Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation

Abstract

a b s t r a c tIn this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green's formulation, the problem can be transformed into a moment problem. Then we propose a modified Tikhonov regularization algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method.

Subspaces of stability in the Cauchy Problem for the Helmholtz equation

“…It appears in several applications such as electromagnetics [24] or acoustics [6]. Computing approximate solutions by various regularizations have been suggested by several authors, e.g., by an initial value approach [22], backpropagation [21], frequency space cut-off [24], iterative methods [19] or Tikhonov regularization [16,27,20,28,23].…”

An alternating iterative procedure for the Cauchy problem for the Helmholtz equation

“…The infimum in (12) is attained for a function u µ ∈ H 1 (Ω). Since u µ L 2 (Ω) = 1, formula (12) shows that…”

“…The Cauchy problem is an ill-posed problem: its solution is unique but does not depend continuously on the Cauchy data; see [6][7][8][9]. A number of methods have been developed for solving the Cauchy problem for the Helmholtz equation such as the potential function method by Sun et al [10], the modified Tikhonov regularization and the truncation method by Qin et al [11,12], the modified regularization method based on the solution given by the method of separation of variables by Wei and Qin [13], the method of approximate solutions by Regińska and Regiński [14] and the alternating boundary element method by Marin et al [15]. For the latter, Marin et al [15] implemented the alternating iterative algorithm formulated in [17] for a purely imaginary number k and in that case, they solved the Cauchy problem for the modified Helmholtz equation, i.e., ∆u − k 2 u = 0.…”

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

“…Some spectral regularization methods and a modified Tikhonov regularization method to stabilize the Cauchy problem for the Helmholtz equation at fixed frequency were proposed by Xiong and Fu [22], while Jin and Marin [23] employed the plane wave method and the SVD to solve stably the same problem. Wei et al [24], Qin and Marin [25], and Qin et al [26] reduced the Cauchy problem associated with Helmholtz-type equations to a moment problem and also provided an error estimate and convergence analysis for the latter. Qin and Wei [27,28] proposed two regularization methods, namely a modified Tikhonov regularization method and a truncation method, for the stable approximate solution to the Cauchy problem for the Helmholtz equation and they also presented convergence and stability results under suitable choices of the regularization parameter.…”