An adaptive finite element method for Fredholm integral equations of the first kind and its verification on experimental data

Abstract

Abstract:We propose an adaptive finite element method for the solution of a linear Fredholm integral equation of the first kind. We derive a posteriori error estimates in the functional to be minimized and in the regularized solution to this functional, and formulate corresponding adaptive algorithms. To do this we specify nonlinear results obtained earlier for the case of a linear bounded operator. Numerical experiments justify the efficiency of our a posteriori estimates applied both to the computationally simulated and experimental backscattered data measured in microtomography. MSC:

Microwave thermometry with potential application in non-invasive monitoring of hyperthermia

“…For improvement of the solution m obtained via minimization of functional (2.4) using formula (2.7), we will apply an adaptive finite element method (AFEM) developed in [23] combined with a balancing principle for choosing the regularization parameter λ, for minimizing the functional…”

“…The regularization parameter can be chosen either by an a priori rule like Tikhonov's regularization strategy [1,2,20] or by a posteriori rules like the balancing principle [18] and Morozov's discrepancy principle [26]. For image quality restoration, an adaptive finite element method, developed in [23], is combined in this study with a balancing principle for optimal choice of the regularization parameter. The main idea of an adaptive finite element method is to obtain better image quality via local mesh refinements through minimization of Tikhonov's functional.…”

An overview of numerical methods for the first kind Fredholm integral equation

“…Combining with the stable linear algebraic equations, the accuracy of the approximate solution is improved by using the improved iteration method. Koshev and Beilina [89] proposed an adaptive finite element method to solve the first kind of linear Fredholm integral equation. The minimum posterior error estimates and regularization solutions are obtained in the functional, and the corresponding adaptive algorithm is given.…”

A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind

“…Numerical methods for obtaining a reasonable approximate solution to the Fredholm integral equation of the first kind have attracted many researchers, and many research results have been achieved; see, for instance, ( [2], Chapter 12), and [3][4][5][6][7]. Due to the ill-posedness nature of the problem, numerical solutions are extremely sensitive to perturbations caused by observation and rounding errors.…”