An iterative shape reconstruction of source functions in a potential problem using the MFS

Abstract

In this work, we address the reconstruction of characteristic source functions in a potential problem, from the knowledge of full and partial boundary data. The inverse problem is formulated as an inverse obstacle problem and two iterative methods are applied. A decomposition method based on the Kirsch-Kress method (requires Cauchy data reconstruction) and a Newton-type method based on the domain derivative (requires the resolution of direct transmission problems) has been applied. For the reconstruction of Cauchy data we use the method of fundamental solutions (MFS) and we show that, for partial data, we can consider only one exterior artificial boundary. We test the domain derivative method using the MFS (for transmission problems) and present theoretical results that justifiy this numerical approximation. The feasibility of these methods will be illustrated by numerical simulations for both full and partial data.

Reconstruction of a source domain from boundary measurements

“…where ∂G k /∂n is expressed from equation (32). This is shown in figures 7(a)-(c) for k = 0, k ′ = 1 and k = 1, for δ = 0.3, R = 1.5 and various degrees of freedom N. From these figures it can be seen that the numerical results are convergent, as the number of degrees of freedom increases.…”

“…We mention that for the potential field our problem and methodology is similar to that developed recently by Martins (2012). However, in our study we investigate Helmhholtztype equations as well.…”

Fourier method for recovering acoustic sources from multi-frequency far-field data

“…Alves, Kress and Serranho [2] used Newton's iterative method in combination with the nonlinear integral equation to determine the locations and intensities of point sources. Martin [23] extended iterative method to identify and reconstruct the characteristic source functions in a potential problem from knowledge of full and partial boundary data. Kress and Rundell [20] reformulated the ISP as an inverse boundary value problem and presented an iterative solution method via boundary integral equations.…”

A Point Source Identification Problem for a Time Fractional Diffusion Equation

“…For parabolic-type differential equation, please see [15][16][17][18][19][20][21][22][23][24][25][26]. For elliptic-type differential equation, one can refer to [27][28][29], though the source identification problem has been well discussed in the classic framework, yet, to the best of the authors' knowledge, there are rare researches in the aspect of the source identification problem associated with fractional differential equation in spite of the physical and practical importance. As indicated in [30][31][32][33], the source identification problem associated with the time fractional diffusion equation is also ill-posed.…”