Résumé : Cette thèse à pour objet l'étude théorique et numérique de solutions dans les équations aux dérivées partielles non linéaires de la physique, en particulier en dynamique des fluides. La présence de discontinuités dans les solutions de ces équations complique la compréhension mathématique des phénomènes mis enjeu et leur traitement numérique, notamment en vue de simulations informatiques . Nous étudions ces équations par une méthode de régularisation dans un espace fonctionnel approprié. Lorsque des schémas numériques construits par des méthodes différentes conduisent à des résultats identiques, ceci jusque dans leurs moindres détails, il semble alors naturel de s'interroger dans quelle mesure ces suites de solutions numériques constituent une approximation d'une solution des équations étudiées. Nous construisons des suites de solutions approchées à partir d'un schéma numérique original,stable et suffisamment simple pour démontrer que ses suites constituent une méthode asymptotique de Maslov au sens des distributions en dimension trois d'espèce. La technique de régularisation employée consiste à étendre les variables réelles du problème ne des variables complexes, ce qui nous permet de construire des familles de solutions particulières que l'on ramène au cas réel en faisant tendre un petit paramètre vers O. Les solutions physiques recherchées apparaissent alors comme valeurs au bord de fonction holomorphes. Nous illustrons les résultats obtenus par des applications en cosmologie dans les cadres Newtoniens et relativistes pour des systèmes sans pression, puis avec pression et auto-gravitation, ainsi que pour le système des gaz parfaits.
This thesis is devoted to the theoretical and numerical study of singular solutions appearing in nonlinear partial differential complicates the mathematical understanding of the phenomena under concem as well as their numerical treatment, in particular in view of computation. These equations are studied by a regularization method in an appropriate functional space. When completely different numerical methods give the same results up to the smallest details one can reasonably expect that these numerical results suggest the existence of a mathematical solution of theses equations. We construct sequences of approximate solutions from an original numerical scheme, which is stable and simple enough to prove that these sequences constitute a Maslov asymptotic method in three space dimension. The regularization technique in use consits in extending the real variables of the problem into complex ones, which perrnits to construct families of particular equations that we bring back to the real case by letting a small paramater tend to zero. The expected physical solutions appear as boundary values of holomorphie functions . Illustrations are given by applications to cosmology in the Newtorian and re1ativistic settings for pressure1ess fluid dynamics, then in presence of self-gravitation and pressure as weil as for the systemof ideal gases

http://www.theses.fr/2011AGUY0478/document

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
We show that the lower limit of a sequence of maximal monotone operators on a reflexive Banach space is a representable monotone operator. As a consequence, we obtain that the variational sum of maximal monotone operators and the variational composition of a maximal monotone operator with a linear continuous operator are both representable monotone operators.
ISSN: 0927-6947

