An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation

Abstract

<p style='text-indent:20px;'>Data completion known as Cauchy problem is one most investigated inverse problems. In this work we consider a Cauchy problem associated with Helmholtz equation. Our concerned is the convergence of the well-known alternating iterative method [<xref ref-type="bibr" rid="b25">25</xref>]. Our main result is to restore the convergence for the classical iterative algorithm (KMF) when the wave numbers are considerable. This is achieved by, some simple modification for the Neumann condition on the under-specified boundary and replacement by relaxed Neumann ones. Moreover, for the small wave number <inline-formula><tex-math id="M111111">\begin{document}$ k $\end{document}</tex-math></inline-formula>, when the convergence of KMF algorithm's [<xref ref-type="bibr" rid="b25">25</xref>] is ensured, our algorithm can be used as an acceleration of convergence.</p><p style='text-indent:20px;'>In this case, we present theoretical results of the convergence of this relaxed algorithm. Meanwhile it, we can deduce the convergence intervals related to the relaxation parameters in different situations. In contrast to the existing results, the proposed algorithm is simple to implement converges for all choice of wave number.</p><p style='text-indent:20px;'>We approach our algorithm using finite element method to obtain an accurate numerical results, to affirm theoretical results and to prove it's effectiveness.</p>