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A mesh adaptation method is proposed for solving optimal control problems with non-smooth control. The original optimal control problem (OCP) is transcribed into a nonlinear programming (NLP) problem by using the Runge-Kutta discretization method, in which the NLP can be solved by using standard nonlinear programming codes. The method employs collocations from the dyadic background points, which used for the second-generation wavelet (SGW) translation simultaneously. The SGW is used to approximate the control variables and get the wavelet coefficients once they are obtained. In regions contain discontinuities, the magnitude of the relevant wavelet coefficients is large than other regions. The corresponding dyadic background points are reserved as the collocation points. Furthermore, the approximation error of the control and/or state variables can be predicted by a given threshold. Thus, the accuracy and efficiency can be balanced in a simple way. Finally, the method is demonstrated by three numerical examples from the open literature.

This thesis focuses on the development of an accurate and efficient method for performing Large-Eddy Simulation (LES) of turbulent flows. An LES approach based upon the Variational Multiscale (VMS) method is considered. VMS produces an a priori scale-separation of the governing equations, in a manner which makes no assumptions on the boundary conditions and mesh uniformity. In order to ensure that scale-separation in wavenumber is achieved, we have chosen to make use of the Second Generation Wavelets (SGW), a polynomial basis which exhibits optimal space-frequency localisation properties. Once scale-separation has been achieved, the action of the subgrid model is restricted to the wavenumber band closest to the cuto. We call this approach wavelet-based VMS-LES (WAV-VMS-LES). This approach has been incorporated within the framework of a high-order incompressible flow solver based upon pressure-stabilised discontinuous Galerkin FEM (DG-FEM). The method has been assessed by performing highly under-resolved LES upon the 3D Taylor-Green Vortex test case at two different Reynolds numbers., Cette thèse se concentre sur le développement d’une méthode précise et efficace pour la simulation des grandes échelles (LES) des écoulements turbulents. Une approche de la LES basée sur la méthode variationnelle multi-échelles (VMS) est considérée. La VMS applique aux équations de la dynamique des fluides une séparation d’échelles a priori sans recours à des hypothèses sur les conditions aux limites ou sur l’uniformité du maillage. Afin d’assurer effectivement une séparation d’échelles dans l’espace des nombres d’onde associé, nous choisissons d’utiliser les ondelettes de deuxième génération (SGW), une base polynomiale qui présente des propriétés de localisation spatiale-fréquence optimales. A partir de la séparation d’échelles ainsi réalisée, l’action du modèle sous-maille est limitée à un intervalle de nombres d’onde proche de la coupure spectrale. Cette approche VMS-LES basée sur les ondelettes est désignée par WAV-VMS-LES. Elle est incorporée dans un solveur d’ordre élevé pour la simulation des écoulements incompressibles sur la base d’une méthode de Galerkin discontinue (DG-FEM) stabilisée pour la pression. La méthode est évaluée par réalisation de LES sur des maillages fortement sous-résolus pour le cas test du tourbillon de Taylor-Green 3D à deux nombres de Reynolds différents.

In this paper, we propose novel adaptive wavelet filter bank structures based on the lifting scheme. The filter banks are nonseparable, based on quincunx sampling, with their properties being pixel-wise adapted according to the local image features. Despite being adaptive, the filter banks retain a desirable number of primal and dual vanishing moments. The adaptation is introduced in the predict stage of the filter bank with an adaptation region chosen independently for each pixel, based on the intersection of confidence intervals (ICI) rule. The image denoising results are presented for both synthetic and real-world images. It is shown that the obtained wavelet decompositions perform well, especially for synthetic images that contain periodic patterns, for which the proposed method outperforms the state of the art in image denoising.

A dynamic adaptive numerical method for solving partial differential equations on the sphere is developed. The method is based on second generation spherical wavelets on almost uniform nested spherical triangular grids, and is an extension of the adaptive wavelet collocation method to curved manifolds. Wavelet decomposition is used for grid adaption and interpola- tion. An OðN Þ hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate Laplace–Beltrami, Jacobian and flux-divergence operators. The accuracy and efficiency of the method is demonstrated using linear and nonlinear examples relevant to geophysical flows. Although the present paper considers only the sphere, the strength of this new method is that it can be extended easily to other curved manifolds by considering appropriate coarse approximations to the desired manifold (here we used the icosahedral approximation to the sphere at the coarsest level). NSERC

Dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time. These methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time. The same time step is used for all spatial locations and all scales: this approach clearly does not fully exploit space–time intermittency. We propose an adaptive wavelet collocation method for solving highly intermittent problems (e.g. turbulence) on a simultaneous space–time computational domain which naturally adapts both the space and time resolution to match the solution. Besides generating a near optimal grid for the full space–time solution, this approach also allows the global time integration error to be controlled. The efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent (1D + t)-dimensional and (2D + t)-dimensional test problems. In particular, we found that the space–time method uses roughly 18 times fewer space–time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method, while achieving similar global accuracy. JMA and NKRK would like to acknowledge support from NSERC and SHARCNET. Partial support for OVV was provided by the National Science Foundation (NSF) under grants no. EAR-0327269 and ACI- 0242457 and National Aeronautics and Space Administration (NASA) under grant no. NAG-1-02116.

In this work, we propose to follow a multi-scale approach for spatial reservoir properties characterization using direct (well observations) and indirect (seismic and production history) data at different resolutions. Two decompositions are used to parameterize the problem: the wavelets and the Gaussian pyramids. Using these parameterizations, we show the advantages of the multi-scale approach with two uncertainty quantification problems based on minimization. The first one concerns the simulation of property fields from a multiple points geostatistics algorithm. It is shown that the multi-scale approach based on Gaussian pyramids improves the quality of the output realizations, the match of the conditioning data and the computational time compared to the standard approach. The second problem concerns the preservation of the prior models during the assimilation of the production history. In order to re-parameterize the problem, we develop a new 3D grid adaptive wavelet transform, which can be used on complex reservoir grids containing dead or zero volume cells. An ensemble-based optimization method is integrated in the multi-scale history matching approach, so that an estimation of the uncertainty is obtained at the end of the optimization. This method is applied on several application examples where we observe that the final realizations better preserve the spatial distribution of the prior models and are less noisy than the realizations updated using a standard approach, while matching the production data equally well., Dans ce travail, nous proposons de suivre une approche multi-échelles pour simuler des propriétés spatiales des réservoirs, permettant d'intégrer des données directes (observation de puits) ou indirectes (sismique et données de production) de résolutions différentes. Deux paramétrisations sont utilisées pour résoudre ce problème: les ondelettes et les pyramides gaussiennes. A l'aide de ces paramétrisations, nous démontrons les avantages de l'approche multi-échelles sur deux types de problèmes d'estimations des incertitudes basés sur la minimisation d'une distance. Le premier problème traite de la simulation de propriétés à partir d'un algorithme de géostatistique multipoints. Il est montré que l'approche multi-échelles basée sur les pyramides gaussiennes améliore la qualité des réalisations générées, respecte davantage les données et réduit les temps de calculs par rapport à l'approche standard. Le second problème traite de la préservation des modèles a priori lors de l'assimilation des données d'historique de production. Pour re-paramétriser le problème, nous développons une transformée en ondelette 3D applicable à des grilles stratigraphiques complexes de réservoir, possédant des cellules mortes ou de volume négligeable. Afin d'estimer les incertitudes liées à l'aspect mal posé du problème inverse, une méthode d'optimisation basée ensemble est intégrée dans l'approche multi-échelles de calage historique. A l'aide de plusieurs exemples d'applications, nous montrons que l'inversion multi-échelles permet de mieux préserver les modèles a priori et est moins assujettie au bruit que les approches standards, tout en respectant aussi bien les données de conditionnement.

We provide a new algorithm for the treatment of the deconvolution problem on the sphere which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. We establish upper bounds for the behavior of our procedure for any $\mathbb {L}_p$ loss. It is important to emphasize the adaptation properties of our procedures with respect to the regularity (sparsity) of the object to recover as well as to inhomogeneous smoothness. We also perform a numerical study which proves that the procedure shows very promising properties in practice as well., Comment: Published in at http://dx.doi.org/10.1214/10-AOS858 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

International audience; We consider inverse problems where one wishes to recover an unknown function from the observation of a transformation of it by a linear operator, corrupted by an additive white noise perturbation. We assume that the operator admits a singular value decomposition where the eigenvalues decay in a polynomial way, and where Jacobi polynomials appear as eigenfunctions. This includes, as an application, the well known Wicksell's problem. We determine the asymptotic rate of the minimax risk for this model in a wide framework, considering (Lp) losses (1 < p < \infty), and Besov-like regularity spaces. We draw a comparison with the minimax rates of the deconvolution problem, which appears as a critical case of the Jacobi-type rates. We also establish some new results on the needlets introduced by Petrushev and Xu (2005) which appear as essential tools in this setting.

