can improve wind estimates during rain, while providing a scatterometer-based rain rate estimate. SWR also affords improved rain flagging for low to moderate rain rates. QuikSCAT-retrieved rain rates correlate well with TRMM PR instantaneous measurements and TMI monthly rain averages. SeaWinds rain measurements can be used to supplement data from other rain-measuring instruments, filling spatial and temporal gaps in coverage.

An accurate PDF of irradiance for a FSO channel is important when designing a laser radar, active laser imaging, or a communications system to operate over the channel. Parameters such as detector threshold level, probability of detection, mean fade time, number of fades, BER, and SNR are derived from the PDF and determine the design constraints of the receiver, transmitter, and corresponding electronics. Current PDF models of irradiance, such as the Gamma-Gamma, do not fully capture the effect of aperture averaging; a reduction in scintillation as the diameter of the collecting optic is increased. The Gamma-Gamma PDF of irradiance is an attractive solution because the parameters of the distribution are derived strictly from atmospheric turbulence parameters; propagation path length, Cn2, l0, and L0. This dissertation describes a heuristic physics-based modeling technique to develop a new PDF of irradiance based upon the optical field. The goal of the new PDF is three-fold: capture the physics of the turbulent atmosphere, better describe aperture averaging effects, and relate parameters of the new model to measurable atmospheric parameters. The modeling decomposes the propagating electromagnetic field into a sum of independent random-amplitude spatial plane waves using an approximation to the Karhunen-Loeve expansion. The scattering effects of the turbulence along the propagation path define the random-amplitude of each component of the expansion. The resulting PDF of irradiance is a double finite sum containing a Bessel function. The newly developed PDF is a generalization of the Gamma-Gamma PDF, and reduces to such in the limit. An experiment was setup and performed to measure the PDF of irradiance for several receiver aperture sizes under moderate to strong turbulence conditions. The propagation path was instrumented with scintillometers and anemometers to characterize the turbulence conditions. The newly developed PDF model and the GG model were compared to histograms of the experimental data. The new PDF model was typically able to match the data as well or better than the GG model under conditions of moderate aperture averaging. The GG model fit the data better than the new PDF under conditions of significant aperture averaging. Due to a limiting scintillation index value of 3, the new PDF was not compared to the GG for point apertures under strong turbulence; a regime where the GG is known to fit data well.

This dissertation investigates uncertainty quantification in dynamic problems. The Advanced Mean Value (AMV) method is used to calculate probabilistic sound power and the sensitivity of elastically supported panels with small uncertainty (coefficient of variation). Sound power calculations are done using Finite Element Method (FEM) and Boundary Element Method (BEM). The sensitivities of the sound power are calculated through direct differentiation of the FEM/BEM/AMV equations. The results are compared with Monte Carlo simulation (MCS). An improved method is developed using AMV, metamodel, and MCS. This new technique is applied to calculate sound power of a composite panel using FEM and Rayleigh Integral. The proposed methodology shows considerable improvement both in terms of accuracy and computational efficiency. In systems with large uncertainties, the above approach does not work. Two Spectral Stochastic Finite Element Method (SSFEM) algorithms are developed to solve stochastic eigenvalue problems using Polynomial chaos. Presently, the approaches are restricted to problems with real and distinct eigenvalues. In both the approaches, the system uncertainties are modeled by Wiener-Askey orthogonal polynomial functions. Galerkin projection is applied in the probability space to minimize the weighted residual of the error of the governing equation. First algorithm is based on inverse iteration method. A modification is suggested to calculate higher eigenvalues and eigenvectors. The above algorithm is applied to both discrete and continuous systems. In continuous systems, the uncertainties are modeled as Gaussian processes using Karhunen-Loeve (KL) expansion. Second algorithm is based on implicit polynomial iteration method. This algorithm is found to be more efficient when applied to discrete systems. However, the application of the algorithm to continuous systems results in ill-conditioned system matrices, which seriously limit its application. Lastly, an algorithm to find the basis random variables of KL expansion for non-Gaussian processes, is developed. The basis random variables are obtained via nonlinear transformation of marginal cumulative distribution function using standard deviation. Results are obtained for three known skewed distributions, Log-Normal, Beta, and Exponential. In all the cases, it is found that the proposed algorithm matches very well with the known solutions and can be applied to solve non-Gaussian process using SSFEM. / Ph. D.

