Waveform inversion via reduced order modeling

Borcea, Liliana; Garnier, Josselin; Mamonov, A.; Zimmerling, Jörn

doi:10.1190/geo2022-0070.1

Cited by 10 publications

(7 citation statements)

References 78 publications

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“…The Lanczos method has been also combined with nonlinear optimization in the Dirichlet and Neumann setting. In particular, in the Dirichlet case, see, [13,15], the input for the optimization is the tridiagonal matrix that the Lanczos orthogonalisation method returns using the ROM projections as input. The reason why it is useful to consider the nonlinear Lanczos transform in terms of the PDE coefficient in a Dirichlet setting is to eliminate the linear relation between boundary measurements and the ROM projections.…”

Section: Solving the Inverse Problem With Nonlinear Optimizationmentioning

confidence: 99%

“…The approach that we take in this paper was inspired by similar works developed for time-domain wave propagation with Dirichlet boundary conditions, see [13], [14] and [15]. For practical applications, however, impedance boundary conditions are of great interest since they are equivalent to the Sommerfeld radiation condition in 1D (and an approximation of the radiation condition in two and three dimensions).…”

Section: Introductionmentioning

confidence: 99%

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Reduced Order Model Based Nonlinear Waveform Inversion for the 1D Helmholtz Equation

Tataris,

Leeuwen

2023

Preprint

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We study a reduced order model (ROM) based waveform inversion method applied to a Helmholtz problem with impedance boundary conditions and variable refractive index. The first goal of this paper is to obtain relations that allow the reconstruction of the Galerkin projection of the continuous problem onto the space spanned by solutions of the Helmholtz equation. The second goal is to study the introduced nonlinear optimization method based on the ROM aimed to estimate the refractive index from reflection and transmission data. Finally we compare numerically our method to the conventional least squares inversion based on minimizing the distance between modelled to measured data.

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Section: Solving the Inverse Problem With Nonlinear Optimizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Reduced Order Model Based Nonlinear Waveform Inversion for the 1D Helmholtz Equation

Tataris,

Leeuwen

2023

Preprint

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“…Because of the sparse structure of the system (tridiagonal finite-difference in 1D problems), the basis functions are localized and depend only weakly on the media perturbations. Rigorous analysis of this property can be obtained using the previously mentioned Marchenko, Gelfand, Levitan and Krein approach and is in progress at the moment [3].…”

Section: Introductionmentioning

confidence: 99%

Lippmann–Schwinger–Lanczos algorithm for inverse scattering problems

Druskin¹,

Moskow²,

Zaslavsky³

2021

Inverse Problems

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Data-driven reduced order models (ROMs) are combined with the Lippmann-Schwinger integral equation to produce a direct nonlinear inversion method. The ROM is viewed as a Galerkin projection and is sparse due to Lanczos orthogonalization. Embedding into the continuous problem, a data-driven internal solution is produced. This internal solution is then used in the Lippmann-Schwinger equation, thus making further iterative updates unnecessary. We show numerical experiments for spectral domain domain data for which our inversion is far superior to the Born inversion and works as well as when the true internal solution is known.

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“…[26]. All of the above ideas on the ROM inversion in time domain wave propagation and in diffusion are elaborated in the following innovative works, [26,8,11,27,8]. All of these great developments and observations motivated us to apply these ideas of the ROM inversion in the frequency domain for a scattering framework.…”

Section: Indirect Methodsmentioning

confidence: 99%

“…The approach that we take in this chapter was initially developed for time-domain wave propagation with Dirichlet boundary conditions, see [11], [47]and [10]. For practical applications, however, impedance boundary conditions are of great interest since they are equivalent to the Sommerfeld radiation condition in 1D (and an approximation of the radiation condition in two and three dimensions).…”

Section: Introductionmentioning

confidence: 99%

Gelfand-Levitan-Marchenko and model order reduction methods in inverse scattering

Tataris

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Inverse scattering problems arise in many applications, especially in imaging. In this thesis we studied frequency domain inverse scattering problems for the Helmholtz and the Schrödinger operators using both classical inverse scattering and modern reduced order model techniques. We started by revisiting the classical Gelfand-Levitan-Marchenko (GLM) integral equation method for solving the inverse Schrödinger scattering problem in 1D. The inverse Schrödinger scattering problem is interesting for imaging purposes, since it is possible to transform the Helmholtz and the (frequency domain) acoustic wave equation to the Schrödinger equation using a coordinate transform. In particular, we considered the GLM method with noise in the data, where we contributed an error bound for the solution of the unregularised GLM equation. We also proposed a regularised total least squares formulation in the infinite dimensional setting and we showed well posedness. Moreover, we studied the 1D scattering problem for the Helmholtz operator and we developed a GLM theory exclusively for the Helmholtz problem. In particular, we derived a generalised GLM equation in the space of tempered distributions for reconstructing the Jost solutions of the Helmholtz operator. To do so, we had to examine the asymptotic behaviour of the Jost solutions of the Helmholtz operator in terms of the wavenumber. After studying classical inverse scattering methods based on the GLM approach, we continued by studying inversion methods based on reduced order models (ROMs). We started with the inverse Schrödinger scattering problem of retrieving the scattering potential in 1D Schrödinger equation using boundary data. For that reason, we proposed a two-step approach inspired by a previously-published ROM-based method. We presented explicit expressions allowing the exact reconstruction of the ROM-matrices from boundary data and proposed a new data-assimilation approach for approximating the state from these matrices. Given the estimates of the states, the scattering potential is obtained by solving a Lippmann-Schwinger type integral equation. Finally, we combined the traditional FWI method with reduced order models and we proposed a new nonlinear inversion method for the inverse Helmholtz scattering problem. In particular, the input of our misfit functional consisted of the stiffness matrix of the ROM projection. In this case, we studied the well posedness of the nonlinear optimization problem and we derived the optimality condition. We finally compared numerically the ROM based FWI method with the conventional FWI method.

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Waveform inversion via reduced order modeling

Cited by 10 publications

References 78 publications

Reduced Order Model Based Nonlinear Waveform Inversion for the 1D Helmholtz Equation

Reduced Order Model Based Nonlinear Waveform Inversion for the 1D Helmholtz Equation

Lippmann–Schwinger–Lanczos algorithm for inverse scattering problems

Gelfand-Levitan-Marchenko and model order reduction methods in inverse scattering

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