Solving Ill-Posed Cauchy Problems by a Krylov Subspace Method

We study the numerical solution of a Cauchy problem for a self-adjoint elliptic partial differential equation $u_{zz}-Lu=0$ in three space dimensions $(x,y,z)$ , where the domain is cylindrical in $z$. Cauchy data are given on the lower boundary and the boundary values on the upper boundary are sought. The problem is severely ill-posed. The formal solution is written as a hyperbolic cosine function in terms of the two-dimensional elliptic operator $L$ (via its eigenfunction expansion), and it is shown that the solution is stabilized (regularized) if the large eigenvalues are cut off. We suggest a numerical procedure based on the rational Krylov method, where the solution is projected onto a subspace generated using the operator $L^{-1}$. This means that in each Krylov step a well-posed two-dimensional elliptic problem involving $L$ is solved. Furthermore, the hyperbolic cosine is evaluated explicitly only for a small symmetric matrix. A stopping criterion for the Krylov recursion is suggested based on the relative change of an approximate residual, which can be computed very cheaply. Two numerical examples are given that demonstrate the accuracy of the method and the efficiency of the stopping criterion.