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

Michael V. Klibanov; Thuy T. Le; Loc H. Nguyen; Anders Sullivan; Lam Nguyen

doi:10.3934/ipi.2021068

Article Contents

2022, Volume 16, Issue 6: 1579-1618. Doi: 10.3934/ipi.2021068

This issue Previous Article A uniqueness theorem for inverse problems in quasilinear anisotropic media Next Article Refined instability estimates for some inverse problems

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

1.
Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
2.
US Army Research Laboratory, 2800 Powder Mill Road, Adelphi, MD 20783-1197, USA

^* Corresponding author: Michael V. Klibanov
^* Corresponding author: Michael V. Klibanov

Dedication: The authors dedicate this paper to the memory of Professor Victor Isakov, one of the world's very top experts in the field of Inverse Problems.

Received: June 2021

Revised: August 2021

Early access: October 2021

Published: December 2022

The first author is supported by the US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044

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Abstract

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.

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Mathematics Subject Classification: 35R30, 78A46.

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Figure 1. The schematic for the data generating and collecting device. A device, called radar, emits an acoustic source and then collect the time-dependent backscattering wave. In the physical experiment, we consider two cases: (a) the target is placed in the air and (b) the target is buried a few centimeters under the ground.

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Figure 2. Illustration of the process of correcting the data near $ (x = 0, t = 0). $ We know that when $ x $ is small, $ c(x) = 1 $. Therefore by (2.5), $ u(x, t) = \frac{1}{2} $ for $ t<| \tau (x)|. $ We, therefore, set $ u(x, t) = \frac{1}{2} $ for $ x $ and $ t $ small. Figure 2a and figure 2b are the graphs of the function $ u(0, t) $ before and after, respectively, this reassignment. These functions are taken from Test 3 in subsection 6.3.

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Figure 3. it The functions $ q^{(0)}(x, 0) $ and $ Q^{(0)}(x, 0) $, $ x \in (\epsilon, M) $ computed by solving (6.3) and using (6.5). The data to compute these functions are taken from the data for Test 1 with $ 5\% $ noise.

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Figure 4. The true spatially distributed dielectric constant function $ c_{\mathrm{true}} $, its initial version $ c_{\mathrm{init}} $ computed by (6.6) and its final reconstruction $ c_{\mathrm{comp}} $ by our convexification method. It is evident that in all cases, the initial solution $ c_{\mathrm{init}} $ computed by (6.6) already carries some information of $ c_{\mathrm{true}} $. The following iterative steps significantly improve the positions of "inclusions" and their values. Especially, in test 5 (Figure 4e), the convexification method successfully reconstructs the curves in the inclusion in the left.

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Figure 5. The case when the target is in the air. The raw and preprocessed data in the first row correspond to the wave scattered from a bush. (A) The time-dependent raw data, (B) The time-dependent backscattering wave after preprocessing, (C) Computed dielectric constant and its maximal value is 6.76. The raw and preprocessed data in the second row correspond to the wave scattered from a wood stake. (D) The time-dependent raw data, (E) The time-dependent backscattering wave after preprocessing, (F) Computed dielectric constant and its maximal value is 2.2. The computed dielectric constants for these two tests meet the expectation since they belong to intervals of their true value, see the last three rows of Table 1.

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Figure 6. The case when the target is buried under the ground. The raw and preprocessed data in the first row correspond to the wave scattered from a metal box. (A) The time-dependent raw data, (B) The time-dependent backscattering wave after preprocessing, (C) Computed dielectric constant and its maximal value is 5.2. The raw and preprocessed data in the second row correspond to the wave scattered from a metal cylinder. The raw and preprocessed data in the third row correspond to the wave scattered from a plastic cylinder. (D) The time-dependent raw data, (E) The time-dependent backscattering wave after preprocessing, (F) Computed dielectric constant and its maximal value is 4.7. Unlike the tests in the first two rows, we choose the upper envelop in third case when preprocessing the data because $ c_{\mathrm{target}} < c_{\mathrm{bckgr}}. $. (G) The time-dependent raw data, (H) The time-dependent backscattering wave after preprocessing, (I) Computed dielectric constant and its minimal value is 0.37. The computed dielectric constants for these three tests meet the expectation since they belong to intervals of their true value, see the last three rows of Table 1.

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Table 1. Computed dielectric constants of five targets

Target	$ c_{\mathrm{bckgr}} $	computed $ c_{\mathrm{rel}} $	$ c_{\mathrm{ bckgr}} $	computed $ c_{\text{target}} $	True $ c_{\text{target}} $
Bush	1	6.76	1	6.76	$ [3, 20] $
Wood stake	1	2.22	1	2.22	$ [2, 6] $
Metal box	4	5.2	$ [3, 5] $	$ [15.6, 26] $	$ [10, 30] $
Metal cylinder	4	4.7	$ [3, 5] $	$ [14.1, 23.5] $	$ [10, 30] $
Plastic cylinder	4	0.37	$ [3, 5] $	$ [1.11, 1.85] $	$ \left[ 1.1, 3.2 \right] $

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