Les deux questions de connexité auxquelles on s'intéresse concernent : – l'espace des feuilletages de codimension 1 sur une variété de dimension 3 ; – l'espace des représentations du groupe Z^2 dans le groupe des difféomorphismes lisses de l'intervalle. Le résultat principal, qu'on démontre dans la seconde partie de la thèse, est le suivant : si deux feuilletages de codimension 1 sur une variété close de dimension 3 ont des sous-fibrés tangents homotopes, on peut les relier par un chemin de feuilletages. Cet énoncé cache une subtilité : si les feuilletages donnés sont lisses, le chemin obtenu peut contenir, près de ses extrémités, des feuilletages qui ne sont que C^1. Cela vient de ce qu'on ne sait pas si l'espace des représentations de Z^2 dans les difféomorphismes de l'intervalle est connexe ou non. En tentant de répondre à cette question, on a montré le phénomène suivant qui fait l'objet de la première partie de la thèse : de nombreux difféomorphismes lisses de R+, sans autre point fixe que l'origine, ont un centralisateur C^infini non dénombrable et dense dans leur centralisateur C^1, lequel est un groupe à un paramètre. On discute également les propriétés arithmétiques de ce sous-groupe.

Für die geodätische Überwachung dreidimensionaler Objekte fehlen in der Regel Informationen zur vollständigen Beschreibung der räumlichen Deformation. Um dieses Informationsdefizit zu beheben, können geotechnische Beobachtungen genutzt werden. In der Dissertation wird die Integration dieser Relativmessungen durch ein erweitertes Krigingverfahren gelöst. Dazu werden für den Signal- und Trendanteil der Beobachtungen die nötigen Kovarianz- und Trendmatrizen hergeleitet. Darüberhinaus wird durch die Darstellung der Beobachtungen als lineare Funktionale des Verschiebungsfeldes die Möglichkeit eröffnet, nicht nur das Verschiebungsfeld zu schätzen sondern auch beliebige lineare Funktionale des Verschiebungsfeldes. Das hergeleitete Verfahren wird an zwei praktischen Beispielen demonstriert.

Zugl.: Koblenz, Landau (Pfalz), Univ., Diss., 2009

Für die geodätische Überwachung dreidimensionaler Objekte fehlen in der Regel Informationen zur vollständigen Beschreibung der räumlichen Deformation. Um dieses Informationsdefizit zu beheben, können geotechnische Beobachtungen genutzt werden. In der Dissertation wird die Integration dieser Relativmessungen durch ein erweitertes Krigingverfahren gelöst. Dazu werden für den Signal- und Trendanteil der Beobachtungen die nötigen Kovarianz- und Trendmatrizen hergeleitet. Darüberhinaus wird durch die Darstellung der Beobachtungen als lineare Funktionale des Verschiebungsfeldes die Möglichkeit eröffnet, nicht nur das Verschiebungsfeld zu schätzen sondern auch beliebige lineare Funktionale des Verschiebungsfeldes. Das hergeleitete Verfahren wird an zwei praktischen Beispielen demonstriert.