Ce travail présente quelques résultats concernant des méthodes numériques déterministes et probabilistes pour des équations aux dérivées partielles à coefficients aléatoires, avec des applications à l'hydrogéologie. On s'intéresse tout d'abord à l'équation d'écoulement dans un milieu poreux en régime stationnaire avec un coefficient de perméabilité lognormal homogène, incluant le cas d'une fonction de covariance peu régulière. On établit des estimations aux sens fort et faible de l'erreur commise sur la solution en tronquant le développement de Karhunen-Loève du coefficient. Puis on établit des estimations d'erreurs éléments finis dont on déduit une extension de l'estimation d'erreur existante pour la méthode de collocation stochastique, ainsi qu'une estimation d'erreur pour une méthode de Monte-Carlo multi-niveaux. On s'intéresse enfin au couplage de l'équation d'écoulement considérée précédemment avec une équation d'advection-diffusion, dans le cas d'incertitudes importantes et d'une faible longueur de corrélation. On propose l'analyse numérique d'une méthode numérique pour calculer la vitesse moyenne à laquelle la zone contaminée par un polluant s'étend. Il s'agit d'une méthode de Monte-Carlo combinant une méthode d'élements finis pour l'équation d'écoulement et un schéma d'Euler pour l'équation différentielle stochastique associée à l'équation d'advection-diffusion, vue comme une équation de Fokker-Planck. / This work presents some results about probabilistic and deterministic numerical methods for partial differential equations with stochastic coefficients, with applications to hydrogeology. We first consider the steady flow equation in porous media with a homogeneous lognormal permeability coefficient, including the case of a low regularity covariance function. We establish error estimates, both in strong and weak senses, of the error in the solution resulting from the truncature of the Karhunen-Loève expansion of the coefficient. Then we establish finite element error estimates, from which we deduce an extension of the existing error estimate for the stochastic collocation method along with an error estimate for a multilevel Monte-Carlo method. We finally consider the coupling of the previous flow equation with an advection-diffusion equation, in the case when the uncertainty is important and the correlation length is small. We propose the numerical analysis of a numerical method, which aims at computing the mean velocity of the expansion of a pollutant. The method consists in a Monte-Carlo method, combining a finite element method for the flow equation and an Euler scheme for the stochastic differential equation associated to the advection-diffusion equation, seen as a Fokker-Planck equation.

Les différentes structures composant les moteurs aéronautiques requièrent des analyses dynamiques afin de prédire leur durée de vie. Pour des raisons d'allègement, les roues aubagées fixes de turbomachines, appelées redresseurs, sont conçus comme des ensembles de secteurs comportant plusieurs aubes. Cette architecture rompt la symétrie cyclique empêchant l'application des méthodes numériques l'exploitant. De plus, les dispersions géométriques et matériaux génèrent un désaccordage involontaire impliquant des zones de forte densité modale, dans lesquelles est observée une amplification de la réponse vibratoire, accrue par le caractère monobloc, et donc peu amorti, des secteurs. Une méthodologie statistique de prédiction du niveau vibratoire d'un secteur de redresseur désaccordé aléatoirement est développée ici. La modélisation des incertitudes est basée sur une approche paramétrique de la théorie probabiliste : des paramètres matériaux aléatoires suivant une loi uniforme sont associés à différentes parties du secteur. Une expansion de Karhunen-Loève permet de réduire le champ stochastique à un petit nombre de variables aléatoires et donc de diminuer les temps de calcul. Les modes stochastiques sont ensuite projetés sur ces espaces aléatoires par le biais de deux méthodes d'interpolation non-intrusives. La première est basée sur une projection sur une base du chaos polynomial tandis que la deuxième est une méthode de régression non-paramétrique (méthode MARS). Afin d'appliquer les deux méthodes de calcul à un modèle industriel, une méthode de double synthèse modale est appliquée permettant de diviser le temps de calcul des modes par un facteur d'environ 300. La sous-structuration adoptée s'adapte à la méthode de modélisation des incertitudes et s'avère robuste vis-à-vis du désaccordage. De plus, les deux méthodes permettent d'obtenir des résultats prédictifs en termes de moments statistiques tout en réduisant les temps de calculs. Enfin, la méthodologie est validée expérimentalement puisque l'enveloppe vibratoire numérique encadre la réponse fréquentielle expérimentale au niveau de la zone des modes d'intérêt. Une stratégie de positionnement des jauges de déformation est proposée à partir d'une distribution statistique des déplacements maximaux à mi-hauteur de veine sur une plage fréquentielle donnée. / Aircraft engine components necessitate extensive dynamical analyses in order to obtain life cycle prediction. In order to lighten the structure, turbomachinery stator bladed disks, called stator vanes, are designed as a set of multiple blades clusters. This architecture implies a loss of cyclic symmetry condition and prevents the use of numerical methods using it. Moreover, geometric dispersions and materials defaults generate an involuntary mistuning involving high modal density areas, in which is observed an amplification of the vibratory response, enhanced by the monobloc character - and hence low damped - of stator vanes. A statistical methodology for predicting the vibratory level of a randomly mistuned industrial stator vanes is developed here. Uncertainties modelization is based on a parametric approach of the probability theory : material random parameters following a uniform distribution are associated with different cluster's parts. A Karhunen-Loeve expansion reduces the stochastic field to a small number of random variables and therefore reduces the computation time. Stochastic modes are then projected on these random spaces through two non-intrusive methods of interpolation. The first is based on a projection on a polynomial chaos basis while the second is non-parametric regression method (MARS method). In order to implement both numerical methods to an industrial model, a double modal synthesis method is applied to divide the calculation time of modes by a factor around 300. The sub-structuring way adopted fits the uncertainties modelization method and is robust towards mistuning. Moreover, both methods yield predictive results in terms of statistical moments while reducing computation time. Finally, the methodology is experimentally validated because the numerical vibratory envelope frames the experimental frequency response at the area of the modes of interest. A positioning strategy of strain gauges is proposed based on a statistical distribution of the maximum displacements in vein halfway over a given frequency range.

