A Regularization Method to Solve a Cauchy Problem for the Two-Dimensional Modified Helmholtz Equation

Abstract

In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the exact solution and approximation solution is given under suitable choices of the regularization parameter. Two numerical experiments show that our procedure is effective and stable with respect to perturbations in the data.

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

“…Problems (2) and 3are both the ill-posed problems, where a small disturbance on the given data can produce a considerable error in the solution [5][6][7], so some regularization techniques are required to overcome its ill-posedness and stabilize numerical computations, please see some regularized strategies in [8,9]. In the past years, we notice that many papers have researched the Cauchy problem of the modified Helmholtz equation and designed some meaningful regularization methods and numerical techniques, such as quasi-reversibility type method [10][11][12][13][14], filtering method [15], iterative method [16], mollification method [17,18], spectral method [19,20], alternating iterative algorithm [21,22], modified Tikhonov method [20,23], Fourier truncation method [12,24], novel trefftz method [25], weighted generalized Tikhonov method [26], and so on.…”

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.…”