Preconditioned iterative methods for the numerical solution of Fredholm equations of the second kind

Abstract

In this paper, we consider the numerical solution of Fredholm integral equations of the second kind: Discretizing the integral equation by a certain quadrature rule, we get the linear system where I is the identity matrix, A is the discretization matrix corresponding to the kernel function a(x,t), and W is a diagonal matrix which depends on the quadrature rule. We propose an approximation scheme based on the polynomial interpolation technique and use the scheme to compute approximation matrices A a of A and matrices B a such that (I+B a W)(I-A a W) ≈ I for sufficiently large N, where N is the number of quadrature points used in the discretization. The approximations A a and B a , and the matrix-vector multiplications and , are obtained in O(N) operations by using the approximation scheme. Hence preconditioned iterative methods such as the preconditioned conjugate gradient method and the residual correction scheme are well suited for the solution of the preconditioned system