Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction

General models of Gibbs Delaunay-Voronoi tessellations, which can be viewed as extensions of Ord's process, are considered. The interaction may occur on each cell of the tessellation and between neighbour cells. The tessellation may also be subjected to a geometric hardcore interaction, forcing the cells not to be too large, too small, or too flat. This setting, natural for applications, introduces some theoretical difficulties since the interaction is not necessarily hereditary. Mathematical results available for studying these models are reviewed and further outcomes are provided. They concern the existence, the simulation and the estimation of such tessellations. Based on these results, tools to handle these objects in practice are presented: how to simulate them, estimate their parameters and validate the fitted model. Some examples of simulated tessellations are studied in details

Phase transitions in Delaunay Potts models

We establish phase transitions for classes of continuum Delaunay multi-type particle systems (continuum Potts models) with infinite range repulsive interaction between particles of different type. In one class of the Delaunay Potts models studied the repulsive interaction is a triangle (multi-body) interaction whereas in the second class the interaction is between pairs (edges) of the Delaunay graph. The result for the edge model is an extension of finite range results in \cite{BBD04} for the Delaunay graph and in \cite{GH96} for continuum Potts models to an infinite range repulsion decaying with the edge length. This is a proof of an old conjecture of Lebowitz and Lieb. The repulsive triangle interactions have infinite range as well and depend on the underlying geometry and thus are a first step towards studying phase transitions for geometry-dependent multi-body systems. Our approach involves a Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. The phase transitions manifest themselves in the percolation of the corresponding random-cluster model. Our proofs rely on recent studies \cite{DDG12} of Gibbs measures for geometry-dependent interactions

A completely random T-tessellation model and Gibbsian extensions

A revised version of this paper has been published in Spatial Statistics, 2013, volume 6, pages 118- 138.In their 1993 paper, Arak, Clifford and Surgailis discussed a new model of random planar graph. As a particular case, that model yields tessellations with only T-vertices (T-tessellations). Using a similar approach involving Poisson lines, a new model of random T-tessellations is proposed. Campbell measures, Papangelou kernels and Georgii-Nguyen-Zessin formulae are translated from point process theory to random T-tessellations. It is shown that the new model shows properties similar to the Poisson point process and can therefore be considered as a completely random T-tessellation. Gibbs variants are introduced leading to models of random T-tessellations where selected features are controlled. Gibbs random T-tessellations are expected to better represent observed tessellations. As numerical experiments are a key tool for investigating Gibbs models, we derive a simulation algorithm of the Metropolis-Hastings-Green family

Contributions à l'étude statistique de la dépendance spatiale dans les champs à longue mémoire sur un réseau, les processus ponctuels et la géométrie aléatoire.

Ce travail d'habilitation porte sur deux thématiques de recherche : la longue mémoire dans les champs aléatoires et les processus ponctuels en interaction. Ces deux domaines ont pour point commun l'étude de la dépendance dans des processus spatiaux. Le premier concerne la forte dépendance dans des processus aléatoires portés par un réseau (comme des séries temporelles ou des images), le second s'intéresse à la dépendance dans la position de points dans l'espace, éventuellement au travers de marques associées, ce qui concerne notamment des objets géométriques en interaction. Ma contribution porte plus spécifiquement sur l'étude de certaines classes de modèles, et sur l'obtention de résultats asymptotiques qui valident ou motivent certaines procédures statistiques.La première partie est consacrée à la longue mémoire. Des modèles de champs à longue mémoire sont tout d'abord présentés. Ils témoignent de la spécificité des champs par rapport aux séries temporelles : la longue mémoire peut émerger de façon isotrope mais aussi anisotrope (par exemple dans une seule direction). Le comportement asymptotique de certaines statistiques en présence de longue mémoire est ensuite étudié. Il s'agit des sommes partielles, du processus empirique, de certaines formes quadratiques. Ces objets sont au coeur de nombreuses procédures statistiques et leur étude est fondamentale. Quelques tests statistiques en présence de longue mémoire sont enfin présentées. Il s'agit de tester la présence de longue mémoire dans des séries temporelles ou des champs aléatoires, ou encore la persistance de cette dernière au cours du temps. La seconde partie traite des processus ponctuels en lien avec la géométrie aléatoire. Les processus ponctuels en interaction peuvent se modéliser de différentes manières. La plus naturelle est sans doute au travers d'un potentiel qui explicite l'interaction précise entre points voisins, conduisant à la classe des modèles de Gibbs. Ce point de vue permet de construire des structures géométriques en interaction, comme des mosaïques de Voronoï dont les cellules interagissent au travers d'un Hamiltonien. Différents modèles de ce type sont présentés. Des méthodes d'inférence pour les processus de Gibbs sont ensuite abordées, principalement au travers de leurs propriétés asymptotiques. La méthode par pseudo-vraisemblance est ainsi étendue au cas d'interactions non-héréditaires, courantes en géométrie aléatoire. La méthode de Takacs-Fiksel est par ailleurs étudiée en détail. Enfin une étude fine des propriétés des résidus d'un processus de Gibbs nous permet de proposer des tests d'adéquations, ce qui est inédit dans ce contexte