A three-dimensional inversion was implemented for magnetotellurics, which is a passive electromagnetic method in geophysics. It exploits natural electromagnetic fields of the Earth, which function as sources. Their interaction with the conductive parts of the subsurface are registered when components of the electric and the magnetic field are measured and evaluated. The all-at-once approach is an inversion scheme that is relatively new to geophysics. In this approach, the objective function – the basis of each inversion – is called the Lagrangian. It consists of three parts: (i) the data residual norm, (ii) the regularisation part, and (iii) the forward problem. The latter is the significant difference to conventional inversion approaches that are built up of a forward calculation part and an inversion part. In the case of all-at-once, the forward problem is incorporated in the objective function and is therefore already taken into account in each inversion iteration. Thus, an explicit forward calculation is obsolete. As an objective function, the Lagrangian shall reach a minimum and therefore its first and second derivatives are evaluated. Hence, the gradient of the Lagrangian and its Hessian are constituent parts of the KKT system – the Newton-type system that is set up in the all-at-once inversion. Conventional inversion approaches avoid the Hessian because it is a large, dense, not positive definite matrix that is challenging to handle. However, it provides additional information to the inversion, which raises hope for a high quality inversion result. As a first step, the inversion was programmed for the more straightforward one-dimensional magnetotelluric case. This was particularly suitable to become familiar with sQMR – a Krylov subspace method which is essential for the three-dimensional case to be able to work with the Hessian and the resulting KKT system. After the implementation and validation of the one-dimensional forward operator, the Lagrangian and its derivatives were set up to complete the inversion, which successfully solved the KKT system. Accordingly, the three-dimensional forward operator also needed to be implemented and validated, which was done using published data from the 3D-2 COMMEMI model. To realise the inversion, the Lagrangian was assembled and its first and second derivatives were validated with a test that exploits the Taylor expansion. Then, the inversion was initially programmed for the Gauss-Newton approximation where second order information is neglected. Since the system matrix of the Gauss-Newton approximation is positive definite, the solution of this system of equations could be carried out by the conventional solver pcg. Based on that, the complete KKT system (Newton\\\'s method) was set up and preconditioned sQMR solved this system of equations.

Tetrahedral meshes are becoming more and more important for geo-modeling applications. The presented work introduces new algorithms for efficient visualization and modeling of tetrahedral meshes. Visualization consists of a generic framework that includes the extraction of geological information like stratigraphic columns, fault block boundaries, simultaneous co-rendering of different attributes and boolean operations of Constructive Solid Geometry with constant complexity. Modeling can be classified into geometric and implicit modeling. Geometric modeling addresses local mesh refinement to increase the numerical resolution of a given mesh. Implicit modeling covers the definition and manipulation of implicitly defined models. A new surface reconstruction method was developed to reconstruct complex, multi-valued surfaces from noisy and sparse data sets as they occur in geological applications. The surface can be bounded and may have discontinuities. Further, this work proposes a new and innovative algorithm for rapid editing of implicitly defined shapes like horizons based on the GeoChron parametrization. The editing is performed interactively on the 3d-volumetric model and geological constraints are respected automatically.

Cette thèse porte sur les invariants des sphères d'homologie entière de dimension 3, et en particulier sur les invariants de type fini pour la filtration de Goussarov-Habiro.<br />Dans une première partie, on étudie la variation d'un invariant de degré 2n après chirurgie le long d'une surface par un élément du 2n-ième terme de la série centrale descendante du groupe de Torelli. Dans le cas d'un commutateur de 2n éléments du groupe de Torelli, on exprime cette variation en fonction de l'homomorphisme de Johnson évalué sur ces 2n éléments et du système de poids de l'invariant.<br /><br />Le calcul des claspers de Goussarov-Habiro donne des équivalences topologiques entre des chirurgies sur des corps en anses plongés dans les variétés. Ce calcul a déjà permis de préciser le comportement des invariants de type fini lors de nombreuses modifications topologiques. La deuxième partie de cette thèse est consacrée à un raffinement de ce calcul. Ce raffinement est ensuite appliqué à l'obtention d'une formule de chirurgie géométrique sur les noeuds pour les invariants de degré 4, c'est-à-dire que l'on exprime la variation d'un tel invariant après chirurgie sur un noeud en fonction d'invariants de courbes tracées au voisinage d'une surface de Seifert de ce noeud.

