Recovering Density and Speed of Sound Coefficients in the 2D Hyperbolic System of Acoustic Equations of the First Order by a Finite Number of Observations

Dmitriy Klyuchinskiy,Maxim Shishlenin,Nikita Novikov

doi:10.3390/math9020199

Abstract

We consider the coefficient inverse problem for the first-order hyperbolic system, which describes the propagation of the 2D acoustic waves in a heterogeneous medium. We recover both the denstity of the medium and the speed of sound by using a finite number of data measurements. We use the second-order MUSCL-Hancock scheme to solve the direct and adjoint problems, and apply optimization scheme to the coefficient inverse problem. The obtained functional is minimized by using the gradient-based approach. We consider different variations of the method in order to obtain the better accuracy and stability of the appoach and present the results of numerical experiments.

Highlights

We deal with the numerical solution of the coefficient inverse problem, which corresponds to the problems of ultrasound tomography
On the mathematical level, such problems are usually considered as inverse problems, when one has to recover the parameters of the model by using some measurement data [5,6,7,8]
The initial approximation was considered as an object without any inclusions, and we seek to identify said inclusions by solving the inverse problem

Summary

Introduction

We deal with the numerical solution of the coefficient inverse problem, which corresponds to the problems of ultrasound tomography. On the mathematical level, such problems are usually considered as inverse problems, when one has to recover the parameters of the model (that in our case describes the propagation of the ultrasound through the object of investigation) by using some measurement data [5,6,7,8]. The inverse problems are known for their ill-posedness and a requirement for a large number of computational resources for the numerical solution. The mathematical models for ultrasound acoustics usually have the form of either the second-order equation or the first-order system of PDE equations. The models, based on the second-order wave equation are usually easier to study, and there are more ways to efficiently solve the direct problem. The first-order system of acoustics, that we consider in this paper, requires more computational resources for solving. We mention several papers [9,10,11,12,13,14], where authors investigated inverse problems for a system of hyperbolic partial differential equations

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