An adaptive multilevel wavelet solver for elliptic equations on an optimal spherical geodesic grid is developed. The method is based on second-generation spherical wavelets on almost uniform optimal spherical geodesic grids. It is an extension of the adaptive multilevel wavelet solver [O. V. Vasilyev and N. K.-R. Kevlahan, J. Comput. Phys., 206 (2005), pp. 412-431] to curved manifolds. Wavelet decomposition is used for grid adaption and interpolation. A hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate the Laplace-Beltrami operator. The optimal spherical geodesic grid [Internat. J. Comput. Geom. Appl., 16 (2006), pp. 75-93] is convergent in terms of local mean curvature and has lower truncation error than conventional spherical geodesic grids. The overall computational complexity of the solver is $O(\mathcal{N})$, where $\mathcal{N}$ is the number of grid points after adaptivity. The accuracy and efficiency of the method is demonstrated for the spherical Poisson equation. Although the present paper considers the sphere, the strength of this new method is that it can be extended easily to other curved manifolds by choosing an appropriate coarse approximation and using recursive surface subdivision.

In this document we present a 3D model-based video coding scheme for streaming static scene video in a compact way but also enabling time and spatial scalability according to network or terminal capability and providing 3D functionalities. The proposed format is based on encoding the sequence of reconstructed models using second generation wavelets, and efficiently multiplexing the resulting geometric, topological, texture and camera motion binary representations. The wavelets decomposition can be adaptive in order to fit to images and scene contents. To ensure time scalability, this representation is based on a common connectivity for all 3D models, which also allows straightforward morphing between successive models ensuring visual continuity at no additional cost. The method proves to be better than previous methods for video encoding of static scenes, even better than state-of-the-art video coders such as H264 (also known as MPEG AVC). Another application of our approach is the fast transmission and real-time visualization of virtual environments obtained by video capture, for virtual or augmented reality, free walk-through in photo-realistic 3D environments, and numerous other image-base applications. / Nous présentons dans ce document un schéma de codage vidéo basé sur des modèles 3D qui permet de compresser efficacement des vidéos de scènes statiques tout en garantissant une scalabilité temporelle et spatiale afin de s'adapter aux capacités du réseau et des terminaux. Le passage par des modèles 3D permettent d'ajouter des fonctionnalités à la vidéo. Le format proposé se base sur l'encodage d'une séquence de modèles 3D extraits à partir de la vidéo en utilisant des ondelettes de seconde génération, et en multiplexant efficacement les représentations binaires résultaants pour la géométrie, la connectivité, la texture et les positions de caméra. La décomposition par ondelettes peut être aadptative afin de s'adapter au contenu des images et de la scène. Afin d'assurer la scalabilité temporelle, cette représentation et basée sur une connectivité commune pour tous les modèles qui permet de plus uu morphing implicite entre les modèles successifs assurant une continuité visuelle. La méthode a permis d'obtenir de meilleurs résultats pour le codage de vidéos de scènes statiques que le codeur vidéo référence de l'état de l'art H264 (également connu sous le nom de MPEG/AVC). Une autre application de notre approche est la transmission rapide et la visualisation temps réel d'environnements virtuels obtenus partir de vidéos pour les réalités augmentée et virtuelle, la navigation photoréalistique dans des environnements 3D et de nombreuses autres applications basées sur les images.

An adaptive multilevel wavelet collocation method for solving multi-dimensional elliptic problems with localized structures is described. The method is based on multi-dimensional second generation wavelets, and is an extension of the dynamically adaptive second generation wavelet collocation method for evolution problems [Int. J. Comp. Fluid Dyn. 17 (2003) 151]. Wavelet decomposition is used for grid adaptation and interpolation, while a hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The multilevel structure of the wavelet approximation provides a natural way to obtain the solution on a near optimal grid. In order to accelerate the convergence of the solver, an iterative procedure analogous to the multigrid algorithm is developed. The overall computational complexity of the solver is O(N), where N is the number of adapted grid points. The accuracy and computational efficiency of the method are demonstrated for the solution of two- and three-dimen- sional elliptic test problems. Partial support for the first author (O.V. Vasilyev) was provided by the National Science Foundation (NSF) under grants No. EAR-0242591, EAR-0327269 and ACI-0242457 and National Aeronautics and Space Administration (NASA) under grant No. NAG-1-02116. This support is gratefully acknowledged. The second author (N.K.-R. Kevlahan) was supported by the NSERC and gratefully acknowledges the use of SHARCNET computational resources.

