An efficient and robust algorithm for source reconstruction in the Helmholtz equation. Issue 13 (23rd December 2021)

• Record Type: Journal Article
• Authors: Charkaoui, Abderrahim; El Badia, Abdellatif; Eddine Alaa, Nour
• Is Part Of: Inverse problems in science and engineering. Volume 29:Issue 13(2021)
• Page Start: 2944
• Page End: 2970
• Publication Date: 2021-12-23
• Subjects: Inverse source problem -- shape optimization -- Laplace equation -- level set method -- Helmholtz equation; 65K10 -- 35R30 -- 49J20 -- 49J50 -- 35J05; Engineering mathematics -- Periodicals; Inverse problems (Differential equations) -- Periodicals; 620.001515357
• DOI: 10.1080/17415977.2021.1960832
• Languages: English
• ISSNs: 1741-5977

Abstract

ABSTRACT: This work proposes an identification method for reconstructing the characteristic source χ D in the Helmholtz equation Δ u + μ u = χ D from boundary measurements. We formulate the inverse source problem into a shape optimization problem by introducing a least-squares cost function. Using the shape optimization techniques, we prove the existence of an optimal solution to the considered shape optimization problem and we calculate the gradient of the cost function with respect to the shape D . By using the Level set method, we present an iterative algorithm to recover numerically the shape D . We develop a new technique to initialize the level set algorithm, which permits capturing different hidden shapes. To examine the validity of the proposed method, we illustrate several numerical experiments with different hidden shapes. By adding a level of noise to the measured data, we evaluate the robustness of our reconstruction algorithm.