Convex bodies: the brunn–minkowski theory

Evolutionary computation and structural design: A survey of the state-of-the-art

The phenological state of vegetation significantly affects exchanges of heat, mass, and momentum between the Earth's surface and the atmosphere. Although current patterns can be estimated from satellites, we lack the ability to predict future trends in response to climate change. We searched the literature for a common set of variables that might be combined into an index to quantify the greenness of vegetation throughout the year. We selected as variables: daylength (photoperiod), evaporative demand (vapor pressure deficit), and suboptimal (minimum) temperatures. For each variable we set threshold limits, within which the relative phenological performance of the vegetation was assumed to vary from inactive (0) to unconstrained (1). A combined Growing Season Index (GSI) was derived as the product of the three indices. Ten-day mean GSI values for nine widely dispersed ecosystems showed good agreement (r>0.8) with the satellite-derived Normalized Difference Vegetation Index (NDVI). We also tested the model at a temperate deciduous forest by comparing model estimates with average field observations of leaf flush and leaf coloration. The mean absolute error of predictions at this site was 3 days for average leaf flush dates and 2 days for leaf coloration dates. Finally, we used this model to produce a global map that distinguishes major differences in regional phenological controls. The model appears sufficiently robust to reconstruct historical variation as well as to forecast future phenological responses to changing climatic conditions.

A generalized, bioclimatic index to predict foliar phenology in response to climate

This paper surveys several topics related to the observation and control of wave propagation phenomena modeled by finite difference methods. The main focus is on the property of observability, corresponding to the question of whether the total energy of solutions can be estimated from partial measurements on a subregion of the domain or boundary. The mathematically equivalent property of controllability corresponds to the question of whether wave propagation behavior can be controlled using forcing terms on that subregion, as is often desired in engineering applications. Observability/controllability of the continuous wave equation is well understood for the scalar linear constant coefficient case that is the focus of this paper. However, when the wave equation is discretized by finite difference methods, the control for the discretized model does not necessarily yield a good approximation to the control for the original continuous problem. In other words, the classical convergence (consistency + stability) property of a numerical scheme does not suffice to guarantee its suitability for providing good approximations to the controls that might be needed in applications. Observability/controllability may be lost under numerical discretization as the mesh size tends to zero due to the existence of high-frequency spurious solutions for which the group velocity vanishes. This phenomenon is analyzed and several remedies are suggested, including filtering, Tychonoff regularization, multigrid methods, and mixed finite element methods. 
We also briefly discuss these issues for the heat, beam, and Schrodinger equations to illustrate that diffusive and dispersive effects may help to retain the observability/controllability properties at the discrete level. We conclude with a list of open problems and future subjects for research.

http://benasque.org/benasque/2005pde/2005pde-talks/023sirev.pdf

Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods

A level set method for structural topology optimization and its applications

Eigenvalues of elliptic operators.- Tools.- The first eigenvalue of the Laplacian-Dirichlet.- The second eigenvalue of the Laplacian-Dirichlet.- The other Dirichlet eigenvalues.- Functions of Dirichlet eigenvalues.- Other boundary conditions for the Laplacian.- Eigenvalues of Schrodinger operators.- Non-homogeneous strings and membranes.- Optimal conductivity.- The bi-Laplacian operator.

Extremum Problems for Eigenvalues of Elliptic Operators

Variation et optimisation de formes

Ce livre est une initiation aux approches modernes de l'optimisation mathematique de formes. Il s'appuie sur les seules connaissances de premiere annee de Master de mathematiques, mais permet deja d'aborder les questions ouvertes dans ce domaine en pleine effervescence. On y developpe la methodologie ainsi que les outils d'analyse mathematique et de geometrie necessaires a l'etude des variations de domaines. On y trouve une etude systematique des questions geometriques associees a l'operateur de Laplace, de la capacite classique, de la derivation par rapport a une forme, ainsi qu'un FAQ sur les topologies usuelles sur les domaines et sur les proprietes geometriques des formes optimales avec ce qui se passe quand elles n'existent pas, le tout avec une importante bibliographie.

Variation et optimisation de formes : une analyse géométrique

Shape Variation and Optimization: A Geometrical Analysis

This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.

/pdf/minimization-problems-for-eigenvalues-of-the-laplacian-4j0sdf2ut6.pdf

Antoine Henrot

Papers

Extremum Problems for Eigenvalues of Elliptic Operators

Variation et optimisation de formes

Variation et optimisation de formes : une analyse géométrique

Shape Variation and Optimization: A Geometrical Analysis

Minimization problems for eigenvalues of the Laplacian