hal-00698802
https://hal.archives-ouvertes.fr/hal-00698802

Éditeur(s) : HAL CCSD
Résumé : Point processes are often used as spatial or spatio-temporal distribution models of occurrences. In this PhD dissertation, we focus first on Cox processes driven by a hidden process associated with a Dirichlet process. This model corresponds to hidden occurrences influencing the stochastic intensity of observed occurrences. We generalize the notion of "Shot noise Cox process" introduced by M{o}ller and develop its bayesian analysis by a Gibbs sampler combined with a Metropolis-Hastings algorithm. We show that our MCMC method is a reversible jump one. The model takes into account a random number of hidden contributions producing effects on the observed point process intensity. Therefore the parameter space has a variable dimension. We focus the statistical inference on the estimation of the hidden contribution expected value, the hidden contribution expected number, the spatial influence and correlation parameters. The contribution equality test and contribution independence test are proposed. Applications in epidemiology and ecology are shown from extit{Rubus fruticosa} data, extit{Ibicella lutea} data and death number data in counties of Georgia, USA. Two situations are considered with respect to available data~: firstly, the spatial positions of occurrences are observed between several pairs of consecutive dates; secondly, counts are carried out over a fixed time interval in several spatial sampling units. Secondly, we focus on point processes with memory introduced by Kagan, Ogata and Vere-Jones, pioneers in statistical seismology. In fact, spatio-temporal point processes play an important role in the studies of earthquake catalogs since they consist of seismic events with their dates and spatial locations. We studied an ETAS (Epidemic Type Aftershock Sequence) model with time independent background intensity and several triggering functions taking into account previous events. We illustrate our approach with a seismicity study of the Lesser Antilles arc. A comparaison study of Gamma, Weibull, Log-Normal and modified Omori law triggering function models is also carried out. We show that the modified Omori law does not fit the Lesser Antilles seismic data and the best adjusted triggering function is the Weibull model. Consequently, the waiting time between aftershocks is weaker in the Lesser Antilles zone compared to the one in regions with seismicity described by the modified Omori law. In other words, aftershock aggregativity is higher in the Lesser Antilles region. The stochastic background intensity following a Dirichlet process centered on a spatial log-normal process is discussed.
Les processus ponctuels sont souvent utilisés comme modèles de répartitions spatiales ou spatio-temporelles d'occurrences. Dans cette thèse, nous nous intéressons tout d'abord à des processus de Cox dirigés par un processus caché associé à un processus de Dirichlet. Ce modèle correspond à des occurrences cachées influençant l'intensité stochastique des occurrences observées. Nous généralisons la notion de " Shot noise Cox process " introduite par Moller et développons le traitement bayésien par un échantillonneur de Gibbs combiné à un algorithme de Metropolis-Hastings. Nous montrons que cette méthode MCMC est à sauts réversibles. Le modèle prend en compte, en effet, un nombre aléatoire de contributions cachées influençant l'intensité du processus ponctuel observé donc a un espace paramétrique de dimension variable. Nous focalisons l'inférence statistique sur l'estimation de la valeur espérée de chaque contribution cachée, le nombre espéré de contributions cachées, le degré d'influence spatiale de ces contributions et leur degré de corrélation. Le test d'égalité des contributions et celui de leur indépendance sont ainsi développés. L'utilité en épidémiologie et en écologie est alors démontrée à partir de données de Rubus fruticosa, Ibicella lutea et de mortalité dans les cantons de Georgia, USA. En termes de données observées, deux situations sont considérées: premièrement, les positions spatiales des occurrences sont observées entre plusieurs paires de dates consécutives; deuxièmement, des comptages sont effectués, au cours d'une période fixée, dans des unités d'échantillonnage spatiales. D'autre part, nous nous intéressons aux processus ponctuels à mémoire introduits par Kagan, Ogata et Vere-Jones, précurseurs de la statistique sismologique. En effet, les processus ponctuels spatio-temporels ont une place importante dans l'étude des catalogues sismiques puisque ces derniers sont généralement constitués d'événements sismiques datés et géo-référencés. Nous avons étudié un modèle ETAS (Epidemic Type Aftershock Sequence) avec une intensité d'arrière-plan indépendante du temps et plusieurs fonctions déclenchantes permettant d'intégrer les événements antérieurs récents. Cette approche est utilisée pour étudier la sismicité de l'arc des Petites Antilles. Une étude comparative des modèles Gamma, Weibull, Log-Normal et loi d'Omori modifiée pour les fonctions déclenchantes est menée. Nous montrons que la loi d'Omori modifiée ne s'ajuste pas aux données sismiques des Petites Antilles et la fonction déclenchante la plus adaptée est le modèle de Weibull. Cela implique que le temps d'attente entre répliques dans la zone des Petites Antilles est plus faible que celui des régions à sismicité décrite par la loi d'Omori modifiée. Autrement dit, l'agrégation des répliques après un événement majeur est plus prononcée dans la zone des Petites Antilles. La possibilité d'inclure une intensité d'arrière-plan suivant un processus de Dirichlet centré sur un processus spatial log-gaussien est discutée.
https://tel.archives-ouvertes.fr/tel-00841146

Droits : info:eu-repo/semantics/OpenAccess
NNT : 2012AGUY0555
tel-00841146
https://tel.archives-ouvertes.fr/tel-00841146
https://tel.archives-ouvertes.fr/tel-00841146/document
https://tel.archives-ouvertes.fr/tel-00841146/file/these_VALMY.pdf

