TY - JOUR
T1 - Inverse problem for the Helmholtz equation with Cauchy data
T2 - Reconstruction with conditional well-posedness driven iterative regularization
AU - Alessandrini, Giovanni
AU - De Hoop, Maarten V.
AU - Faucher, Florian
AU - Gaburro, Romina
AU - Sincich, Eva
N1 - Publisher Copyright:
© 2019 EDP Sciences, SMAI.
PY - 2019/5/1
Y1 - 2019/5/1
N2 - 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.
AB - 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.
KW - Helmholtz equation
KW - Inverse problems
KW - Reconstruction algorithm
KW - Stability and convergence of numerical methods
UR - http://www.scopus.com/inward/record.url?scp=85068354876&partnerID=8YFLogxK
U2 - 10.1051/m2an/2019009
DO - 10.1051/m2an/2019009
M3 - Article
AN - SCOPUS:85068354876
SN - 2822-7840
VL - 53
SP - 1002
EP - 1030
JO - Mathematical Modelling and Numerical Analysis
JF - Mathematical Modelling and Numerical Analysis
IS - 3
ER -