Iterative algorithms for processing experimental data
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MA DIAGNOSTICS
Iterative Algorithms for Processing Experimental Data K. K. Tretiak* Institute of Plasma Physics, National Science Center Kharkiv Institute of Physics and Technology, Akademichna vul. 1, Kharkiv, 61108 Ukraine *e-mail: [email protected] Received July 30, 2015; in final form, February 2, 2016
Abstract—The need to solve linear and nonlinear integral equations arise, e.g., in recovering plasma parameters from the data of multichannel diagnostics. The paper presents an iterative method for solving integral equations with a singularity at the upper limit of integration. The method consists in constructing successive approximations and calculating the integral by quadrature formulas in each integration interval. An example of application of the iterative algorithm to numerically solve an integral equation similar to those arising in recovering the plasma density profile from reflectometry data is presented. DOI: 10.1134/S1063780X16100081
1. INTRODUCTION Mathematical modeling of physical processes is an important tool of scientific, engineering, and technological research [1]. By means of numerical modeling, physical processes described by sophisticated mathematical models can be analyzed qualitatively and, frequently, quantitatively. Mathematical modeling plays an important role when comparing theoretical and experimental results. Various numerical methods are used to process and analyze experimental data. Both linear and nonlinear integral equations occur in solving inverse problems that arise when the plasma parameters are impossible or difficult to measure directly. For example, the Abel integral equation has received wide application in problems of plasma diagnostics, astrophysics, and optical diffraction [2, 3]. The Abel equation belongs to the class of ill-posed problems of mathematical physics: the experimental data are measured with a certain error; hence, when solving the problem, the equation turns out to be satisfied for a wide class of functions within measurement errors. Let us consider some of the existing numerical methods for solving the Abel equation. It is worth mentioning one of the simplest and most illustrative methods, which was proposed by Pearce [4] for the cylindrically symmetric case. The Pearce method is based on dividing the cross section of the plasma configuration into ring zones and approximating the distribution of the sought-for functions in the Abel integral equation by a stepwise function. In this case, the problem of recovering the radial profile of the plasma density is reduced to solving a system of linear equations of the form An = g , where n is the vector of the densities in the steps, the vector g is the set of the lin-
ear densities measured along different chords, and A is the matrix of linear coefficients. In [4], the coefficients were calculated for the number of zones j = 25. In this method, the matrix A is ill posed; as a result, for a fixed error in measuring the linear density, the error of density recovering increases without bound as the number
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