Shape optimization method for an inverse geometric source problem and stability at critical shape

Lekbir Afraites; Chorouk Masnaoui; Mourad Nachaoui

doi:10.3934/dcdss.2021006

Article Contents

2022, Volume 15, Issue 1: 1-21. Doi: 10.3934/dcdss.2021006

This issue Previous Article Preface to the special issue "Partial Differential Equations, Optimization and their Applications" Next Article A new coupled complex boundary method (CCBM) for an inverse obstacle problem

Shape optimization method for an inverse geometric source problem and stability at critical shape

Laboratory of Mathematics and Applications (LMA), Faculty of Sciences and Techniques, Sultan Moulay Slimane University, Beni Mellal, Morocco

^* Corresponding author: lekbir.afraites@gmail.com
^* Corresponding author: lekbir.afraites@gmail.com

Received: July 31, 2020

Revised: October 31, 2020

Early access: January 2021

Published: January 2022

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Abstract

This work deals with a geometric inverse source problem. It consists in recovering inclusion in a fixed domain based on boundary measurements. The inverse problem is solved via a shape optimization formulation. Two cost functions are investigated, namely, the least squares fitting, and the Kohn-Vogelius function. In this case, the existence of the shape derivative is given via the first order material derivative of the state problems. Furthermore, using the adjoint approach, the shape gradient of both cost functions is characterized. Then, the stability is investigated by proving the compactness of the Hessian at the critical shape for both considered cases. Finally, based on the gradient method, a steepest descent algorithm is developed, and some numerical experiments for non-parametric shapes are discussed.

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Mathematics Subject Classification: Primary: 49Q10, 65N20; Secondary: 35J25, 35R35, 65N30.

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Figure 1. Reconstruction of the first domain by Kohn-Vogelius and least squares methods

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Figure 2. Evolution of the Kohn-Vogelius and least squares cost functions and their gradients according to number of iterations

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Figure 3. Reconstruction of the second domain by Kohn-Vogelius and least squares methods

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Figure 4. Evolution of the Kohn-Vogelius and least squares cost functions, and their associated gradients according to the number of iterations

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Figure 5. Reconstruction of the third domain by Kohn-Vogelius and least squares methods

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Figure 6. Reconstruction of the fourth domain by Kohn-Vogelius and least squares methods

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Figure 7. Reconstruction of the fifth domain by Kohn-Vogelius and least squares methods

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Figure 8. Reconstruction of the first domain by Kohn-Vogelius and least squares methods with noise

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Figure 9. Reconstruction of the second domain by Kohn-Vogelius and least squares methods with noise

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Figure 10. Reconstruction of the third domain by Kohn-Vogelius and least squares methods with noise

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Figure 11. Reconstruction of the fourth domain by Kohn-Vogelius and least squares methods with noise

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