Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.

Rigorous derivation of a nonlinear diffusion equation as fast-reaction limit of a continuous coagulation-fragmentation model with diffusion

Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [CDF2], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters.

A mixed finite element method for nonlinear diffusion equations

We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model.

A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics

To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [13, 12], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by conductance-based, linear, integrate-and-fire point neurons. A diffusion approximation then led to a nonlinear Fokker-Planck equation for the probability density function of neuronal membrane potentials and synaptic conductances. In this work, we propose a deterministic numerical scheme for a Fokker-Planck model of an excitatory-only network. Our numerical solver allows us to obtain the time evolution of probability distribution functions, and thus, the evolution of all possible macroscopic quantities that are given by suitable moments of the probability density function. We show that this deterministic scheme is capable of capturing the bistability of stationary states observed in Monte Carlo simulations. Moreover, the transient behavior of the firing rates computed from the Fokker-Planck equation is analyzed in this bistable situation, where a bifurcation scenario, of asynchronous convergence towards stationary states, periodic synchronous solutions or damped oscillatory convergence towards stationary states, can be uncovered by increasing the strength of the excitatory coupling. Finally, the computation of moments of the probability distribution allows us to validate the applicability of a moment closure assumption used in [13] to further simplify the kinetic theory.

Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states

Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter.

Determination of chemical composition and its relationship with optical properties of Ti-N and Ti-V-N sputtered thin films

In the investigation of thin films of transition metal nitrides, an essential role is played by the accurate determination of their chemical composition. Actually the chemical composition depends on the deposition parameters and influences the optical properties. These relations are illustrated in thin films of TiNx and (Ti1-yVy)N-x deposited by reactive magnetron sputtering from composite targets of the elements. By variation of the nitrogen partial pressure and the target composition, different samples have been obtained. The chemical composition has been measured by electron probe microanalysis at low irradiation voltages. The optical properties are evaluated by ex-situ ellipsometry. Using the screened Drude model, they are correlated with the differences in composition. Adding vanadium or nitrogen in Ti-N is shown to have the same effect on the optical properties.

Line search multilevel optimization as computational methods for dense optical flow

We evaluate the performance of different optimization techniques developed in the context of optical flowcomputation with different variational models. In particular, based on truncated Newton methods (TN) that have been an effective approach for large-scale unconstrained optimization, we develop the use of efficient multilevel schemes for computing the optical flow. More precisely, we evaluate the performance of a standard unidirectional multilevel algorithm - called multiresolution optimization (MR/OPT), to a bidrectional multilevel algorithm - called full multigrid optimization (FMG/OPT). The FMG/OPT algorithm treats the coarse grid correction as an optimization search direction and eventually scales it using a line search. Experimental results on different image sequences using four models of optical flow computation show that the FMG/OPT algorithm outperforms both the TN and MR/OPT algorithms in terms of the computational work and the quality of the optical flow estimation.

Comparability of clinical wear measurements by optical 3D laser scanning in two different centers.

OBJECTIVE: The purpose of this study was to compare the use of different variables to measure the clinical wear of two denture tooth materials in two analysis centers. METHODS: Twelve edentulous patients were provided with full dentures. Two different denture tooth materials (experimental material and control) were placed randomly in accordance with the split-mouth design. For wear measurements, impressions were made after an adjustment phase of 1-2 weeks and after 6, 12, 18, and 24 months. The occlusal wear of the posterior denture teeth of 11 subjects was assessed in two study centers by use of plaster replicas and 3D laser-scanning methods. In both centers sequential scans of the occlusal surfaces were digitized and superimposed. Wear was described by use of four different variables. Statistical analysis was performed after log-transformation of the wear data by use of the Pearson and Lin correlation and by use of a mixed linear model. RESULTS: Mean occlusal vertical wear of the denture teeth after 24 months was between 120μm and 212μm, depending on wear variable and material. For three of the four variables, wear of the experimental material was statistically significantly less than that of the control. Comparison of the two study centers, however, revealed correlation of the wear variables was only moderate whereas strong correlation was observed among the different wear variables evaluated by each center. SIGNIFICANCE: Moderate correlation was observed for clinical wear measurements by optical 3D laser scanning in two different study centers. For the two denture tooth materials, wear measurements limited to the attrition zones led to the same qualitative assessment.

HC12 HS Charges and Optical Voucher Values (PDF 133KB)

Information on HS charges and optical voucher values

HC12 HS Charges and Optical Voucher Values (PDF 470 KB)

Information on HS charges and optical voucher values

Active Learning of Very-High Resolution Optical Imagery with SVM: Entropy vs Margin Sampling

An active learning method is proposed for the semi-automatic selection of training sets in remote sensing image classification. The method adds iteratively to the current training set the unlabeled pixels for which the prediction of an ensemble of classifiers based on bagged training sets show maximum entropy. This way, the algorithm selects the pixels that are the most uncertain and that will improve the model if added in the training set. The user is asked to label such pixels at each iteration. Experiments using support vector machines (SVM) on an 8 classes QuickBird image show the excellent performances of the methods, that equals accuracies of both a model trained with ten times more pixels and a model whose training set has been built using a state-of-the-art SVM specific active learning method