Preconditioning the differential emission measure (T-e) inverse problem

In an inverse problem of any kind, poor conditioning of the inverse operato r decreases the numerical stability of any unregularized solution in the pr esence of data noise. In this paper we show that the numerical stability of the differential emission measure (DEM) inverse problem can be considerabl y improved by judicious choice of the integral operator. Specifically, we f ormulate a combinatorial optimization problem where, in a preconditioning s tep, a subset of spectral lines is selected in such a way as to minimize ex plicitly the condition number of the discretized integral operator. We tack le this large combinatorial optimization problem using a genetic algorithm. We apply this preconditioning technique to a synthetic data set comprising of solar UV/EUV emission lines in the SOHO SUMER/CDS wavelength range. Fol lowing which we test the same hypothesis on lines observed by the Harvard S -055 EUV spectroheliometer. On performing the inversion we see that the tem perature distribution in the emitting region of the solar atmosphere is rec overed with considerably better stability and smaller error bars when our p reconditioning technique is used, in both synthetic and "real" cases, even though this involves the analysis of fewer spectral lines than in the "All- lines" approach. The preconditioning step leads to regularized inversions t hat compare favorably to inversions by singular value decomposition, while providing greater flexibility in the incorporation of physically and/or obs ervationally based constraints in the line selection process.