Axel Osses

Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights

Baudouin and Puel (2002 Inverse Problems 18 1537-54), investigated some inverse problems for the evolution Schr¨odinger equation bymeans of Carleman inequalities proved under a strict pseudoconvexity condition. We showhere that new Carleman inequalities for the Schr¨odinger equationmay be derived under a relaxed pseudoconvexity condition, which allows us to use degenerate weights with a spatial dependence of the type ψ(x) = x * e, where e is some fixed direction in RN. As a result, less restrictive boundary or internal observations are allowed to obtain the stability for the inverse problem consisting in retrieving a stationary potential in the Schr¨odinger equation from a single boundary or internal measurement.

Added mass and damping in fluid-structure interaction

Abstract This paper is concerned with the added mass matrix for a mechanical structure vibrating in an incompressible liquid. It is shown in particular that this matrix does not depend on viscosity and, from this fact, can be calculated as if the fluid is perfect. The viscous effect on the mechanical system can then be represented by a damping term of type time-convolution. The presence of a flowing fluid around the structure leads to additional damping terms proportional to the fluid density.

Constraining surface emissions of air pollutants using inverse modelling: method intercomparison and a new two-step two-scale regularization approach

When constraining surface emissions of air pollutants using inverse modelling one often encounters spurious corrections to the inventory at places where emissions and observations are colocated, referred to here as the colocalization problem. Several approaches have been used to deal with this problem: coarsening the spatial resolution of emissions; adding spatial correlations to the covariance matrices; adding constraints on the spatial derivatives into the functional being minimized; and multiplying the emission error covariance matrix by weighting factors. Intercomparison of methods for a carbon monoxide inversion over a city shows that even though all methods diminish the colocalization problem and produce similar general patterns, detailed information can greatly change according to the method used ranging from smooth, isotropic and short range modifications to not so smooth, non-isotropic and long range modifications. Poisson (non-Gaussian) and Gaussian assumptions both show these patterns, but for the Poisson case the emissions are naturally restricted to be positive and changes are given by means of multiplicative correction factors, producing results closer to the true nature of emission errors. Finally, we propose and test a new two-step, two-scale, fully Bayesian approach that deals with the colocalization problem and can be implemented for any prior density distribution.

A Rotated Multiplier Applied to the Controllability of Waves, Elasticity, and Tangential Stokes Control

A new family of multipliers with rotated direction is introduced. This technique is applied to obtain new results concerning controllability of waves, elasticity, and Stokes equations. The boundary exact controllability for the wave equation and the dynamic elasticity system is reviewed generalizing the classical exit condition in the case of explicit observability constants. Approximate controllability for the Stokes system is also studied using a boundary control acting only on the tangential component of the velocity. A geometric sufficient condition of exit generalized type is deduced.

A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem

We consider a transmission wave equation in two embedded domains in , where the speed is a1 > 0 in the inner domain and a2 > 0 in the outer domain. We prove a global Carleman inequality for this problem under the hypothesis that the inner domain is strongly convex and a1 > a2. As a consequence of this inequality, uniqueness and Lipschitz stability are obtained for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient from a single time-dependent Neumann boundary measurement.

Rotated weights in global Carleman estimates applied to an inverse problem for the wave equation

In this paper, we establish geometrical conditions in order to solve an inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data from a single time-dependent Neumann boundary measurement on a suitable part of the boundary. We prove the uniqueness and the stability results for this problem when a Neumann measurement is only located on a part of the boundary satisfying a rotated exit condition. The strategy consists of introducing an angle-type dependence in the weight functions used to obtain global Carleman estimates for the wave equation and combination of several of these estimates and then apply it to the inverse problem.

Modeling heterocyst pattern formation in cyanobacteria

BackgroundTo allow the survival of the population in the absence of nitrogen, some cyanobacteria strains have developed the capability of differentiating into nitrogen fixing cells, forming a characteristic pattern. In this paper, the process by which cyanobacteria differentiates from vegetative cells into heterocysts in the absence of nitrogen and the elements of the gene network involved that allow the formation of such a pattern are investigated.MethodsA simple gene network model, which represents the complexity of the differentiation process, and the role of all variables involved in this cellular process is proposed. Specific characteristics and details of the systems behavior such as transcript profiles for ntcA, hetR and patS between consecutive heterocysts were studied.ResultsThe proposed model is able to capture one of the most distinctive features of this system: a characteristic distance of 10 cells between two heterocysts, with a small standard deviation according to experimental variability. The systems response to knock-out and over-expression of patS and hetR was simulated in order to validate the proposed model against experimental observations. In all cases, simulations show good agreement with reported experimental results.ConclusionA simple evolution mathematical model based on the gene network involved in heterocyst differentiation was proposed. The behavior of the biological system naturally emerges from the network and the model is able to capture the spacing pattern observed in heterocyst differentiation, as well as the effect of external perturbations such as nitrogen deprivation, gene knock-out and over-expression without specific parameter fitting.

Homogenization of Elastic Media with Gaseous Inclusions

We study the asymptotic behavior of a system modeling a composite material made of an elastic periodically perforated support, with period

Controls Insensitizing the Observation of a Quasi-geostrophic Ocean Model

\varepsilon > 0

Analysis and evolution of air quality monitoring networks using combined statistical information indexes

, and a perfect gas placed in each of these perforations, as