Recovery of wave speeds and density of mass across a heterogeneous smooth interface from acoustic and elastic wave reflection operators

Bhattacharyya, Sombuddha; Hoop, Maarten V. de; Katsnelson, Vitaly; Uhlmann, Günther

doi:10.1007/s13137-022-00199-1

Cited by 3 publications

(1 citation statement)

References 32 publications

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“…Analyzing these problems under realistic assumptions is subject to current research and often requires the use of sophisticated mathematical tools, see e.g. Rachele (2000), Stefanov et al (2021), Bhattacharyya et al (2022) for the application of microlocal techniques to the recovery of material parameters from certain type of boundary measurements. Furthermore, numerical methods which make optimal use of the latest analytical results and lead to provably convergent and reliable solutions of the inverse problems are in high demand.…”

Section: Introductionmentioning

confidence: 99%

Unique continuation for the Lamé system using stabilized finite element methods

Burman

Preuß

2023

Int J Geomath

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We introduce an arbitrary order, stabilized finite element method for solving a unique continuation problem subject to the time-harmonic elastic wave equation with variable coefficients. Based on conditional stability estimates we prove convergence rates for the proposed method which take into account the noise level and the polynomial degree. A series of numerical experiments corroborates our theoretical results and explores additional aspects, e.g. how the quality of the reconstruction depends on the geometry of the involved domains. We find that certain convexity properties are crucial to obtain a good recovery of the wave displacement outside the data domain and that higher polynomial orders can be more efficient but also more sensitive to the ill-conditioned nature of the problem.

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Section: Introductionmentioning

confidence: 99%

Unique continuation for the Lamé system using stabilized finite element methods

Burman

Preuß

2023

Int J Geomath

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The solid-fluid transmission problem

Eptaminitakis,

Stefanov

2024

Trans. Amer. Math. Soc.

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We study microlocally the transmission problem at the interface between an isotropic linear elastic solid and a linear inviscid fluid. We set up a system of evolution equations describing the particle displacement and velocity in the solid, and pressure and velocity in the fluid, coupled by suitable transmission conditions at the interface. We show well-posedness for the coupled system and study the problem microlocally, constructing a parametrix for it using geometric optics. This construction describes the reflected and transmitted waves, including mode converted ones, related to incoming waves from either side. We also study formation of surface Scholte waves. Finally, we prove that under suitable assumptions, we can recover the s- and the p-speeds, as well as the speed of the liquid, from boundary measurements.

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Recovery of Piecewise Smooth Density and Lamé Parameters from High Frequency Exterior Cauchy Data

Bhattacharyya¹,

Hoop²,

Katsnelson³

et al. 2022

SIAM J. Imaging Sci.

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In this paper, we study an inverse problem for an acoustic-gravitational system whose principal symbol is identical to that of an acoustic wave operator. The displacement vector of a gas or liquid between the unperturbed and perturbed flow is denoted by u(t, x). It satisfies a partial differential equation (PDE) system with a principal symbol corresponding to an acoustic wave operator, but with additional terms to account for a global gravitational field and self-gravitation. These factors make the operator nonlocal, as it depends on the wave speed and density of mass. We assume that all parameters are piecewise smooth in R 3 (i.e., smooth everywhere except for jump discontinuities across closed hypersurfaces called interfaces) but unknown inside a bounded domain D. We are given the solution operator for this acoustic-gravitational system, but only outside D and only for initial data supported outside D. Using high-frequency waves, we prove that the piecewise smooth wave speed and density are uniquely determined by this map under certain geometric conditions.

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Recovery of wave speeds and density of mass across a heterogeneous smooth interface from acoustic and elastic wave reflection operators

Cited by 3 publications

References 32 publications

Unique continuation for the Lamé system using stabilized finite element methods

Unique continuation for the Lamé system using stabilized finite element methods

The solid-fluid transmission problem

Recovery of Piecewise Smooth Density and Lamé Parameters from High Frequency Exterior Cauchy Data

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