Use of weak quasi-solutions of the Fredholm first-kind equation in problems with scarce data

摘要

This paper addresses the situation where the available data are insufficient to obtain a stable approximate solution. Instead solving the problem directly, it is proposed to convert the linear integral Fredholm first-kind equation to the equation for the anti-derivative of the unknown function. The Lp-compact of monotone bounded functions is the natural set of well-posedness for the modified equation. This transformation does not impose any a priori restriction on the class of solutions. In many applications a weak solution provides sufficient information on the unknown function. A numerical example is presented for application to the important problem in biology of reconstruction of age composition from size distribution data for a population.