78 research outputs found
Reconstruction of cracks and material losses by perimeter-like penalizations and phase-field methods: numerical results
We numerically implement the variational approach for reconstruction in the
inverse crack and cavity problems developed by one of the authors. The method
is based on a suitably adapted free-discontinuity problem. Its main features
are the use of phase-field functions to describe the defects to be
reconstructed and the use of perimeter-like penalizations to regularize the
ill-posed problem.
The numerical implementation is based on the solution of the corresponding
optimality system by a gradient method. Numerical simulations are presented to
show the validity of the method.Comment: 15 pages, 12 figure
Stable determination of a scattered wave from its far-field pattern: the high frequency asymptotics
We deal with the stability issue for the determination of outgoing
time-harmonic acoustic waves from their far-field patterns. We are especially
interested in keeping as explicit as possible the dependence of our stability
estimates on the wavenumber of the corresponding Helmholtz equation and in
understanding the high wavenumber, that is frequency, asymptotics.
Applications include stability results for the determination from far-field
data of solutions of direct scattering problems with sound-soft obstacles and
an instability analysis for the corresponding inverse obstacle problem.
The key tool consists of establishing precise estimates on the behavior of
Hankel functions with large argument or order.Comment: 49 page
Corrigendum to ``Determining a sound-soft polyhedral scatterer by a single far-field measurement''
In the paper, G. Alessandrini and L. Rondi, ``Determining a sound-soft
polyhedral scatterer by a single far-field measurement'', Proc. Amer. Math.
Soc. 133 (2005), pp. 1685-1691, on the determination of a sound-soft polyhedral
scatterer by a single far-field measurement, the proof of Proposition 3.2 is
incomplete. In this corrigendum we provide a new proof of the same proposition
which fills the previous gap.Comment: 3 page
Continuity properties of Neumann-to-Dirichlet maps with respect to the H-convergence of the coefficient matrices
We investigate the continuity of boundary operators, such as the Neumann-to-Dirichlet map, with respect to the coefficient matrices of the underlying elliptic equations. We show that for nonsmooth coefficients the correct notion of convergence is the one provided by H-convergence (or G-convergence for symmetric matrices). We prove existence results for minimum problems associated to variational methods used to solve the so-called inverse conductivity problem, at least if we allow the conductivities to be anisotropic. In the case of isotropic conductivities we show that on certain occasions existence of a minimizer may fail
Reconstruction of material losses by perimeter penalization and phase-field methods
We treat the inverse problem of determining material losses, such as
cavities, in a conducting body, by performing electrostatic measurements at the
boundary. We develop a numerical approach, based on variational methods, to
reconstruct the unknown material loss by a single boundary measurement of
current and voltage type.
The method is based on the use of phase-field functions to model the material
losses and on a perimeter-like penalization to regularize the otherwise
ill-posed problem.We justify the proposed approach by a convergence result, as
the error on the measurement goes to zero.Comment: 28 page
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