Bochner identities for Kählerian gradients

We discuss algebraic properties for the symbols of geometric first order differential operators on Kähler manifolds. Through a study of the universal enveloping algebra and higher Casimir elements, we know a lot of relations for the symbols, which induce Bochner identities for the operators. As applications, we have vanishing theorems, eigenvalue estimates, and so on.     

A representation-theoretical proof of Bransonʼs classification of elliptic generalized gradients

The purpose of this paper is to present a new proof of Branson?s classification (Branson, 1997 [3]), of minimal elliptic sums of generalized gradients. The advantage of this proof is that it is local, being mainly based on representation theory and on the relationship between ellipticity and refined Kato inequalities. This approach is promising for the classification of elliptic generalized gradients of G-structures, for other subgroups G of the special orthogonal group.     

A new proof of Branson��s classification of elliptic generalized gradients

We give a representation theoretical proof of Branson??s classification (J Funct Anal 151(2):334?C383, 1997), of minimal elliptic sums of generalized gradients. The original proof uses tools of harmonic analysis, which as powerful as they are, seem to be specific for the structure groups SO(n) and Spin(n). The different approach we propose is a local one, based on the relationship between ellipticity and optimal Kato constants and on the representation theory of ${\mathfrak{so}(n)}$ . Optimal Kato constants for elliptic operators were computed by Calderbank et?al. (J Funct Anal 173(1):214?C255, 2000). We extend their method to all generalized gradients (not necessarily elliptic) and recover Branson??s result, up to one special case. The interest of this method is that it is better suited to be applied for classifying elliptic sums of generalized gradients of G-structures, for other subgroups G of the special orthogonal group.     

On the global convergence rate of the gradient descent method for functions with Hölder continuous gradients

The gradient descent method minimizes an unconstrained nonlinear optimization problem with \({\mathcal {O}}(1/\sqrt{K})\), where K is the number of iterations performed by the gradient method. Traditionally, this analysis is obtained for smooth objective functions having Lipschitz continuous gradients. This paper aims to consider a more general class of nonlinear programming problems in which functions have Hölder continuous gradients. More precisely, for any function f in this class, denoted by \({{\mathcal {C}}}^{1,\nu }_L\), there is a \(\nu \in (0,1]\) and \(L>0\) such that for all \(\mathbf{x,y}\in {{\mathbb {R}}}^n\) the relation \(\Vert \nabla f(\mathbf{x})-\nabla f(\mathbf{y})\Vert \le L \Vert \mathbf{x}-\mathbf{y}\Vert ^{\nu }\) holds. We prove that the gradient descent method converges globally to a stationary point and exhibits a convergence rate of \({\mathcal {O}}(1/K^{\frac{\nu }{\nu +1}})\) when the step-size is chosen properly, i.e., less than \([\frac{\nu +1}{L}]^{\frac{1}{\nu }}\Vert \nabla f(\mathbf{x}_k)\Vert ^{\frac{1}{\nu }-1}\). Moreover, the algorithm employs \({\mathcal {O}}(1/\epsilon ^{\frac{1}{\nu }+1})\) number of calls to an oracle to find \({\bar{\mathbf{x}}}\) such that \(\Vert \nabla f({{\bar{\mathbf{x}}}})\Vert <\epsilon \).     

On mixed finite element approximations of shape gradients in shape optimization with the Navier–Stokes equation

For shape optimization of fluid flows governed by the Navier–Stokes equation, we investigate effectiveness of shape gradient algorithms by analyzing convergence and accuracy of mixed finite element approximations to both the distributed and boundary types of shape gradients. We present convergence analysis with a priori error estimates for the two approximate shape gradients. The theoretical analysis shows that the distributed formulation has superconvergence property. Numerical results with comparisons are presented to verify theory and show that the shape gradient algorithm based on the distributed formulation is highly effective and robust for shape optimization.     

Considérations sur les gradients généralisés de G. Fichera et E. De Giorgi

Résumé Dans notre ? Cours de Varenna ? (15–25 Ao?t 1954) nous avons comparé les notions de dérivé et d'intégrant, eu particulier dans la théorie des fonctions de cellule. Parmi les exemples examinés figurent au chapitre IV les gradients généralisés deG. Fichera et d'E. De Giorgi. Dans la présente note nous établissons quelques résultats qui y étaient formulés seulement comme conjectures. à Mauro Picone pour son 70^mo anniversaire.