An Adaptive Finite Element Method for Fredholm Integral Equations of the First Kind and its Verification on Experimental Data

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 of this functional, and formulate corresponding adaptive algorithm. To do it we specify nonlinear results obtained in [6, 7, 8, 9, 18] for the case of the 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. 1. Introduction. In this work we consider a problem of the solution of a Fredholm integral equation of the first kind and propose an adaptive finite element method to solve it. Such problems arise in many applications of astrophysics [13], astronomy [3], image processing of smeared and defocused photography [19