Dans cette thèse on étudie un invariant topologique des variétés de dimension 3, la torsion de Reidemeister, comme un objet global sur les variétés de caractères du groupe fondamental dans SL(2,C). Dans le cas du complexe cohomologique associé à la représentation adjointe, on définit la torsion « adjointe » comme une forme différentielle méromorphe sur la variété des caractères. On reliera l’apparition de pôles ou de zéros à :-des singularités de la variété des caractères-la topologie de certaines surfaces incompressibles plongées, produites via la théorie de Culler-Shalen.On obtiendra, comme conséquence de ces résultats, une formule reliant le genre de ces surfaces incompressibles, et celui de la variété des caractères.Dans le cas du complexe standard, la torsion « acyclique » est une fonction méromorphe sur la variété des caractères. Une étude poussée des pôles apparaissant aux points à l’infini nous permettra, entre autre, de donner des conditions suffisantes pour que la torsion soit non constante. / In this PhD dissertation, we study a topological invariant of 3-manifolds, namely the Reidemeister torsion, as globally defined on character varieties of the fundamental group in SL(2,C). The « adjoint » torsion will be the torsion of the cohomological complex associated to the adjoint representation. We explain that it can be seen as a meromorphic differential form on the character variety, and we aim to understand its poles and zeros. They will be related with -singular points of the character variety -the topology of incompressible surfaces embedded in the 3-manifold, provided by the Culler-Shalen theory. As an application, we prove a relation between the genus of those incompressible surface and the genus of the character variety. The « acyclic » torsion of the standard complex is a rational function on the character variety. We study its poles at infinity in the character variety, and we give sufficient conditions for this torsion to be non constant.

Tetrahedral meshes are becoming more and more important for geo-modeling applications. The presented work introduces new algorithms for efficient visualization and modeling of tetrahedral meshes. Visualization consists of a generic framework that includes the extraction of geological information like stratigraphic columns, fault block boundaries, simultaneous co-rendering of different attributes and boolean operations of Constructive Solid Geometry with constant complexity. Modeling can be classified into geometric and implicit modeling. Geometric modeling addresses local mesh refinement to increase the numerical resolution of a given mesh. Implicit modeling covers the definition and manipulation of implicitly defined models. A new surface reconstruction method was developed to reconstruct complex, multi-valued surfaces from noisy and sparse data sets as they occur in geological applications. The surface can be bounded and may have discontinuities. Further, this work proposes a new and innovative algorithm for rapid editing of implicitly defined shapes like horizons based on the GeoChron parametrization. The editing is performed interactively on the 3d-volumetric model and geological constraints are respected automatically.

A three-dimensional inversion was implemented for magnetotellurics, which is a passive electromagnetic method in geophysics. It exploits natural electromagnetic fields of the Earth, which function as sources. Their interaction with the conductive parts of the subsurface are registered when components of the electric and the magnetic field are measured and evaluated. The all-at-once approach is an inversion scheme that is relatively new to geophysics. In this approach, the objective function – the basis of each inversion – is called the Lagrangian. It consists of three parts: (i) the data residual norm, (ii) the regularisation part, and (iii) the forward problem. The latter is the significant difference to conventional inversion approaches that are built up of a forward calculation part and an inversion part. In the case of all-at-once, the forward problem is incorporated in the objective function and is therefore already taken into account in each inversion iteration. Thus, an explicit forward calculation is obsolete. As an objective function, the Lagrangian shall reach a minimum and therefore its first and second derivatives are evaluated. Hence, the gradient of the Lagrangian and its Hessian are constituent parts of the KKT system – the Newton-type system that is set up in the all-at-once inversion. Conventional inversion approaches avoid the Hessian because it is a large, dense, not positive definite matrix that is challenging to handle. However, it provides additional information to the inversion, which raises hope for a high quality inversion result. As a first step, the inversion was programmed for the more straightforward one-dimensional magnetotelluric case. This was particularly suitable to become familiar with sQMR – a Krylov subspace method which is essential for the three-dimensional case to be able to work with the Hessian and the resulting KKT system. After the implementation and validation of the one-dimensional forward operator, the Lagrangian and its derivatives were set up to complete the inversion, which successfully solved the KKT system. Accordingly, the three-dimensional forward operator also needed to be implemented and validated, which was done using published data from the 3D-2 COMMEMI model. To realise the inversion, the Lagrangian was assembled and its first and second derivatives were validated with a test that exploits the Taylor expansion. Then, the inversion was initially programmed for the Gauss-Newton approximation where second order information is neglected. Since the system matrix of the Gauss-Newton approximation is positive definite, the solution of this system of equations could be carried out by the conventional solver pcg. Based on that, the complete KKT system (Newton\\\'s method) was set up and preconditioned sQMR solved this system of equations.