Éditeur(s) : HAL CCSD Springer
Résumé : The analysis of three different regions of the South- Pyrenean fold-and-thrust belt reveals that during the Tertiary compression the hydrological system was compartmentalised in time and space. During the early-middle Eocene, when the thrust front affected marine soft-sediments in the Ainsa basin, the thrust fault zones were dominated by formation fluids derived from Eocene marine waters trapped in the underlying Eocene marls, although influences of meteoric waters were also present. During the middle-late Eocene, when the thrust front emplaced marine rocks over continental redbeds in the eastern Catalan basin (L'Escala thrust), the thrust fault zones were dominated by meteoric fluids. These fluids flowed preferentially along these faults, draining laterally the meteoric fluids and acting as barriers hindering their flowing towards more external parts of the belt. During the Oligocene, the most external part of the fold-and-thrust belt in the eastern Catalan basin developed on top of a salt detachment horizon. The thrust front affected continental materials of late Eocene-Oligocene age. At this moment, the thrusts were conduits for meteoric fluids arriving from the surface and also for evolved meteoric fluids migrating over short distance upwards after being in contact with the underlying evaporitic beds. Most of the fractures show a similar sequence of microfractures. Microfractures of stage 1 formed when the sediment was poorly lithified. Microfractures of stage 2 represent the main episode of vein formation which developed when a progressive induration of the host sediment occurred. During microfracture stage 3, formed in an extensional regime, the host sediment was more indurated. The repetition of this sequence of microfractures in different fracture generations of the same outcrop indicates that the sediment induration was restricted to the vicinity of the vein. Away from the veins, the sediment remained poorly lithified during the entire deformation cycle. Calcite cement within the host rock precipitated later than the syn-compressive veins, when the sediment was more indurated. Elemental geochemistry and stable isotopes of the calcite veins indicates that early generation of microfractures is infilled by local fluids (either marine or meteoric), whereas external fluids (meteoric or evolved meteoric) infilled the main compressive stage of microfractures. The hot temperature of these fluids (157°C to 183°C in the Atiart-Arro example) indicates their circulation through deep parts of the thrust belt. The progressive increase of the 87Sr/86Sr ratio through time is due to the progressive uplift, exposure and erosion of the internal Pyrenean Axial Zone.
Thrust belts and foreland basins, from fold kinematics to hydrocarbon systems

hal-00408001
https://hal.archives-ouvertes.fr/hal-00408001
DOI : 10.1007/978-3-540-69426-7_5

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
The aim of this study is the approximation of a solution x* of a ganeralized equation 0 in f(x)+F(x) in Banach spaces, where f is a single-valued function whose second order Frechet derivative satisfies an Hölder condition and F stands for a set-valed map with closed graph. Using a fixed point theorem and the Aubin property of F, we show the existence and the superquadratic convergence of a sequence derived from a midpoint method.
ISSN: 0167-8019

hal-00642118
https://hal.archives-ouvertes.fr/hal-00642118

Éditeur(s) : HAL CCSD
Résumé : International audience
Starting from a description of various generalized function algebras based on sequence spaces, we develop the general framework for considering linear problems with singular coefficients or non linear problems. Therefore, we prove functorial properties of those algebras and show how weak equalities, in the sense of various associations, can be described in this setting.
International Journal of Mathematics and Mathematical Sciencess

hal-00761083
https://hal.univ-antilles.fr/hal-00761083
https://hal.univ-antilles.fr/hal-00761083/document
https://hal.univ-antilles.fr/hal-00761083/file/0210249v1.pdf

