In this paper, we study the performance of Full Waveform Inversion (FWI) from time-harmonic Cauchy data via conditional well-posedness driven iterative regularization. The Cauchy data can be obtained with dual sensors measuring the pressure and the normal velocity. We define a novel misfit functional which, adapted to the Cauchy data, allows the independent location of experimental and computational sources. The conditional well-posedness is obtained for a hierarchy of subspaces in which the inverse problem with partial data is Lipschitz stable. Here, these subspaces yield piecewise linear representations of the wave speed on given domain partitions. Domain partitions can be adaptively obtained through segmentation of the gradient. The domain partitions can be taken as a coarsening of an unstructured tetrahedral mesh associated with a finite element discretization of the Helmholtz equation. We illustrate the effectiveness of the iterative regularization through computational experiments with data in dimension three. In comparison with earlier work, the Cauchy data do not suffer from eigenfrequencies in the configurations.
Accepté le :
DOI : 10.1051/m2an/2019009
Mots-clés : Inverse problems, Helmholtz equation, stability and convergence of numerical methods, reconstruction algorithm
@article{M2AN_2019__53_3_1005_0,
author = {Alessandrini, Giovanni and De Hoop, Maarten V. and Faucher, Florian and Gaburro, Romina and Sincich, Eva},
title = {Inverse problem for the {Helmholtz} equation with {Cauchy} data: reconstruction with conditional well-posedness driven iterative regularization},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1005--1030},
publisher = {EDP-Sciences},
volume = {53},
number = {3},
year = {2019},
doi = {10.1051/m2an/2019009},
zbl = {1418.65164},
mrnumber = {3973923},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2019009/}
}
TY - JOUR AU - Alessandrini, Giovanni AU - De Hoop, Maarten V. AU - Faucher, Florian AU - Gaburro, Romina AU - Sincich, Eva TI - Inverse problem for the Helmholtz equation with Cauchy data: reconstruction with conditional well-posedness driven iterative regularization JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 1005 EP - 1030 VL - 53 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2019009/ DO - 10.1051/m2an/2019009 LA - en ID - M2AN_2019__53_3_1005_0 ER -
%0 Journal Article %A Alessandrini, Giovanni %A De Hoop, Maarten V. %A Faucher, Florian %A Gaburro, Romina %A Sincich, Eva %T Inverse problem for the Helmholtz equation with Cauchy data: reconstruction with conditional well-posedness driven iterative regularization %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 1005-1030 %V 53 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2019009/ %R 10.1051/m2an/2019009 %G en %F M2AN_2019__53_3_1005_0
Alessandrini, Giovanni; De Hoop, Maarten V.; Faucher, Florian; Gaburro, Romina; Sincich, Eva. Inverse problem for the Helmholtz equation with Cauchy data: reconstruction with conditional well-posedness driven iterative regularization. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 3, pp. 1005-1030. doi : 10.1051/m2an/2019009. http://www.numdam.org/articles/10.1051/m2an/2019009/
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