The regularizing properties of multistep methods for first kind Volterra integral equations with smooth kernels
This work provides theoretical and practical guidance for choosing step sizes in numerical solutions of ill-posed Volterra equations, benefiting researchers in inverse problems and numerical analysis.
The paper analyzes the regularizing properties of quadrature methods generated by linear multistep methods for solving first-kind Volterra integral equations with smooth kernels under noisy data, proposing an adaptive step-size selection via the balancing principle that reduces computational cost. Numerical results demonstrate the effectiveness of the approach.
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.