Éditeur(s) : HAL CCSD
Résumé : The convex feasibility problem (CFP) is to find a member of the intersection of finitely many closed convex sets in Euclidean space. When the intersection is empty, one can minimize a proximity function to obtain an approximate solution to the problem. The split feasibility problem (SFP) and the split equality problem (SEP) are generalizations of the CFP. The approximate SFP (ASFP) and approximate SEP (ASEP) involve finding only approximate solutions to the SFP and SEP, respectively. We present here the SSEA, a simultaneous iterative algorithm for solving the ASEP. When this algorithm is applied to the ASFP it resembles closely, but is not equivalent to, the CQ algorithm. The SSEA involves orthogonal projection onto the given closed convex sets. The relaxed SSEA (RSSEA) is an easily implementable variant of the SSEA that uses orthogonal projection onto half-spaces at each step to solve the SEP. The perturbed version of the SSEA (PSSEA) is similar to the RSSEA, but uses orthogonal projection onto a sequence of epi-convergent closed convex sets.
https://hal.univ-antilles.fr/hal-00776640

Droits : info:eu-repo/semantics/OpenAccess
hal-00776640
https://hal.univ-antilles.fr/hal-00776640
https://hal.univ-antilles.fr/hal-00776640/document
https://hal.univ-antilles.fr/hal-00776640/file/CBAM10.pdf

Éditeur(s) : HAL CCSD Elsevier
Résumé : Pb, Hf, Nd and Sr isotopes of basaltic lavas from the two Reunion Island volcanoes are reported in order to examine the origin of the sources feeding these volcanoes and to detect possible changes through time. Samples, chosen to cover the whole lifetime of the two volcanoes (from 2 Ma to present), yield a chemically restricted (compared to 0113 lavas) but complex distribution. Reunion plume isotopic characteristics have been defined on the basis of the composition of uncontaminated shield-building lavas from the Piton de la Fournaise volcano. The average epsilon(Nd), epsilon(Hf), Sr-87/Sr-86 and Pb-206/Pb-204, Pb-207/Pb-204, and Pb-208/Pb-204 isotope ratios calculated for this component are +4.4, +9.1, 0.70411, 18.97, 15.59 and 39.03, respectively. In Pb-Pb isotope space, each volcano defines a distinct linear trend but slight variation are also detected within the various volcanic sequences. The Piton des Neiges volcano yields a distinct and significantly more scattered isotopic distribution than Piton de la Fournaise for both Pb, Hf and Nd isotope tracers. A principal component analysis of the Pb isotope data from Piton de la Fournaise reveals a major contribution of the C and EM-1 components (with a clear Dupal flavor) as main components for the modem Reunion plume. The same components have been identified for Piton des Neiges but with a stronger participation of a depleted mantle component and a weaker EM-1 contribution. The compositional change of the lavas erupted by the Piton des Neiges and Piton de la Fournaise volcanoes is attributed to the impingement of two small-scale blobs of plume material at the base of the Reunion lithosphere. Compared to other hot-spots worldwide, in particular Hawaii and Kerguelen, magmas beneath Reunion are generated from a considerably more homogeneous, compositionally more primitive plume higher in Pb-206. Although shallow-level contamination processes have been locally detected they did not alter significantly the composition of the plume magmas. This is tentatively attributed to mantle dynamics producing small, high-velocity blobs that ascend rapidly through the lithosphere, and to the lack of a well-developed magma chamber at depth in the lithosphere.
ISSN: 0012-821X

hal-00412196
https://hal.archives-ouvertes.fr/hal-00412196
DOI : 10.1016/j.epsl.2007.11.018

