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

Abstract

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. (USSR Comput Math Math Phys 31 (1991), 45-52) applied to the Cauchy problem for the two-dimensional modified Helmholtz equation. The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is selected according to the generalized cross-validation criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the modified Helmholtz equation in two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.

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

“…Theorem 7. Let the exact solution of (3) is given in (7), the regularization solution v δ β is defined by (20), the error data ψ δ satisfies (21). We suppose that v satisfies…”

“…Theorem 8. Let the exact solution of (3) is given in (7), the regularization solution v δ β is defined in (20), the error data ϕ δ satisfies (21). We assume v satisfies the a priori bound (45), and the regularization parameter is chosen by an a posteriori rule (51), then we have…”

“…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.…”

Efficient numerical methods for boundary data and right‐hand side reconstructions in elliptic partial differential equations

“…The method of fundamental solutions is a powerful, accurate, easy to implement and low cost numerical meshless method and has been applied for solving a wide class of stationary and time dependent equations . To present a numerical study, we apply this method for solving IP1–IP3 and investigate its advantages or probable drawbacks in comparison with the Ritz–Galerkin method.…”

“…To construct the approximate solution for the unknown function defined over , we utilize the finite expansions in terms of fundamental solution of two‐dimensional Laplace equation . The approximate solution possesses the general form where are the source points settled out of the , and c j are the unknown parameters . The truncated series substantially satisfies Eq.…”