Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations

Abstract

The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers $$k^2$$ k 2 , in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of $$k^2$$ k 2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

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

“…, where X = H −1/2 (Γ 1 ) and Y = H −1/2 (Γ 0 ), here we consider K * K, which maps from H −1/2 (Γ 1 ) to H −1/2 (Γ 1 ), and thus the normal equation can be solved using the Krylov subspace method. The adjoint operator is required here and we show in [1] that under certain positivity assumption we have an expression for K * .…”

“…It is proved in [1] that those iterations converges to the exact solution of (1.12) (for the data without noise).…”

Solving the Cauchy problem for the Helmholtz equation using cubic smoothing splines

“…For our work we avoid issues related to non-uniqueness of the solution by only considering small values of the wave-number, i.e. k 2 ≤ π 2 (1 + 1/a 2 ), see [1].…”

“…However the residual does not necessarily tend to zero as λ → 0. In our work we compute the function v(x, t) by a standard finite difference scheme, see [1,2] for details. The computed L-curve is displayed in Fig.…”

“…In [24], the problem was regularized using a modified Tikhonov method. We also refer to [1][2][3][4]19,26], and the references therein, for more regularization techniques that have been successfully applied to the problem.…”