TY - JOUR

T1 - Continuous approximation methods for the regularization and smoothing of integral transforms

AU - Bennell, R. P.

AU - Mason, J. C.

PY - 1989

Y1 - 1989

N2 - 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* = ∑jϕj, where aj are parameters and ɑj are certain basis functions; the values of aj 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 {ϕj}, including a novel algorithm when {ϕj} 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.

AB - 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* = ∑jϕj, where aj are parameters and ɑj are certain basis functions; the values of aj 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 {ϕj}, including a novel algorithm when {ϕj} 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.

UR - http://www.scopus.com/inward/record.url?scp=84881051214&partnerID=8YFLogxK

U2 - 10.1216/RMJ-1989-19-1-51

DO - 10.1216/RMJ-1989-19-1-51

M3 - Article

AN - SCOPUS:84881051214

SN - 0035-7596

VL - 19

SP - 51

EP - 66

JO - Rocky Mountain Journal of Mathematics

JF - Rocky Mountain Journal of Mathematics

IS - 1

ER -