Weighted Conjugate Gradient-Type Methods for Solving Quadrature Discretization of Fredholm Integral Equations of the First Kind

Abstract

A variant of conjugate gradient-type methods, called weighted conjugate gradient (WCG), is given to solve quadrature discretization of various first-kind Fredholm integral equations with continuous kernels. The WCG-type methods use a new inner product instead of the Euclidean one arising from discretization of $$L^2$$ -inner product by the quadrature formula. On this basis, the proposed algorithms generate a sequence of vectors which are approximations of solution at the quadrature points. Numerical experiments on a few model problems are used to illustrate the performance of the new methods compared to the CG-type methods.