We introduce and discuss shape based models for finding the best interpolation data when reconstructing missing regions in images by means of solving the Laplace equation. The shape analysis is done in the framework of Γ-convergence, from two different points of view. First, we propose a continuous PDE model and get pointwise information on the "importance" of each pixel by a topological asymptotic method. Second, we introduce a finite dimensional setting into the continuous model based on fat pixels (balls with positive radius), and study by Γ-convergence the asymptotics when the radius vanishes. In this way, we obtain relevant information about the optimal distribution of the best interpolation pixels. We show that the resulting optimal data sets are identical to sets that can also be motivated using level set ideas and approximation theoretic considerations. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.

Link to this article: http://journals.cambridge.org/abstract_S0956792501004582How to cite this article: ZAKARIA BELHACHMI, BERNARD BRIGHI and KHALID TAOUS (2001). On a family of differential equations for boundary layer approximations in porous media.Free convection along a vertical flat plate embedded in a porous medium is considered, within the framework of boundary layer approximations. In some cases, similarity solutions can be obtained by solving a boundary value problem involving an autonomous third-order nonlinear equation, depending on a parameter related to the temperature on the wall. The paper deals with existence and uniqueness questions to this problem, for every value of the parameter.

The standard approach for photoacoustic imaging with variable speed of sound is time reversal, which consists in solving a well-posed final-boundary value problem backwards in time. This paper investigates the iterative Landweber regularization algorithm, where convergence is guaranteed by standard regularization theory, notably also in cases of trapping sound speed or for short measurement times. We formulate and solve the direct and inverse problem on , what is common in standard photoacoustic imaging, but not for time-reversal algorithms, . We both the direct and adjoint photoacoustic operator an interior and an exterior equation . The prior is solved using a Galerkin scheme in space and finite difference discretization in time, while the latter boundary integral equation. We therefore use a BEM-FEM approach for numerical solution of the forward operators. We analyze this method, prove convergence, and provide numerical tests. Moreover, we compare the approach to time reversal.

We consider the Stokes problem in an axisymmetric three-dimensional domain with data which are axisymmetric and have angular component equal to zero. We observe that the solution is also axisymmetric and the velocity has also zero angular component, hence the solution satisfies a system of equations in the meridian domain. The weak three-dimensional problem reduces to a two-dimensional one with weighted integrals. The latter is discretized by Taylor-Hood type finite elements. A weighted Clément operator is defined and approximation results are proved. This operator is then used to derive the discrete inf-sup condition and optimal a priori error estimates. Mathematics Subject Classification