Publications

2013

• A conformal mapping method in inverse obstacle scattering
  
  • Haddar Houssem
  • Kress Rainer
  Complex Variables and Elliptic Equations, Taylor & Francis, 2013. Akduman, Haddar and Kress [1, 5, 11] have employed a conformal map- ping technique for the inverse problem to reconstruct a perfectly conducting inclusion in a homogeneous background medium from Cauchy data for elec- trostatic imaging, that is, for solving an inverse boundary value problem for the Laplace equation. We propose an extension of this approach to inverse obstacle scattering for time-harmonic waves, that is, to the solution of an inverse boundary value problem for the Helmholtz equation. The main idea is to use the conformal mapping algorithm in an iterative procedure to ob- tain Cauchy data for a Laplace problem from the given Cauchy data for the Helmholtz problem. We present the foundations of the method together with a convergence result and exhibit the feasibility of the method via numerical examples.
• Adaptation and migration of a population between patches
  
  • Mirrahimi Sepideh
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2013, 18, pp.753-768. A Hamilton-Jacobi formulation has been established previously for phenotypically structured population models where the solution concentrates as Dirac masses in the limit of small diffusion. Is it possible to extend this approach to spatial models? Are the limiting solutions still in the form of sums of Dirac masses? Does the presence of several habitats lead to polymorphic situations? We study the stationary solutions of a structured population model, while the population is structured by continuous phenotypical traits and discrete positions in space. The growth term varies from one habitable zone to another, for instance because of a change in the temperature. The individuals can migrate from one zone to another with a constant rate. The mathematical modeling of this problem, considering mutations between phenotypical traits and competitive interaction of individuals within each zone via a single resource, leads to a system of coupled parabolic integro-differential equations. We study the asymptotic behavior of the stationary solutions to this model in the limit of small mutations. The limit, which is a sum of Dirac masses, can be described with the help of an effective Hamiltonian. The presence of migration can modify the dominant traits and lead to polymorphic situations. (10.3934/dcdsb.2013.18.753)
  DOI : 10.3934/dcdsb.2013.18.753
• Ergodic Control and Polyhedral approaches to PageRank Optimization
  
  • Fercoq Olivier
  • Akian Marianne
  • Bouhtou Mustapha
  • Gaubert Stéphane
  IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2013, 58 (1), pp.134--148. We study a general class of PageRank optimization problems which involve finding an optimal outlink strategy for a web site subject to design constraints. We consider both a continuous problem, in which one can choose the intensity of a link, and a discrete one, in which in each page, there are obligatory links, facultative links and forbidden links. We show that the continuous problem, as well as its discrete variant when there are no constraints coupling different pages, can both be modeled by constrained Markov decision processes with ergodic reward, in which the webmaster determines the transition probabilities of websurfers. Although the number of actions turns out to be exponential, we show that an associated polytope of transition measures has a concise representation, from which we deduce that the continuous problem is solvable in polynomial time, and that the same is true for the discrete problem when there are no coupling constraints. We also provide efficient algorithms, adapted to very large networks. Then, we investigate the qualitative features of optimal outlink strategies, and identify in particular assumptions under which there exists a "master" page to which all controlled pages should point. We report numerical results on fragments of the real web graph. (10.1109/TAC.2012.2226103)
  DOI : 10.1109/TAC.2012.2226103
• The Moutard transformation and two-dimensional multi-point delta-type potentials
  
  • Novikov Roman
  • Taimanov Iskander
  Russian Mathematical Surveys, Turpion, 2013, 68 (5), pp.957–959. In the framework of the Moutard transformation formalism we find multi-point delta-type potentials for two-dimensional Schrodinger operators and their isospectral deformations on the zero energy level. In particular, these potentials are "reflectionless" in the sense of the Faddeev generalized "scattering" data.
• Diffraction of Bloch Wave Packets for Maxwell's Equations
  
  • Allaire Grégoire
  • Palombaro Mariapia
  • Rauch Jeffrey
  Communications in Contemporary Mathematics, World Scientific Publishing, 2013, 15 (6), pp.1350040. We study, for times of order 1/h, solutions of Maxwell's equations in an O(h^2) modulation of an h-periodic medium. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order h. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation. A weak ray average hypothesis guarantees stability. Compared to earlier work on scalar wave equations, the generator is no longer elliptic. Coercivity holds only on the complement of an infinite dimensional kernel. The system structure requires many innovations. (10.1142/S0219199713500405)
  DOI : 10.1142/S0219199713500405
• Lineage selection and the maintenance of sex
  
  • de Vienne Damien M.
  • Giraud Tatiana
  • Gouyon Pierre-Henri
  PLoS ONE, Public Library of Science, 2013, 8 (6), pp.e66906. Sex predominates in eukaryotes, despite its short-term disadvantage when compared to asexuality. Myriad models have suggested that short-term advantages of sex may be sufficient to counterbalance its twofold costs. However, despite decades of experimental work seeking such evidence, no evolutionary mechanism has yet achieved broad recognition as explanation for the maintenance of sex. We explore here, through lineage-selection models, the conditions favouring the maintenance of sex. In the first model, we allowed the rate of transition to asexuality to evolve, to determine whether lineage selection favoured species with the strongest constraints preventing the loss of sex. In the second model, we simulated more explicitly the mechanisms underlying the higher extinction rates of asexual lineages than of their sexual counterparts. We linked extinction rates to the ecological and/ or genetic features of lineages, thereby providing a formalisation of the only figure included in Darwin's "The origin of species". Our results reinforce the view that the long-term advantages of sex and lineage selection may provide the most satisfactory explanations for the maintenance of sex in eukaryotes, which is still poorly recognized, and provide figures and a simulation website for training and educational purposes. Short-term benefits may play a role, but it is also essential to take into account the selection of lineages for a thorough understanding of the maintenance of sex. (10.1371/journal.pone.0066906)
  DOI : 10.1371/journal.pone.0066906