Éditeur(s) : HAL CCSD
Résumé : Subduction is one of the major geodynamical process active during the Earth evolution. The characterization of arc magmas, their evolution through space and time, and the relationship between arc magmatism and continental crust forming processes, constitute the main research topics in petrology and geochemistry since the last three decades. This study focuses on an exhumed portion of the Kohistan Arc (N Pakistan), which is an exceptional ‘in situ' laboratory to study and follow through time the processes active on subduction zones. Based on a pluri-geochemical approach, this study allows unraveling two major issues : 1/ The lack of a cogenetic relationship between the arc root (i.e. ultramafic rocks) and the overlying gabbroic crustal section. The Jijal ultramafic section does not bear a crustal origin and was emplaced at ~117 Ma via a magma-rock reaction between REE-depleted ‘boninite-like' melts and the pre-existing lithospheric mantle with Indian-MORB affinities; 2/ A three-stage geodynamical model covering the arc evolution through ~30 Ma. This model starts with the onset of the subduction followed by the building of the volcanic arc (1st stage). The 2nd stage corresponds to a major thermal event, characterized by abundant magma underplating and by granulitisation of the arc base. Recycling of the residual/cumulative lower crust into the convective asthenospheric mantle was efficient during this stage, and probably related to thermo-mechanical erosion processes. The last stage, between ~95 and 85 Ma, seals the end of the intra-oceanic subduction and corresponds to an “amagmatic” period. This step is followed by a voluminous, but short, magmatic pulse at ~85 Ma, occurring just before the arc-continent collision. At last, this study emphasizes the more mafic composition of the investigated crustal arc section when compared to the bulk continental crust. Nevertheless the arc section composition nicely fits with the lower continental crust average.
Les zones de subduction représentent des zones d'évolution clés dans la compréhension des processus géologiques majeurs actifs sur notre planète. En particulier, la caractérisation et l'évolution des magmas générés dans les zones d'arcs océaniques et la relation entre leur genèse et les processus de croissance crustale sont des axes primordiaux. Cette étude est focalisée sur la portion d'arc océanique exhumée du Kohistan (Nord Pakistan) qui constitue un laboratoire exceptionnel pour appréhender cette problématique. Basée sur une approche géochimique multi-méthodes (éléments majeurs et en trace, isotopes), cette étude a permis d'établir : 1/ l'absence de relation génétique directe entre les roches ultrabasiques de la racine de l'arc et la section crustale sus-jacente. La formation de la séquence ultrabasique de Jijal se produit à ~117 Ma via une réaction de type magma-roche entre des liquides de type boninitique (appauvris en terres rares), et le manteau lithosphérique de type MORB-Indien ; 2/ un modèle géodynamique en trois stades majeurs résumant l'évolution de l'arc océanique et de la subduction sur une période d'environ 30 Ma. Ce modèle débute par l'initiation de la subduction et la formation de l'arc volcanique s.s. (1er stade) ; le 2ème stade correspond à un évènement thermique majeur. Il est représenté par un sous-placage important de magmas et une granulitisation intense de la base de l'arc. Le recyclage de la croûte inférieure cumulative et résiduelle, dans le manteau sous-jacent, se produit durant cette étape vraisemblablement suite à des processus d'érosion thermo-mécanique. Le dernier stade, entre 95 Ma et 85 Ma, scelle la fin du fonctionnement de la zone de subduction et correspond à une période amagmatique suivie par une brève reprise du magmatisme avant la collision de l'arc après 85 Ma. Enfin, une modélisation numérique comparative avec la croûte continentale globale met en évidence la composition significativement plus basique de la croûte de l'arc insulaire du Kohistan. De fortes similitudes sont en revanche observées entre la section d'arc étudiée et la croûte continentale inférieure.
https://tel.archives-ouvertes.fr/tel-00437770

tel-00437770
https://tel.archives-ouvertes.fr/tel-00437770
https://tel.archives-ouvertes.fr/tel-00437770/document
https://tel.archives-ouvertes.fr/tel-00437770/file/thesedhuime.pdf

Éditeur(s) : HAL CCSD
Résumé : International audience
The author proposes a generalized proximal method for solving an equilibrium problem which consists in finding x 2 K such that F(x, y) 0, for all y 2 K, where K is a nonempty, convex and closed set of a real Hilbert space X, and F:K ×K !Ris a given bifunction with F(x, x) = 0 for all x 2 K. The weak convergence of the sequence generated by the method is proved under the assumptions of monotonicity and convexity, with respect to the second argument y (for every fixed x 2 K), of the bifunction F. Replacing the assumption of monotonicity with the one of strong monotonicity on F, a strong convergence result is obtained. A second strong convergence theorem is proved under the hypotheses of monotonicity and a new assumption of "co-Lipschitz continuity at 0" of the operator F. Applications to convex optimization, to the problem of finding a zero of a maximal monotone operator and to Nash equilibria problems are provided.
Journal of Natural Geometry

hal-00773194
https://hal.univ-antilles.fr/hal-00773194