Heuristic parameter choice rule for solving linear ill-posed integral equations in finite dimensional space

Abstract

A new heuristic parameter choice rule is proposed, which is an important process in solving the linear ill-posed integral equation. Based on multiscale Galerkin projection, we establish the error upper bound between the approximate solution obtained by this rule and the exact solution. Under certain conditions, we prove that the approximate solution obtained by this rule can reach the optimal convergence rate. Since the computational cost will be very large when the dimension of space increases, we analyze a special m-dimensional integral operator that can be transformed to m one-dimensional integral operator, which can reduce the computational cost greatly. Numerical experiments show that the proposed heuristic rule is promising among the known heuristic parameter choice rules.