Convexification for a 1D hyperbolic coefficient inverse problem with single measurement data

Alexey Smirnov; Michael Klibanov; Loc Nguyen

doi:10.3934/ipi.2020042

Article Contents

2020, Volume 14, Issue 5: 913-938. Doi: 10.3934/ipi.2020042

This issue Previous Article Nonlocal regularized CNN for image segmentation Next Article On the simultaneous recovery of the conductivity and the nonlinear reaction term in a parabolic equation

Convexification for a 1D hyperbolic coefficient inverse problem with single measurement data

Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, USA

^* Corresponding author: Michael Klibanov (mklibanv@uncc.edu)
^* Corresponding author: Michael Klibanov (mklibanv@uncc.edu)

Received: February 2020

Revised: May 2020

Published: July 2020

The work was supported by US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044

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Abstract

A version of the convexification numerical method for a Coefficient Inverse Problem for a 1D hyperbolic PDE is presented. The data for this problem are generated by a single measurement event. This method converges globally. The most important element of the construction is the presence of the Carleman Weight Function in a weighted Tikhonov-like functional. This functional is strictly convex on a certain bounded set in a Hilbert space, and the diameter of this set is an arbitrary positive number. The global convergence of the gradient projection method is established. Computational results demonstrate a good performance of the numerical method for noisy data.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 35L10.

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Figure 1. The rectangle $ D(x,t) = \left\{ (\xi ,\tau ):\left\vert \xi \right\vert <\tau <t-\left\vert x-\xi \right\vert \right\} $ and the triangle $ Tr $

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Figure 3. Numerical reconstructions (the black marked dots) of functions $ a(x) $ (the solid lines). Noise level $ \xi = 0.1 $.

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Figure 2. The comparison of noiseless and noisy data. Figure 2(A) shows the norm of the functional (6.2) for each iteration of the gradient descent for the test function depicted on Figure 3(A). Figure 2(D) corresponds to our test for $ \lambda $ = 0; see the text.

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Figure 4. Limiting testing of different values of the parameter $ \lambda $ for the test function of Test 1, see comments in the text. The data are noiseless.

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Figure 5. Testing of different values of the parameter $ \alpha , $ see comments in the text. Solid line is the correct function of Test 1. The data are noiseless.

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Table 6.2. Summary of numerical results. Here $ \left\Vert \cdot\right\Vert _{\infty } $ denotes the $ L_{\infty } $

Test	$ n^{\ast} $	Error	$ \Vert J^h_{\lambda,\beta,\mu}(w_0) \Vert_{\infty} $	$ \Vert J^h_{\lambda,\beta,\mu}(w_{n^{\ast}}) \Vert_{\infty} $
1	30	0.1628	2570	2.7465
2	33	0.2907	34.42	0.22
3	51	0.0804	3.12	0.0007
4	41	0.3222	0.82	0.0003

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