Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum

Abstract

We investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the result of conditional stability under an a priori assumption for an exact solution. A generalized Tikhonov method is proposed to solve this problem, we select the regularization parameter by a priori and a posteriori rules and derive the convergence results of sharp type for this method. The corresponding numerical experiments are implemented to verify that our regularization method is practicable and satisfied.

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

“…Notably, Callé et al [18] innovatively combined the MFS with fading regularization to stabilize the numerical solution for the two-dimensional Helmholtz equation. However, the landscape is rich with varied approaches to this ill-posed inverse problem: Qin and Wei's quasi-reversibility method [34], mollification regularization using the de la Vallée Poussin kernel [14], the truncation method [42], and the generalized Tikhonov method [43] deserve mention.…”

Correction: Zhang, H.; Zhang, X. Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum. Mathematics 2019, 7, 667

“…The authors wish to make the following corrections to this paper [1]: 1. In the original paper, the value of the parameter γ is greater than 0.…”