Continuous approximation methods for the regularization and smoothing of integral transforms

Continuous approximation methods are described for obtaining a numerical solution ƒ(t) to an integral transform g(s) = ʃ K(s, t)ƒ(t)dt, where the given function g(s) may be affected by noise and where the problem may be ill-posed. The approximate solution is expressed in the linear form ƒ* = ∑_jϕ_j, where a_j are parameters and ɑ_j are certain basis functions; the values of a_j are determined by the minimization of a regularising/smoothing measure, which takes account of both the discrete 1₂ error in the integral transform and the continuous L₂ norm of ƒ* or of one of its derivatives. A Generalized Cross-Validation technique, based on the work of G. Wahba, is used for determining the smoothing parameter, and efficient algorithms are developed for three specific sets of basis functions {ϕ_j}, including a novel algorithm when {ϕ_j} are chosen to be a set of eigenfu net ions. Numerical examples are given to compare the merits of the various algorithms. In the case where the function g(s) is not affected by noise, the established "Method of Truncated Solutions" is adopted and an improved version of this method, based on B-splines, is described and then tested on numerical examples.