Indirect Abel Inversions by Power Series and SALS

Abstract

Derivative-free Abel inversions named here as 'Indirect' method is provedto be far superior to conventional 'Direct' method involving derivative term, by numerical experiments. With powerful least-square fitting method named SALS developed at computer center of Tokyo University, and simple power-series expansions of emission coefficient e(r) is sufficient without any iterations nor deconvolutions. 10-test functions often used in literatures are treated and the result of numerical experiments show reconstructions reached to the ultimate of computer precisions.

Comparison of Different Methods of Abel Inversion Using Computer Simulated and Experimental Side-On Data

“…Thus we looked for numerical methods which do not perform the calculations recurrently but in one single step (e.g. [10]). …”

“…Very many numerical methods have been developed and used for performing these calculations e.g. [2][3][4][5][6][7][8][9][10][11]. In Sect.…”

A new Abel inversion by means of the integrals of an input function with noise

“…To overcome this drawback, it is therefore most desirable to obtain an accurate, stable and fast method for computing Abel inversion, and to date many methods have been proposed for this purpose. One approach is to employ the curve fitting techniques such as the least-squares polynomial fit [8][9][10], which are able to affect the true curve due to the treatment of smoothing experimental data. Another approach is to expand the desired inverse with respect to a chosen basis including various orthogonal polynomials [11,12], and then determine coefficients by inserting the expansion into Abel's integral equations.…”

“…In this section, to examine the effectiveness of the suggested method, several test examples used frequently in plasma diagnostics to assess the performance of Abel inversion are given as follows Example 1 [10,27,29] g(r) =…”