Testing the integrity of some cavity – the Cauchy problem and the range test

For a number of applications testing the structural integrity of some cavity is of importance. A particular application we have in mind is the monitoring of the structural integrity of the fusion reactor ITER by electromagnetic waves, but the methods developed in this work can be applied to a collection of rather general settings.We use the solution of the Cauchy problem by potential methods and the range test to test the integrity of the boundary of some cavity using acoustic waves. The main idea of this approach is to test whether the scattered field can be analytically extended into the interior of some test domains and to calculate this extension. If the extension is possible, then we might reconstruct the field either by the inversion of Green's formula, a Green's approach incorporating the Dirichlet-to-Neumann map or a single-layer approach. If it is not the case, then the integral equations which arise from these approaches do not have solutions and we prove that in principle we can test this by observing the norm of the reconstruction density. As an alternative new approach to the range test we show that also the approximation error can be used as a discriminating criterion. We will show numerical results for the above cases, which provide a prove of concept to show the practicability of the method. For our application, the approximation error has turned out to be a more precise indicator for some singularity than the norm of the approximation density.