Entropically driven self-assembly of pear-shaped nanoparticles

This thesis addresses the entropically driven colloidal self-assembly of pear-shaped particle ensembles, including the formation of nanostructures based on triply periodic minimal surfaces, in particular of the Ia3d gyroid. One of the key results is that the formation of the Ia3d gyroid, re-ported earlier in the so-called pear hard Gaussian overlap (PHGO) approximation and confirmed here, is due to a slight non-additivity of that potential; this phase does not form in pears with true hard-core potential. First, we computationally study the PHGO system and present the phase diagram of pears with an aspect ratio of 3 in terms of global density and particle shape (degree of taper), containing gyroid, isotropic, nematic and smectic phases. We confirm that it is adequate to interpret the gyroid as a warped smectic bilayer phase. The collective behaviour to arrange into interdigitated sheets with negative Gauss curvature, from which the gyroid results, is investigated through correlations of (Set-)Voronoi cells and local curvature. This geometric arrangement within the bilayers suggests a fundamentally different stabilisation mechanism of the pear gyroid phase compared to those found in both lipid-water and di-block copolymer systems forming the Ia3d gyroid. The PHGO model is only an approximation for hard-core interactions, and we additionally investigate, by much slower simulations, pear-assemblies with true hard-core interactions (HPR). We find that HPR phase diagram only contains isotropic and nematic phases, but neither gyroid nor smectic phases. To understand this shape sensitivity more profoundly, the depletion interactions of both models are studied in two pear-shaped colloids dissolved in a hard sphere solvent. The HPR particles act as one would expect from a geometric analysis of the excluded-volume minimisation, whereas the PHGO particles show deviations from this expectation. These differences are attributed to the unusual angle dependency of the (non-additive) contact function and, more so, to small overlaps induced by the approximation. For the PHGO model, we further demonstrate that the addition of a small concentration of hard spheres ("solvent") drives the system towards a Pn3m diamond phase. This result is explained by the greater spatial heterogeneity of the diamond geometry compared to the gyroid where additional material is needed to relieve packing frustration. In contrast to copolymer systems, however, the solvent mostly aggregates near the diamond minimal surface, driven by the non-additivity of the PHGO pears. At high solvent concentrations, the mixture phase separates into “inverse” micelle-like structures with the blunt ends at the micellar centres and thin ends pointing out-wards. The micelles themselves spontaneously cluster, indicative of a hierarchical self-assembly process for bicontinuous structures. Finally, we develop a density functional for hard solids of revolution (including pears) within the framework of fundamental measure theory. It is applied to low-density ensembles of pear-shaped particles, where we analyse their response near a hard substrate. A complex orientational ordering close to the wall is predicted, which is directly linked to the particle shape and gives insight into adsorption processes of asymmetric particles. This predicted behaviour and the differences between the PHGO and HPR model are confirmed by MC simulations

A study of a novel modular variable geometry frame arranged as a robotic surface

The novel concept of a variable geometry frame is introduced and explored through a three-dimensional robotic surface which is devised and implemented using triangular modules. The link design is optimized using surplus motor dimensions as firm constraints, and round numbers for further arbitrary constraints. Each module is connected by a passive six-bar mechanism that mimics the constraints of a spherical joint at each triangle intersection. A three dimensional inkjet printer is used to create a six-module prototype designed around surplus stepper motors powered by an old computer power supply as a proof-of-concept example. The finite element method is applied to the static and dynamic loading of this device using linear three dimensional (6 degrees of freedom per node) beam elements to calculate the cartesian displacement and force and the angular displacement and torque at each joint. In this way, the traditional methods of finding joint forces and torques are completely bypassed. An efficient algorithm is developed to linearly combine local stiffness matrices into a full structural stiffness matrix for the easy application of loads. This is then decomposed back into the local matrices to easily obtain joint variables used in the design and open-loop control of the surface. Arbitrary equation driven surfaces are approximated ensuring that they are within the joints limits. Moving shapes are then calculated by considering the initial position of the surface, the desired position of the surface, and intermediate shapes at discrete times along the desired path. There are no sensors on the prototype, but feedback models and state estimators are developed for future use. These models include shape sampling methods derived from existing meshing algorithms, trajectory planning using sinusoidal acceleration profiles, spline-based path approximation to allow lower curvature paths able to be traversed more quickly and/or able to be travelled with a constant velocity and optimized by iteratively calculating actuator saturation with no discontinuities, and the optimal tracking of a desired path (modeled with a time-varying ricatti equation)

Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction

International audienceGeneral models of Gibbs Delaunay-Voronoi tessellations, which can be viewed as extensions of Ord's process, are considered. The interaction may occur on each cell of the tessellation and between neighbour cells. The tessellation may also be subjected to a geometric hardcore interaction, forcing the cells not to be too large, too small, or too flat. This setting, natural for applications, introduces some theoretical difficulties since the interaction is not necessarily hereditary. Mathematical results available for studying these models are reviewed and further outcomes are provided. They concern the existence, the simulation and the estimation of such tessellations. Based on these results, tools to handle these objects in practice are presented: how to simulate them, estimate their parameters and validate the fitted model. Some examples of simulated tessellations are studied in details

Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction

General models of Gibbs Delaunay-Voronoi tessellations, which can be viewed as extensions of Ord's process, are considered. The interaction may occur on each cell of the tessellation and between neighbour cells. The tessellation may also be subjected to a geometric hardcore interaction, forcing the cells not to be too large, too small, or too flat. This setting, natural for applications, introduces some theoretical difficulties since the interaction is not necessarily hereditary. Mathematical results available for studying these models are reviewed and further outcomes are provided. They concern the existence, the simulation and the estimation of such tessellations. Based on these results, tools to handle these objects in practice are presented: how to simulate them, estimate their parameters and validate the fitted model. Some examples of simulated tessellations are studied in detail.Gibbs point process Random tessellations Stochastic geometry Pseudo-likelihood estimator Spatial statistics