Nous considérons les tests d'ajustement de type Cramér-von Mises pour tester l'hypothèse que le processus de diffusion observé est un "switching diffusion", c'est-à-dire un processus de diffusion à changement de régime dont la dérive est de type signe. Ces tests sont basés sur la fonction de répartition empirique et la densité empirique. Il est montré que les distributions limites des tests statistiques proposés sont définis par des fonctionnelles de type intégrale des processus Gaussiens continus. Nous établissons les développements de Karhunen-Loève des processus limites correspondants. Ces développements nous permettent de simplifier le problème du calcul des seuils. Nous étudions le comportement de ces statistiques sous les alternatives et nous montrons que ces tests sont consistants. Pour traiter les hypothèses de base composite nous avons besoin de connaître le comportement asymptotique des estimateurs statistiques des paramètres inconnus, c'est pourquoi nous considérons le problème de l'estimation des paramètres pour le processus de diffusion à changement de régime. Nous supposons que le paramètre inconnu est à deux dimensions et nous décrivons les propriétés asymptotiques de l'estimateur de maximum de vraisemblance et de l'estimateur bayésien dans ce cas. L'utilisation de ces estimateurs nous ramène à construire les tests de type Cramér-von Mises correspondants et à étudier leurs distributions limites. Enfin, nous considérons deux tests de type Cramér-von Mises de processus de diffusion ergodiques dans le cas général. Il est montré que pour le choix de certaines des fonctions de poids ces tests sont asymptotiquement " distribution-free ". Pour certains cas particuliers, nous établissons les expressions explicites des distributions limites de ces statistiques par le calcul direct de la transformée de Laplace.

The thesis consists of four papers on numerical complexityanalysis of weak approximation of ordinary and partialstochastic differential equations, including illustrativenumerical examples. Here by numerical complexity we mean thecomputational work needed by a numerical method to solve aproblem with a given accuracy. This notion offers a way tounderstand the efficiency of different numerical methods. The first paper develops new expansions of the weakcomputational error for Itˆo stochastic differentialequations using Malliavin calculus. These expansions have acomputable leading order term in a posteriori form, and arebased on stochastic flows and discrete dual backward problems.Beside this, these expansions lead to efficient and accuratecomputation of error estimates and give the basis for adaptivealgorithms with either deterministic or stochastic time steps.The second paper proves convergence rates of adaptivealgorithms for Itˆo stochastic differential equations. Twoalgorithms based either on stochastic or deterministic timesteps are studied. The analysis of their numerical complexitycombines the error expansions from the first paper and anextension of the convergence results for adaptive algorithmsapproximating deterministic ordinary differential equations.Both adaptive algorithms are proven to stop with an optimalnumber of time steps up to a problem independent factor definedin the algorithm. The third paper extends the techniques to theframework of Itˆo stochastic differential equations ininfinite dimensional spaces, arising in the Heath Jarrow Mortonterm structure model for financial applications in bondmarkets. Error expansions are derived to identify differenterror contributions arising from time and maturitydiscretization, as well as the classical statistical error dueto finite sampling. The last paper studies the approximation of linear ellipticstochastic partial differential equations, describing andanalyzing two numerical methods. The first method generates iidMonte Carlo approximations of the solution by sampling thecoefficients of the equation and using a standard Galerkinfinite elements variational formulation. The second method isbased on a finite dimensional Karhunen- Lo`eve approximation ofthe stochastic coefficients, turning the original stochasticproblem into a high dimensional deterministic parametricelliptic problem. Then, adeterministic Galerkin finite elementmethod, of either h or p version, approximates the stochasticpartial differential equation. The paper concludes by comparingthe numerical complexity of the Monte Carlo method with theparametric finite element method, suggesting intuitiveconditions for an optimal selection of these methods. 2000Mathematics Subject Classification. Primary 65C05, 60H10,60H35, 65C30, 65C20; Secondary 91B28, 91B70. / QC 20100825