Abstract
This paper presents a stable method for the identification of sources located on the separation interface of two homogeneous media (where one of them is contained by the other one), from measurement yielded by those sources on the exterior boundary of the media. This is an ill-posed problem because numerical instability is presented, i.e., minimal errors in the measurement can result in significant changes in the solution. To obtain the proposed stable method the identification problem is categorized into three subproblems, two of which present numerical instability and regularization methods must be applied to obtain their solution in a stable form. To manage the numerical instability due to the ill-posedness of these subproblems, the Tikhonov regularization and sequential smoothing methods are used. We illustrate this methodology in a circular and irregular region to demonstrate the feasibility of the proposed method, which yields convergent and stable solutions for input data with and without noise.
Highlights
Source identification problems are widely investigated in many research fields [1,2,3,4,5,6,7,8,9], and they are modeled by boundary value problems for which the analysis of the associated forward problem and its corresponding inverse problem must be considered, see, e.g., [10,11]
This method is used to handle the numerical instability associated to ill-posed problems, which depends on a parameter α called the regularization parameter that is chosen in terms of the error δ that is known [33]
The problem of identifying sources defined on separation interfaces in nonhomogeneous media can be solved in a stable form using the proposed algorithm, which can be applied to regions with simple and complex geometries
Summary
Source identification problems are widely investigated in many research fields [1,2,3,4,5,6,7,8,9], and they are modeled by boundary value problems for which the analysis of the associated forward problem and its corresponding inverse problem must be considered, see, e.g., [10,11]. The finite element method is one of the most important numerical methods for solving differential equations [28] It has been used for solving the inverse source problem using the gradient conjugate method and a control approach [19,29,30]. The Cauchy problem is ill-posed because it presents numerical instability, i.e., minimal errors in the measurement can yield significant changes in the solution To manage such instability, we employed the algorithm proposed in [29], where a penalization method was applied (equivalent to the Tikhonov regularization [33,34]); the conjugate gradient method; and the finite element method, where the regularization parameter was determined by the Tikhonov criterion.
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