Seminars

Many applied optimization problems are often modeled via tensors and polynomials. For example, data compression or denoising can be cast as the problem of low-rank approximation of a matrix (more generally, a tensor). Finding the maximal cardinality of a stable set in a finite graph reduces to minimizing a multivariate polynomial over the Euclidean sphere. Understanding the structure of such optimization problems naturally leads to research questions in (real) algebraic geometry with interesting (and sometimes unexpected) connections to other fields.
In the talk I will give a general overview of my research in this area presenting some of the techniques and ideas from algebraic/computational/convex/differential geometry, probability, harmonic analysis, approximation theory, etc.

The construction of a geological numerical model is a key step in the study and exploration of the subsurface. These models are constructed from seismic or well data, which consist of data points associated with values corresponding to their geological ages. This task involves constructing an implicit function, known also as stratigraphic function, which interpolates this set of data points. Often the available data are sparse and noisy, which makes this task difficult, mainly for reservoirs where the geological structures are complex with distinct discontinuities and unconformities. To address this, the interpolation problem is typically supplemented with a regularization term that enforces a regular behaviour of the implicit function.

In this thesis, we propose a new method to compute the stratigraphic function that represents geological layers in arbitrary settings. In this method, the data are interpolated by piecewise quadraticPowell-Sabin splines and the function can be regularized via many regularization energies. The method is discretized in finite elements on a triangular mesh conforming to the geological faults. Compared to classical interpolation methods, the use of piecewise quadratic splines has two major advantages. First, a better handling of stratigraphic surfaces with strong curvatures. Second, a reduction in mesh resolution, while generating surfaces of higher smoothness and regularity.

The regularization of the function is the most difficult component of any implicit modeling approach. Often, classical methods produce inconsistent geological models, in particular for data with high thickness variation, and bubble effects are generally observed. To handle this problem, we introduce two new regularization energies that are linked to two fundamental PDEs, in their general form with spatially varying coefficients. These PDEs are the anisotropic diffusion equation and the equation that describes the bending of an anisotropic thin plate. In the first approach, the diffusion tensor is introduced and iteratively adapted to the variations and anisotropy of the data. In the second, the rigidity tensor is iteratively adapted to the variations and anisotropy in the data. We demonstrate the effectiveness of the proposed methods in 2D, specifically on cross-sections of geological models with complex fault networks and thickness variations in the layers.

Key words: Implicit modeling - Structural modeling - High thickness variations - Splines - anisotropic PDE - Interpolation.

Jury:

Reviewers

* Géraldine Morin, Professor, IRIT - N7 - Toulouse INP Université de Toulouse.
* Guillaume Caumon, Professor, RING, ENSG-GeoRessources, Université de Lorraine.

Examiners

* Boniface Nkonga, Professor, Université Côte d’Azur (president of the jury)
* Colin Daly, Doctor, SLB Schlumberger, Oilfield UK
* Bernard Mourrain, Research Director, INRIA d’Université Côte d’Azur

Guest

* Azeddine Benabbou, Doctor, SLB Schlumberger Services Pétroliers Montpellier

Supervised by: Bernard Mourrain
Co-supervised by: Azeddine Benabbou

[see  https://intranet.inria.fr/en/News/PhD-Defense-of-Ayoub-Belhachmi-Aromath-Team]