Diagonality Measures of Hermitian Positive-Definite Matrices with Application to the Approximate Joint Diagonalization Problem

In this paper, we introduce properly-invariant diagonality measures of Hermitian positive-definite matrices. These diagonality measures are defined as distances or divergences between a given positive-definite matrix and its diagonal part. We then give closed-form expressions of these diagonality measures and discuss their invariance properties. The diagonality measure based on the log-determinant $\alpha$-divergence is general enough as it includes a diagonality criterion used by the signal processing community as a special case. These diagonality measures are then used to formulate minimization problems for finding the approximate joint diagonalizer of a given set of Hermitian positive-definite matrices. Numerical computations based on a modified Newton method are presented and commented

A Double-Strand Elastic Rod Theory

Abstract.: Motivated by applications in the modeling of deformations of the DNA double helix, we construct a continuum mechanics model of two elastically interacting elastic strands. The two strands are described in terms of averaged, or macroscopic, variables plus an additional small, internal or microscopic, perturbation. We call this composite structure a birod. The balance laws for the macroscopic configuration variables of the birod can be cast in the form of a classic Cosserat rod model with coupling to the internal balance laws through the constitutive relations. The internal balance laws for the microstructure variables also take a mathematical form analogous to that for a Cosserat rod, but with coupling to the macroscopic system through terms corresponding to distributed force and couple load

TERNARY QUARTIC APPROACH FOR POSITIVE 4TH ORDER DIFFUSION TENSORS REVISITED

International audienceIn Diffusion Magnetic Resonance Imaging (D-MRI), the 2nd order diffusion tensor has given rise to a widely used tool – Diffusion Tensor Imaging (DTI). However, it is known that DTI is limited to a single prominent diffusion direction and is inaccurate in regions of complex fiber structures such as crossings. Various other approaches have been introduced to recover such complex tissue micro-geometries, one of which is Higher Order Cartesian Tensors. Estimating a positive diffusion function has also been emphasised mathematically, since diffusion is a physical quantity. Recently there have been efforts to estimate 4th order diffusion tensors from DiffusionWeighted Images (DWIs), which are capable of describing crossing configurations with the added property of a positive diffusion function. We take up one such, the Ternary Quartic approach, and reformulate the estimation equation to facilitate the estimation of the non-negative 4th order diffusion tensor. With our modified approach we test on synthetic, phantom and real data and confirm previous results

A fast algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems

In this paper, we develop a new algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems. Numerical experiments are given in order to illustrate the validity and efficiency of our algorithm.The authors would like to thank the supports of the Portuguese Funds through FCT–Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013

An algebraic condition and an algorithm for the internal contact between two ellipsoids

4th International Conference on Discrete Element Methods, Brisbane, AUSTRALIA, AUG, 2007International audiencePurpose - The purpose of this paper is to present a new method for the detection and resolution of the contact point between two ellipsoids. Numerical simulations of ellipsoidal particles in a rotary cylinder are also presented. Desigri/methodology/approach - An algebraic condition is developed for the internal contact between two ellipsoids and an efficient contact detection algorithm for overlapping ellipsoids is implemented. Findings - This method was found to have the advantages of effectiveness and speed in the detection and resolution of the contact point. Originality/value - The dynamics of granular materials are of great importance in many industries dealing with powders and grains, such as pharmaceutical, chemical, and food industries. The main difficulty of such simulations is the excessive CPU time required for a large number of particles. In the discrete element method, contact detection between grains is the most expensive step in solving a nonlinear system for determination of the contact point, the normal vector and the overlap distance between ellipsoids. The numerical behavior and the optimization of the new algorithm presented in this paper are important also

Numerical convergence and stability of mixed formulation with X-FEM cut-off

National audienceSee http://hal.archives-ouvertes.fr/docs/00/59/26/89/ANNEX/r_KJ9QU562.pd

The derivation of homogenized diffusion kurtosis models for diffusion MRI

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Numerical convergence and stability of mixed formulation with X-fem cut-off

International audienceIn this paper we are concerned with the mathematical and numerical analysis of convergence and stability of the mixed formulation for incompressible elasticity in cracked domains. The objective is to extend the X-FEM cut-off analysis done in the case of compressible elasticity to the incompressible one. A mathematical proof of the inf-sup condition of the discrete mixed formulation with X-FEM is established for some enriched fields. We also give a mathematical result of quasi-optimal error estimate. Finally, we validate these results with numerical tests

