Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation

Abstract

The Cauchy problem for the Helmholtz equation appears in various applications. The problem is severely ill-posed and regualrization is needed to obtain accurate solutions. We start from a formulation of this problem as an operator equation on the boundary of the domain and consider the equation in (H 1/2) * spaces. By introducing an artificial boundary in the interior of the domain we obtain an inner product for this Hilbert space in terms of a quadratic form associated with the Helmholtz equation; perturbed by an integral over the artificial boundary. The perturbation guarantees positivity property of the quadratic form. This inner product allows an efficient evaluation of the adjoint operator in terms of solution of a well-posed boundary value problem for the Helmholtz equation with transmission boundary conditions on the artificial boundary. In an earlier paper we showed how to take advantage of this framework to implement the conjugate gradient method for solving the Cauchy problem. In this work we instead use the Conjugate gradient method for minimizing a Tikhonov functional. The added penalty term regularize the problem and gives us a regularization parameter that can be used to easily control the stability of the numerical solution with respect to measurement errors in the data. Numerical tests show that the proposed algorithm works well.

Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations

“…For future work, we will investigate how to improve the rate of convergence for very large values of μ 0 and μ 1 using methods such as the conjugate gradient method or the generalized minimal residual method. We will also investigate implementing Tikhonov regularization based on the Dirichlet-Robin alternating procedure, see [5]. Also a stopping rule for inexact data will be developed.…”

“…On the other hand, the alternating iterative algorithm, in its basic form, suffers from slow convergence, see [4], and in the presence of noise additional regularization techniques have to be implemented, see, e.g. [5]. Thus, a practically useful form of the alternating algorithm tends to be more complicated than the variant analyzed in this paper.…”

Accelerated Dirichlet-Robin Alternating Algorithm for Solving the Cauchy Problem for an Elliptic Equation using Krylov Subspaces

“…The rate of convergence were quite slow. Therefore, acceleration methods were considered [6,8]. Furthermore, it was also shown that the Dirichlet-Robin algorithm can be represented as a Landweber iteration for certain operator equation defined on functions on the boundary of the domain.…”

On numerical resolution of an inverse Cauchy problem modeling the airflow in the bronchial tree

“…Then many research works have been conducted to propose different regularization approaches, we can not list all of them here. The most popular one is Tikhonov's regularization method (Tikhonov et al 1977;Feng et al 2011;Cimetiere et al 2001;Cheng et al 2016;Berntsson et al 2017), which transforms the original ill-posed problem into a well posed one by minimizing the L 2 -norm of the solution subjected to constraint equations. Other methods have been proposed to regularize the Cauchy problem, we can mention for instance the alternating method (Berntsson et al 2014;Kozlov et al 1991;Jourhmane and Nachaoui 1999), the universal regularization method (Klibanov 2015(Klibanov , 2006(Klibanov , 2013, the technic of fundamental solution (Chen et al 2019;Fairweather and Karageorghis 1998;Marin and Cipu 2017;Yeih et al 2015), the fixed point approach based on domain decomposition like methods (Chakib et al 2018).…”