An optimization method for the inverse scattering problem of the biharmonic wave

Yan Chang; Yukun Guo

doi:10.3934/cac.2023010

Article Contents

2023, Volume 1, Issue 2: 168-182. Doi: 10.3934/cac.2023010

This issue Previous Article Stability in determination of states for the mean field game equations Next Article

An optimization method for the inverse scattering problem of the biharmonic wave

Yan Chang and
Yukun Guo^,

School of Mathematics, Harbin Institute of Technology, Harbin, China

^*Corresponding author: Yukun Guo
^*Corresponding author: Yukun Guo

Received: May 2023

Revised: June 2023

Early access: June 2023

Published: June 2023

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Abstract

We consider an inverse scattering problem of reconstructing the shape of a cavity in an infinitely thin plate, where the out-of-plane displacement is governed by the two-dimensional biharmonic wave equation. We first approximate the out-of-plane displacement of the plate by potentials over an auxiliary circle and then reformulate the inverse problem into an optimization problem. The convergence properties are established to show that when the regularization parameter tends to zero, the minimizer of the optimization problem tends to a solution of the inverse scattering problem. Numerical experiments are designed to verify the performance of the proposed method.

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Mathematics Subject Classification: Primary: 65N21, 65R30; Secondary: 31A30.

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Figure 1. Illustration of the inverse scattering problem

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Figure 2. Reconstruction of the starfish-shaped cavity subject to different observation points

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Figure 3. Reconstruction of the kite-shaped cavity subject to different regularization parameters and the number of source points

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Table 1. Computing time and relative errors for recovering the starfish

$ N_r $	$ 256 $	$ 128 $	$ 64 $	$ 32 $
CPU ($ s $)	$ 146.50 $	$ 84.00 $	$ 78.09 $	$ 54.70 $
$ E_D $	$ 9.8\% $	$ 11.50\% $	$ 12.02\% $	$ 15.94\% $

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Table 2. Computing time and reconstruction errors for recovering the kite

$ \delta $	$ 1\% $	$ 5\% $	$ 10\% $	$ 20\% $
CPU ($ s $)	$ 79.89 $	$ 83.95 $	$ 80.79 $	$ 77.08 $
$ E_D $	$ 10.58\% $	$ 10.53\% $	$ 10.55\% $	$ 10.70\% $

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