The principal aim of this thesis is to propose methods for the reconstruction of functions from noisy, random or deterministic nonequispaced data. Two of them rely on first generation wavelets. They consist in a preconditioning / interpolation on a equispaced design of the randomly designed data, folowed by wavelet shrinkage. We show that the resulting estimates are near-minimax on Holder class functions, respectively on Besov balls. We also investigate a method relying on second generation, design adapted wavelets. As a first step, yet independent, we prove that under some conditions the irregular Lagrange subdivision schemes converge and produce functions having the same number of continuous derivatives as the limit functions of the regular schemes of the same degree. Then we show the existence of design adapted multiresolution analysis and wavelet biorthogonal systems, constructed by average-interpolating subdivision. We conclude with some numerical simulations, illustrating the finite sample behaviour of the three methods., Cette thèse porte sur la restauration des signaux bruités observés sur des plans d'expérience aléatoires. Trois méthodes sont proposées. Dans les deux premières, on se ramène (soit par préconditionnement des données initiales, soit par régression polynomiale locale), à un problème de régression sur grille régulière. Des majorations asymptotiques de l'erreur d'estimation sont données pour les deux méthodes, sur des classes de fonctions holderiennes pour la première et sur des boules d'espaces de Besov pour la deuxième. La vitesse de décroissance de l'erreur est dans les deux cas très proche de la vitesse optimale. Un troisième algorithme concerne les plans d'expérience déterministes et utilise les ondelettes adaptées à la grille. Elles sont construites par des schémas de subdivision non réguliers, dont on étudie la convergence et les propriétés. Des nombreuses simulations et une étude comparative illustrent le comportement des trois algorithmes quand ils sont appliqués à des échantillons de taille finie.

More efficient coding, modelling and image analysis generate the need for search for better methods of the multiresolution image analysis, i.e. for more efficient wavelet filter bank structures. This thesis is based on a previous research of one-dimensional filter bank structures that had the possibility to adapt filter parameters to the properties of the analyzed signal. In this thesis we present the construction of a two-dimensional adaptive wavelet filter bank that is based on a lifting scheme. The filter bank is nonseparable, based on a quincunx polyphase decomposition and nonseparable filters. The lifting scheme has been chosen since it allows for an easy construction of space varying filter banks with a perfect reconstruction property. The proposed filter bank adapts to the analyzed image for every pixel in all decomposition levels while still preserving the good properties of the wavelet decomposition. A number of vanishing moments is guaranteed by the fixed part of the filter bank. Without degrading the overall filter bank properties, the variable part can be changed in order to adapt to the analyzed image. The paper explores various one-dimensional and two-dimensional adaptation methods based on the least squares criterion. Adaptation results have been shown for a number of synthetic and real-world images. Effects of lossy image reconstruction and impact of filter coefcients' quantization to the efficiency of the image decomposition have been presented. Učinkovitije kodiranje, modeliranje i analiza slika stvaraju potrebu za traženjem boljih metoda višerezolucijskog razlaganja slika odnosno potrebu za učinkovitijim strukturama dvodimenzionalnih wavelet filtarskih slogova. Rad se temelji na prethodnom istraživanju struktura jednodimenzionalnih filtarskih slogova koji su imali mogućnost adaptacije filtarskih parametara svojstvima signala. U ovom radu prikazujemo izvedbu dvodimenzionalnog adaptivnog wavelet filtarskog sloga koji se temelji na shemi podizanja. Filtarski slog je neseparabilan, temelji se na quincunx polifaznom razlaganju i neseparabilnim filtrima. Shema podizanja je odabrana jer omogućava jednostavnu izvedbu prostorno promjenjivih filtarskih slogova sa svojstvima savršene rekonstrukcije. Predloženi filtarski slog se prilagođuje analiziranoj slici u svakom slikovnom elementu u svim razinama razlaganja u isto vrijeme zadržavajući dobra svojstva wavelet razlaganja. Dovoljan broj nul-momenata pridruženih wavelet funkcija zagarantiran je nepromjenjivim dijelom filtarskog sloga. Promjenjivi dio filtarskog sloga može se prilagođivati svojstvima slike bez narušavanja cjelokupnih svojstava filtarskog sloga. Istražene su razne jednodimenzionalne i dvodimenzionalne metode adaptacije zasnovane na kriteriju najmanjih kvadrata. Rezultati adaptacije prikazani su za razne sintetske i realne slike. Prezentirati su rezultati rekonstrukcije slike s gubicima i utjecaj kvantizacije filtarskih koeficijenata na učinkovitost adaptivnog razlaganja slike.