Abstract
In this article, we study a novel computational technique for the efficient numerical solution of the inverse boundary identification problem with uncertain data in two dimensions. The method essentially relies on a posteriori error indicators consisting of the Tikhonov regularized solutions obtained by the method of fundamental solutions (MFS) and the given data for the problem in hand. For a desired accuracy, the a posteriori error estimator chooses the best possible combination of a complete set of fundamental solutions determined by the location of the sources that are arranged in a particular manner on a pseudo-boundary at each iteration. Also, since we are interested in a stable solution, an adaptive stochastic optimization strategy based on an error-balancing criterion is used, so as to avoid unstable regions where the stability contributions may be relatively large. These ideas are applied to two benchmark problems and are found to produce efficient and accurate results.
Highlights
In the context of heat transfer, the boundary identification problem is a special class of inverse geometric problem for determining the location and shape of the boundary of a conducting body by means of thermal measurements, performed on an accessible part of the boundary
For inverse problems such as those involving boundary identification, which suffer from stability issues, it is a nontrivial task, since they are ill-posed in the sense of Hadamard [8], i.e., any small change in the input data can lead to a drastic change in the solution; it is imperative to use regularization techniques for stabilizing the computations
We have considered the efficient numerical solution of the inverse boundary identification problem in 2D
Summary
In the context of heat transfer, the boundary identification problem is a special class of inverse geometric problem for determining the location and shape of the boundary of a conducting body by means of thermal measurements, performed on an accessible part of the boundary. An efficient numerical approximation depends on a reliable method to determine its quality and an efficient algorithm for the solution of the discrete problem. This can be achieved by obtaining a sharp a posteriori error estimator, which depends on the given data in the problem and the numerical solution. For inverse problems such as those involving boundary identification, which suffer from stability issues, it is a nontrivial task, since they are ill-posed in the sense of Hadamard [8], i.e., any small change in the input data can lead to a drastic change in the solution; it is imperative to use regularization techniques for stabilizing the computations
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