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

Andreas Tataris; Tristan van Leeuwen

doi:10.1007/s10440-024-00700-y

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

Andreas Tataris^*, Tristan van Leeuwen

^*Corresponding author for this work

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Abstract

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.

Original language	English
Article number	11
Number of pages	24
Journal	Acta Applicandae Mathematicae
Volume	194
Issue number	1
DOIs	https://doi.org/10.1007/s10440-024-00700-y
Publication status	Published - 2024

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

Keywords

Full waveform inversion
Helmholtz equation
Nonlinear inversion
Reduced order models

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N2 - 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.

AB - 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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KW - Reduced order models

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

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