Accelerated Dirichlet–Robin alternating algorithms for solving the Cauchy problem for the Helmholtz equation

Abstract

The Cauchy problem for Helmholtz equation, for moderate wave number $k^{2}$, is considered. In the previous paper of Achieng et al. (2020, Analysis of Dirichlet–Robin iterations for solving the Cauchy problem for elliptic equations. Bull. Iran. Math. Soc.), a proof of convergence for the Dirichlet–Robin alternating algorithm was given for general elliptic operators of second order, provided that appropriate Robin parameters were used. Also, it has been noted that the rate of convergence for the alternating iterative algorithm is quite slow. Thus, we reformulate the Cauchy problem as an operator equation and implement iterative methods based on Krylov subspaces. The aim is to achieve faster convergence. In particular, we consider the Landweber method, the conjugate gradient method and the generalized minimal residual method. The numerical results show that all the methods work well. In this work, we discuss also how one can approach non-symmetric differential operators by using similar operator equations and model problems which are used for symmetric differential operators.

Regularization methods for solving Cauchy problems for elliptic and degenerate elliptic equations

“…The downside for the Landweber iteration is a slow rate of convergence, see [8,26]. Thus, apart from Landweber iteration, other accelerating algorithms can be used to solve the ill-posed problem (1.4), such as the Conjugate gradient method, see [2,4] .…”

“…where u 0 (f, η) solves problem (2.5), see [2]. Note that we use the symbol N for the operator in this paper instead of T , see Section 1.1.…”

Reconstruction of solutions of Cauchy problems for elliptic equations in bounded and unbounded domains using iterative regularization methods

“…We note that the conjugate gradient method converges for self-adjoint and positive definite operators. This method has been implemented to solve Cauchy problems for elliptic equations, see [6,8,35], among others.…”

“…Strategies for improving the convergence rate of the alternating iterative algorithms have also been developed and investigated, see [6,8], etc. Besides, in the presense of noisy data, a stopping rule must be added to achieve convergence of the alternating iterative method.…”

Accelerated Dirichlet-Robin Alternating Algorithm for Solving the Cauchy Problem for an Elliptic Equation using Krylov Subspaces

“…The operator K has to be modified. We also don't need the adjoint operator and we can pick any scalar product, for instance the L 2 scalar product, see [5].…”