Limitations of Regularization Methods for the Reconstruction of Electron Velocity Distribution Function

The extraction of electron energy distribution functions (EEDFs) from Langmuir probe data is a discrete ill-posed problem. This problem rises due to the integral relationship between electron current and the probe voltage known as the Druyvesteyn relation. There have been a number of methods for the solution of this ill-posed problem ranging from data smoothing to a priori solution conditioning. Such methods include truncated singular value decomposition, truncated generalized singular value decomposition, and various regularization techniques. When these methods are extended to solve for similar integral relationships between electron current and electron distributions, complications arise due to their slightly different integral characteristics. For example, the electron velocity distribution function (EVDF) presents a similar ill-posed integral relationship. However, the EVDF integral presents an additional complication of rank deficiency that can` make accurate solutions of the inverse problem extremely challenging. In this paper, the ill-posed and rank deficiency problems of EEDF and EVDF reconstructions, respectively, are compared to highlight these challenges.