Adaptive anadromic regularization method for the Cauchy problem of the Helmholtz equation

Abstract

In this research work, we investigate the Cauchy problem for the Helmholtz equation. Considering the completion data problem in a bounded cylindrical domain with Neumann and Dirichlet conditions on a part of the boundary. An immediate approximation of missing boundary data is obtained using a method that factorizes the boundary value problem. This factorization uses the Neumann to Dirichlet (NtD) or Dirichlet to Neumann (DtN) operators that satisfy the Riccati equation. Some singularities appear in the solution of the Riccati equation for a particular length of the waveguide of the Helmholtz equation. We elaborate a new numerical method called "adaptive anadromic regularization method" that can solve these operators beyond the singularity. In addition, we introduce a scaling matrix technique to the linear matrix equations associated with the Riccati equations to generate normalized solutions. Our numerical procedure not only approximates missing boundary data, but also provides an error estimate that allows efficient time stepping. Numerical tests proved to confirm the theory even in the presence of high noise levels.