Numerical boundary identification for Helmholtz-type equations

Abstract

We study the identification of an unknown portion of the boundary of a two-dimensional domain occupied by a material satisfying Helmholtz-type equations from additional Cauchy data on the remaining known portion of the boundary. This inverse geometric problem is approached using the boundary element method (BEM) in conjunction with the Tikhonov first-order regularization procedure, whilst the choice of the regularization parameter is based on the L-curve criterion. The numerical results obtained show that the proposed method produces a convergent and stable solution.

The Plane Waves Method for Numerical Boundary Identification

“…We finally mention that the results obtained with the PWM in Figures 8 and 9 are comparable to those in Figure 10(b) in [38] which were obtained using the MFS, parametrization of ∂Ω 2 by a function y = f (x) with regularization using the NAG [42] Fortran routine E04UNF from the initial guess y = 0 for −1 < x < 1. This is to be expected since the PWM may be viewed as an asymptotic version of the MFS as the source points move further away from the simply connected bounded domain Ω, [2].…”

“…Prior to this paper, the inverse geometric problem (2.1a)-(2.1c) has been solved using the BEM in [37] and the MFS in [38], and it is the purpose of this study to develop the PWM for solving the same problem. In addition, we make a comparison between the MFS and the PWM and also investigate a physical example with g 3 = 0.…”

“…For Helmholtztype equations such problems have been solved using the BEM in [37] and the MFS in [7,38]. The paper is organized as follows.…”

The method of fundamental solutions for a biharmonic inverse boundary determination problem

“…In (17), (x i 2 ) (exact) and (x i 2 ) (λ) are the exact and numerical values, respectively, of the y-coordinates of the boundary γ represented by the graph of the function y. We can then plot the objective function F, given by (16) and the accuracy error E, given by (17), as functions of the regularization parameter λ. Then we expect that the optimal choice of λ returns the smallest values of both F and E. Of course, in the absence of an analytical solution for γ available, the error E cannot be calculated, but it is included herein in order to further justify the optimal choice of the regularization parameter λ given by the discrepancy principle, as it will be illustrated later on in Figs.…”

“…Prior to this study, Lesnic et al [16] and Marin [17], respectively, proposed a numerical technique to solve inverse Laplace and Helmholtz boundary determination problems in potential corrosion damage. They discretised the problem using the boundary element method (BEM) and solved the resulting nonlinear algebraic equations by minimizing the Tikhonov regularized function.…”