A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data

In this paper the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data is considered. This problem is severely ill-posed, the solution does not depend continuously on the data. An approximate method based on the a posteriori Fourier regularization in the frequency space is analyzed. Some crucial information about the regularization parameter hidden in the a posteriori choice rule are found, and some sharp error estimates between the exact solution and its regularization approximate solution are proved. Numerical examples show the effectiveness of the method. A comparison of numerical effect between the a posteriori and the a priori Fourier method is also taken into account.