Determination of the characteristic directions of lossless linear optical elements

We show that the problem of finding the primary and secondary characteristic directions of a linear lossless optical element can be reformulated in terms of an eigenvalue problem related to the unimodular factor of the transfer matrix of the optical device. This formulation makes any actual computation of the characteristic directions amenable to pre-implemented numerical routines, thereby facilitating the decomposition of the transfer matrix into equivalent linear retarders and rotators according to the related Poincare equivalence theorem. The method is expected to be useful whenever the inverse problem of reconstruction of the internal state of a transparent medium from optical data obtained by tomographical methods is an issue.Comment: Replaced with extended version as published in JM

Relaxation property for the adaptivity for ill-posed problems

Adaptive finite element method (adaptivity) is known to be an effective numerical tool for some ill-posed problems. The key advantage of the adaptivity is the image improvement with local mesh refinements. A rigorous proof of this property is the central part of this paper. In terms of coefficient inverse problems with single measurement data, the authors consider the adaptivity as the second stage of a two-stage numerical procedure. The first stage delivers a good approximation of the exact coefficient without an advanced knowledge of a small neighborhood of that coefficient. This is a necessary element for the adaptivity to start iterations from. Numerical results for the two-stage procedure are presented for both computationally simulated and experimental data