The regularizing properties of multistep methods for first kind Volterra integral equations with smooth kernels

TL;DR

This paper investigates how multistep quadrature methods can regularize solutions to first kind Volterra integral equations with smooth kernels, especially under noisy data, analyzing step size choices and adaptive strategies.

Contribution

It introduces an analysis of regularizing properties of multistep methods with a focus on step size selection and adaptive balancing principles for noisy integral equations.

Findings

Regularization effectiveness depends on step size choice.

Adaptive balancing principle can reduce computational effort.

Numerical experiments confirm theoretical insights.

Abstract

We study quadrature methods for solving Volterra integral equations of the first kind with smooth kernels under the presence of noise in the right-hand sides, with the quadrature methods being generated by linear multistep methods. The regularizing properties of an a priori choice of the step size are analyzed, with the smoothness of the involved functions carefully taken into consideration. The balancing principle as an adaptive choice of the step size is also studied. It is considered in a version which sometimes requires less amount of computational work than the standard version of this principle. Numerical results are included.