Statistical approach to the solution of first-kind integral equations arising in the study of materials and their properties

Abstract

There are a number of problems arising when studying the properties of materials, which require for their solution the inversion of a Fredholm first-kind integral equation. Examples include the determination of the distribution of adsorption energies on the surface of a solid and the evaluation of the distribution of pore radii of a solid from diffusion data. Such equations are, in practice, notoriously difficult to solve. This paper describes a general methodology for solving equations of this type. The method combines the ideas of regularization with a quadratic programming algorithm for minimizing quadratic expressions subject to non-negativity constraints. The condition of non-negativity is essential if we are to recover distribution functions for physical attributes of a solid. The method proposed is tested on simulated data for which the true solution to the equation is already known and on real data arising in each of the two situations described above. The method is shown to perform well in recovering the true solution for the simulated data and to produce results in the real data situations that are consistent with the data observed and with observations of related physical quantities.