Résolution de problèmes inverses en géodésie physique

Ce travail traite de deux problèmes de grande importances en géodésie physique. Le premier porte sur la détermination du géoïde sur une zone terrestre donnée. Si la terre était une sphère homogène, la gravitation en un point, serait entièrement déterminée à partir de sa distance au centre de la terre, ou de manière équivalente, en fonction de son altitude. Comme la terre n'est ni sphérique ni homogène, il faut calculer en tout point la gravitation. A partir d'un ellipsoïde de référence, on cherche la correction à apporter à une première approximation du champ de gravitation afin d'obtenir un géoïde, c'est-à-dire une surface sur laquelle la gravitation est constante. En fait, la méthode utilisée est la méthode de collocation par moindres carrés qui sert à résoudre des grands problèmes aux moindres carrés généralisés. Le seconde partie de cette thèse concerne un problème inverse géodésique qui consiste à trouver une répartition de masses ponctuelles (caractérisées par leurs intensités et positions), de sorte que le potentiel généré par eux, se rapproche au maximum d'un potentiel donné. Sur la terre entière une fonction potentielle est généralement exprimée en termes d'harmoniques sphériques qui sont des fonctions de base à support global la sphère. L'identification du potentiel cherché se fait en résolvant un problème aux moindres carrés. Lorsque seulement une zone limitée de la Terre est étudiée, l'estimation des paramètres des points masses à l'aide des harmoniques sphériques est sujette à l'erreur, car ces fonctions de base ne sont plus orthogonales sur un domaine partiel de la sphère. Le problème de la détermination des points masses sur une zone limitée est traitée par la construction d'une base de Slepian qui est orthogonale sur le domaine limité spécifié de la sphère. Nous proposons un algorithme itératif pour la résolution numérique du problème local de détermination des masses ponctuelles et nous donnons quelques résultats sur la robustesse de ce processus de reconstruction. Nous étudions également la stabilité de ce problème relativement au bruit ajouté. Nous présentons quelques résultats numériques ainsi que leurs interprétations.This work focuses on the study of two well-known problems in physical geodesy. The first problem concerns the determination of the geoid on a given area on the earth. If the Earth were a homogeneous sphere, the gravity at a point would be entirely determined from its distance to the center of the earth or in terms of its altitude. As the earth is neither spherical nor homogeneous, we must calculate gravity at any point. From a reference ellipsoid, we search to find the correction to a mathematical approximation of the gravitational field in order to obtain a geoid, i.e. A surface on which gravitational potential is constant. The method used is the method of least squares collocation which is the best for solving large generalized least squares problems. In the second problem, We are interested in a geodetic inverse problem that consists in finding a distribution of point masses (characterized by their intensities and positions), such that the potential generated by them best approximates a given potential field. On the whole Earth a potential function is usually expressed in terms of spherical harmonics which are basis functions with global support. The identification of the two potentials is done by solving a least-squares problem. When only a limited area of the Earth is studied, the estimation of the point-mass parameters by means of spherical harmonics is prone to error, since they are no longer orthogonal over a partial domain of the sphere. The point-mass determination problem on a limited region is treated by the construction of a Slepian basis that is orthogonal over the specified limited domain of the sphere. We propose an iterative algorithm for the numerical solution of the local point mass determination problem and give some results on the robustness of this reconstruction process. We also study the stability of this problem against added noise. Some numerical tests are presented and commented.RENNES1-Bibl. électronique (352382106) / SudocSudocFranceF

A Nash-game approach to joint image restoration and segmentation

International audienceWe propose a game theory approach to simultaneously restore and segment noisy images. We define two players: one is restoration, with the image intensity as strategy, and the other is segmentation with contours as strategy. Cost functions are the classical relevant ones for restoration and segmentation, respectively. The two players play a static game with complete information, and we consider as solution to the game the so-called Nash Equilibrium. For the computation of this equilibrium we present an iterative method with relaxation. The results of numerical experiments performed on some real images show the relevance and efficiency of the proposed algorithm