An effective numerical strategy for retrieving all characteristic parameters of an elastic scatterer from its FFP measurements

Abstract A new computational strategy is proposed for determining all elastic scatterer characteristics including the shape, the material properties (Lame coefficients and density), and the location from the knowledge of far-field pattern (FFP) measurements. The proposed numerical approach is a multi-stage procedure in which a carefully designed regularized iterative method plays a central role. The adopted approach is critical for recognizing that the different nature and scales of the sought-after parameters as well as the frequency regime have different effects on the scattering observability. Identification results for two-dimensional elastic configurations highlight the performance of the designed solution methodology.

[1]  Peter Monk,et al.  Near field sampling type methods for the inverse fluid--solid interaction problem , 2011 .

[2]  Rainer Kress,et al.  Inverse scattering for surface impedance from phase-less far field data , 2011, J. Comput. Phys..

[3]  Dan Givoli,et al.  Combined arrival-time imaging and time reversal for scatterer identification , 2017 .

[4]  Y. M. Chen,et al.  An efficient numerical method for exterior and interior inverse problems of Helmholtz equation , 1991 .

[5]  Peter Monk,et al.  Recent Developments in Inverse Acoustic Scattering Theory , 2000, SIAM Rev..

[6]  Olha Ivanyshyn,et al.  Shape reconstruction of acoustic obstacles from the modulus of the far field pattern , 2007 .

[7]  R. Djellouli,et al.  Characterization of partial derivatives with respect to material parameters in a fluid–solid interaction problem , 2018, Journal of Mathematical Analysis and Applications.

[8]  A. Kirsch,et al.  A simple method for solving inverse scattering problems in the resonance region , 1996 .

[9]  William Rundell,et al.  Inverse Obstacle Scattering with Modulus of the Far Field Pattern as Data , 1997 .

[10]  R. Djellouli,et al.  Fréchet differentiability of the elasto‐acoustic scattered field with respect to Lipschitz domains , 2017 .

[11]  Charbel Farhat,et al.  On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method , 2002 .

[12]  William Rundell,et al.  A quasi-Newton method in inverse obstacle scattering , 1994 .

[13]  Integral equation methods in inverse obstacle scattering , 1995 .

[14]  Mansor Nakhkash,et al.  An analytical approach to estimate the number of small scatterers in 2D inverse scattering problems , 2012 .

[15]  Otmar Scherzer,et al.  Lamé Parameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems , 2017, SIAM J. Imaging Sci..

[16]  Patrick Amestoy,et al.  Hybrid scheduling for the parallel solution of linear systems , 2006, Parallel Comput..

[17]  Michael V. Klibanov,et al.  Convexification Method for an Inverse Scattering Problem and Its Performance for Experimental Backscatter Data for Buried Targets , 2019, SIAM J. Imaging Sci..

[18]  Marc Bonnet,et al.  Inverse material identification in coupled acoustic-structure interaction using a modified error in constitutive equation functional , 2014, Computational mechanics.

[19]  R. Djellouli,et al.  On the existence and the uniqueness of the solution of a fluid-structure interaction scattering problem , 2014 .

[20]  Patrick Amestoy,et al.  A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling , 2001, SIAM J. Matrix Anal. Appl..

[21]  Rainer Kress,et al.  Identification of sound-soft 3D obstacles from phaseless data , 2010 .

[22]  A. Fichtner.  Full Seismic Waveform Modelling and Inversion , 2011 .

[23]  Johannes Elschner,et al.  An Optimization Method in Inverse Acoustic Scattering by an Elastic Obstacle , 2009, SIAM J. Appl. Math..

[24]  Armin Lechleiter,et al.  Identifying Lamé parameters from time-dependent elastic wave measurements , 2017 .

[25]  Xiaodong Liu,et al.  The linear sampling method for inhomogeneous medium and buried objects from far field measurements , 2016 .

[26]  Gerhard Kristensson,et al.  Inverse problems for acoustic waves using the penalised likelihood method , 1986 .

[27]  A. Bayliss,et al.  Radiation boundary conditions for wave-like equations , 1980 .

[28]  Roland Griesmaier,et al.  A multifrequency MUSIC algorithm for locating small inhomogeneities in inverse scattering , 2016, 1607.04017.

[29]  D. Colton,et al.  Qualitative Methods in Inverse Electromagnetic Scattering Theory: Inverse Scattering for Anisotropic Media. , 2017, IEEE Antennas and Propagation Magazine.

