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Solution of the Cauchy problem for the wave equation using iterative regularization.(English)Zbl 07484734

Summary: We propose a regularization method based on the iterative conjugate gradient method for the solution of a Cauchy problem for the wave equation in one dimension. This linear but ill-posed Cauchy problem consists of finding the displacement and flux on a hostile and inaccessible part of the medium boundary from Cauchy data measurements of the same quantities on the remaining friendly and accessible part of the boundary. This inverse boundary value problem is recast as a least-squares minimization problem that is solved by using the conjugate gradient method whose iterations are stopped according to the discrepancy principle for obtaining stable reconstructions. The objective functional associated is proved Fréchet differentiable and a formula for its gradient is derived. The well-posed direct, adjoint and sensitivity problems present in the conjugate gradient method are solved by using a finite-difference method. Two numerical examples to illustrate the accuracy and stability of the proposed numerical procedure are thoroughly presented and discussed.

MSC:

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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