Abstract

       It is in the problematic of solving inverse problems governed by Helmholtz type equations that this thesis takes place. The objective is  to compute the solution of the Cauchy problem for the Helmholtz equation without constraint on the wave number.

       One of the most widespread approaches in the simulation of Cauchy problems is the Kozlov-Maz’iya-Fomin (KMF) algorithm. It is an approach where the solution is sought as the limit of a sequence of solutions to well-posed problems. It has been widely used in solving this kind of  problems for elliptic equations thanks to the simplicity of its implementation and its regularizing character. But several researchers have pointed out that this method is not applicable to certain Cauchy problems governed by the Helmholtz equation. Recently, Kozlov, one of the creators of this technique, as well as co-authors showed that this method converges only if the wave number is lower than a certain value which depends on the domain. It is mainly this difficulty that we  circumvented in his thesis, we proposed iterative algorithms with a regularizing character, easy to execute, efficient and whose convergence does not deteriorate when the wave number becomes large.

       The existence of interval of convergence and interval of acceleration is proved. The algorithms are approached using finite element method. The obtained accurate numerical results  ensure theoretical results and proves their effectiveness.