[30]  Izar Azpiroz,et al.  Contribution to the numerical reconstruction in inverse elasto-acoustic scattering , 2018 .

[31]  Paul A. Martin,et al.  Fluid-Solid Interaction: Acoustic Scattering by a Smooth Elastic Obstacle , 1995, SIAM J. Appl. Math..

[32]  Assad A. Oberai,et al.  Solution of the time‐harmonic viscoelastic inverse problem with interior data in two dimensions , 2012 .

[33]  Michael V. Klibanov,et al.  Convexification of a 3-D coefficient inverse scattering problem , 2018, Comput. Math. Appl..

[34]  Albert Tarantola,et al.  Inverse problem theory - and methods for model parameter estimation , 2004 .

[35]  Florian Faucher,et al.  Localization of small obstacles from back-scattered data at limited incident angles with full-waveform inversion , 2018, J. Comput. Phys..

[36]  Dan Givoli,et al.  Obstacle identification using the TRAC algorithm with a second‐order ABC , 2018, International Journal for Numerical Methods in Engineering.

[37]  Fioralba Cakoni,et al.  Qualitative Methods in Inverse Scattering Theory: An Introduction , 2005 .

[38]  Xavier Antoine,et al.  Bayliss-Turkel-like radiation conditions on surfaces of arbitrary shape , 1999 .

[39]  Michael V. Klibanov,et al.  Reconstruction Procedures for Two Inverse Scattering Problems Without the Phase Information , 2015, SIAM J. Appl. Math..

[40]  W. Tobocman,et al.  Inverse acoustic wave scattering in two dimensions from impenetrable targets , 1989 .

[41]  Lars Mönch,et al.  A Newton method for solving the inverse scattering problem for a sound-hard obstacle , 1996 .

[42]  Bo Zhang,et al.  Recovering scattering obstacles by multi-frequency phaseless far-field data , 2016, J. Comput. Phys..

[43]  Francois Le Chevalier,et al.  Principles of Radar and Sonar Signal Processing , 2002 .

[44]  V. Morozov.  On the solution of functional equations by the method of regularization , 1966 .

[45]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[46]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .

[47]  Eli Turkel,et al.  Simultaneous Scatterer Shape Estimation and Partial Aperture Far-Field Pattern Denoising , 2012 .

[48]  Zhiming,et al.  An Adaptive Uniaxial Perfectly Matched Layer Method for Time-Harmonic Scattering Problems , 2008 .

[49]  Ross D. Murch,et al.  CORRIGENDUM: Newton-Kantorovich method applied to two-dimensional inverse scattering for an exterior Helmholtz problem , 1988 .

[50]  Gregory Beylkin,et al.  Linearized inverse scattering problems in acoustics and elasticity , 1990 .

[51]  R. Djellouli,et al.  Efficient DG‐like formulation equipped with curved boundary edges for solving elasto‐acoustic scattering problems , 2014 .

[52]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[53]  Yukun Guo,et al.  Uniqueness results on phaseless inverse acoustic scattering with a reference ball , 2018, Inverse Problems.

[54]  Julien Diaz,et al.  Long-term stable acoustic absorbing boundary conditions for regular-shaped surfaces , 2013 .

[55]  Dianne P. O'Leary,et al.  The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems , 1993, SIAM J. Sci. Comput..

[56]  A. N. Tikhonov,et al.  REGULARIZATION OF INCORRECTLY POSED PROBLEMS , 1963 .

[57]  David Colton,et al.  The three dimensional inverse scattering problem for acoustic waves , 1982 .

[58]  Ekaterina Iakovleva,et al.  A MUSIC Algorithm for Locating Small Inclusions Buried in a Half-Space from the Scattering Amplitude at a Fixed Frequency , 2005, Multiscale Model. Simul..

[59]  Houssem Haddar,et al.  On Simultaneous Identification of the Shape and Generalized Impedance Boundary Condition in Obstacle Scattering , 2012, SIAM J. Sci. Comput..

[60]  G. Wahba.  Spline models for observational data , 1990 .

[61]  Rabia Djellouli,et al.  Mathematical Determination of the Fréchet Derivative with Respect to the Domain for a Fluid-Structure Scattering Problem: Case of Polygonal-Shaped Domains , 2018, SIAM J. Math. Anal..