arXiv subjects

Elie Bretin

Publications and source records attributed to Elie Bretin.

21 records · Page 2Linked to original sources

arXiv2010

Transient Anomaly Imaging in Visco-Elastic Media Obeying a Frequency Power-Law

In this work, we consider the problem of reconstructing a small anomaly in a viscoelastic medium from wave-field measurements. We choose Szabo's model to describe the viscoelastic properties of the medium. Expressing the ideal elastic field without any viscous effect in terms of the measured field in a viscous medium, we generalize the imaging procedures, such as time reversal, Kirchhoff Imaging and Back propagation, for an ideal medium to detect an anomaly in a visco-elastic medium from wave-field measurements.

math.AP↗

arXiv2010

Consistency result for a non monotone scheme for anisotropic mean curvature flow

In this paper, we propose a new scheme for anisotropic motion by mean curvature in $\R^d$. The scheme consists of a phase-field approximation of the motion, where the nonlinear diffusive terms in the corresponding anisotropic Allen-Cahn equation are linearized in the Fourier space. In real space, this corresponds to the convolution with a kernel of the form \[ K_{ϕ,t}(x) = \F^{-1}\left[ e^{-4π^2 t ϕ^o(ξ)} \right](x). \] We analyse the resulting scheme, following the work of Ishii-Pires-Souganidis on the convergence of the Bence-Merriman-Osher algorithm for isotropic motion by mean curvature. The main difficulty here, is that the kernel $K_{ϕ,t}$ is not positive and that its moments of order 2 are not in $L^1(\R^d)$. Still, we can show that in one sense the scheme is consistent with the anisotropic mean curvature flow.

math.NA↗

arXiv2009

A modified phase field approximation for mean curvature flow with conservation of the volume

This paper is concerned with the motion of a time dependent hypersurface that evolves by mean curvature flow with a a volume constraint. Phase field approximation of this motion leads to the well known nonlocal Allen--Cahn equation. Here we propose a modified version of this equation, and we show that it has better volume preserving properties than the classical one, even in the presence of an additional forcing term.

math.NA↗