scholarly journals A Modification of Gradient Descent Method for Solving Coefficient Inverse Problem for Acoustics Equations

Computation ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 73
Author(s):  
Dmitriy Klyuchinskiy ◽  
Nikita Novikov ◽  
Maxim Shishlenin
Keyword(s):  
Inverse Problem ◽  
Acoustic Waves ◽  
Gradient Descent ◽  
Descent Method ◽  
Godunov Method ◽  
Order System ◽  
Gradient Descent Method ◽  
Coefficient Inverse Problem ◽  
First Order ◽  
Acoustics Equations

We investigate the mathematical model of the 2D acoustic waves propagation in a heterogeneous domain. The hyperbolic first order system of partial differential equations is considered and solved by the Godunov method of the first order of approximation. This is a direct problem with appropriate initial and boundary conditions. We solve the coefficient inverse problem (IP) of recovering density. IP is reduced to an optimization problem, which is solved by the gradient descent method. The quality of the IP solution highly depends on the quantity of IP data and positions of receivers. We introduce a new approach for computing a gradient in the descent method in order to use as much IP data as possible on each iteration of descent.

scholarly journals On the modeling of ultrasound wave propagation in the frame of inverse problem solution

2021 ◽  
Vol 2099 (1) ◽  
pp. 012044
Author(s):  
N S Novikov ◽  
D V Klyuchinskiy ◽  
M A Shishlenin ◽  
S I Kabanikhin
Keyword(s):  
Inverse Problem ◽  
Gradient Descent ◽  
Finite Volume Methods ◽  
Adjoint Problem ◽  
Order System ◽  
Ultrasound Wave ◽  
Problem Solution ◽  
Coefficient Inverse Problem ◽  
Acoustic Sounding ◽  
First Order

Abstract In this paper we consider the inverse problem of detecting the inclusions inside the human tissue by using the acoustic sounding wave. The problem is considered in the form of coefficient inverse problem for first-order system of PDE and we use the gradient descent approach to recover the coefficients of that system. The important part of the sceme is the solution of the direct and adjoint problem on each iteration of the descent. We consider two finite-volume methods of solving the direct problem and study their Influence on the performance of recovering the coefficients.

scholarly journals Convexification-based globally convergent numerical method for a 1D coefficient inverse problem with experimental data

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Michael V. Klibanov ◽  
Thuy T. Le ◽  
Loc H. Nguyen ◽  
Anders Sullivan ◽  
Lam Nguyen
Keyword(s):  
Inverse Problem ◽  
Gradient Descent ◽  
Regularization Parameter ◽  
Descent Method ◽  
Gradient Descent Method ◽  
Convexity Property ◽  
Coefficient Inverse Problem ◽  
True Solution ◽  
Cost Functional ◽  
Bounded Set

<p style='text-indent:20px;'>To compute the spatially distributed dielectric constant from the backscattering computationally simulated ane experimentally collected data, we study a coefficient inverse problem for a 1D hyperbolic equation. To solve this inverse problem, we establish a new version of the Carleman estimate and then employ this estimate to construct a cost functional, which is strictly convex on a convex bounded set of an arbitrary diameter in a Hilbert space. The strict convexity property is rigorously proved. This result is called the convexification theorem and it is the central analytical result of this paper. Minimizing this cost functional by the gradient descent method, we obtain the desired numerical solution to the coefficient inverse problems. We prove that the gradient descent method generates a sequence converging to the minimizer starting from an arbitrary point of that bounded set. We also establish a theorem confirming that the minimizer converges to the true solution as the noise in the measured data and the regularization parameter tend to zero. Unlike the methods, which are based on the optimization, our convexification method converges globally in the sense that it delivers a good approximation of the exact solution without requiring a good initial guess. Results of numerical studies of both computationally simulated and experimentally collected data are presented.</p>

Numerics of acoustical 2D tomography based on the conservation laws

2020 ◽  
Vol 28 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Sergey I. Kabanikhin ◽  
Dmitriy V. Klyuchinskiy ◽  
Nikita S. Novikov ◽  
Maxim A. Shishlenin
Keyword(s):  
Mathematical Modeling ◽  
Inverse Problem ◽  
Conservation Laws ◽  
Speed Of Sound ◽  
Gradient Method ◽  
Acoustic Waves ◽  
Order System ◽  
First Order ◽  
Waves Propagation ◽  
Gradient Of The Functional

AbstractWe investigate the mathematical modeling of the 2D acoustic waves propagation, based on the conservation laws. The hyperbolic first-order system of partial differential equations is considered and solved by the method of S. K. Godunov. The inverse problem of reconstructing the density and the speed of sound of the medium is considered. We apply the gradient method to reconstruct the parameters of the medium. The gradient of the functional is obtained. Numerical results are presented.

scholarly journals 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

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 199
Author(s):  
Dmitriy Klyuchinskiy ◽  
Nikita Novikov ◽  
Maxim Shishlenin
Keyword(s):  
Inverse Problem ◽  
Finite Number ◽  
Hyperbolic System ◽  
Speed Of Sound ◽  
Acoustic Waves ◽  
Coefficient Inverse Problem ◽  
Optimization Scheme ◽  
First Order ◽  
Acoustic Equations ◽  
Gradient Based

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.

scholarly journals On the recovering of acoustic attenuation in 2D acoustic tomography

2021 ◽  
Vol 2099 (1) ◽  
pp. 012046
Author(s):  
M A Shishlenin ◽  
N S Novikov ◽  
D V Klyuchinskiy
Keyword(s):  
Inverse Problem ◽  
Gradient Method ◽  
Conjugate Problem ◽  
Order System ◽  
Acoustic Tomography ◽  
Coefficient Inverse Problem ◽  
Acoustic Attenuation ◽  
First Order ◽  
Gradient Of The Functional ◽  
The Cost

Abstract The inverse problem of recovering the acoustic attenuation in the inclusions inside the human tissue is considered. The coefficient inverse problem is formulated for the first-order system of PDE. We reduce the inverse problem to the optimization of the cost functional by gradient method. The gradient of the functional is determined by solving a direct and conjugate problem. Numerical results are presented.

Image restoration algorithm based on primal-dual hybrid gradient descent method

2009 ◽  
Vol 29 (4) ◽  
pp. 987-989
Author(s):  
Hui ZHANG ◽  
Li-zhi CHENG ◽  
Zai-xin ZHAO
Keyword(s):  
Image Restoration ◽  
Gradient Descent ◽  
Descent Method ◽  
Gradient Descent Method ◽  
Restoration Algorithm ◽  
Primal Dual
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