Two approximate methods of a Cauchy problem for the Helmholtz equation

Abstract

Abstract.In this paper, we consider a Cauchy problem for the Helmholtz equation at fixed frequency, especially we give the optimal error bound for the ill-posed problem. Within the framework of general regularization theory, we present some spectral regularization methods and a modified Tikhonov regularization method to stabilize the problem. Moreover, Hölder-type stability error estimates are proved for these regularization methods. According to the regularization theory, the error estimates are order optimal. Some numerical results are reported. 35R25, 35R30. Mathematical subject classification:

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].…”

Boundary element–minimal error method for the Cauchy problem associated with Helmholtz-type equations

“…Marin et al [31,32,34], is employed in order to discretise the integral equation (42). If the boundaries int and out are discretised into N int and N out constant boundary elements, respectively, such that N = N int + N out , then on applying the boundary integral equation (42) at each node/collocation point, we arrive at the following system of linear algebraic equations…”

“…The Cauchy problem for the Helmholtz equation by employing the Fourier transformation method has been addressed by Tumakov [40], whilst Marin et al [35] have applied the dual reciprocity BEM (DRBEM), along with the zeroth-order Tikhonov regularization method, to the Cauchy problem for Helmholtz-type equations with variable coefficients. Some spectral regularization methods and a modified Tikhonov regularization method to stabilize the Cauchy problem for the Helmholtz equation at fixed frequency have been proposed by Xiong and Fu [42], whilst Jin and Marin [15] have employed the plane wave method and the SVD to solve stably the same problem. Recently, Wei et al [41] and Qin et al [38,39] have 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.…”

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

“…The numerical solution for the Cauchy problem for two-and three-dimensional Helmholtz-type equations by employing the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method and SVD, was investigated by Marin and Lesnic [19] and Marin [20], and Jin and Zheng [21], respectively. 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.…”