Convergence Analysis of a Class of Regularization Methods with a Novel Discrete Scheme for Solving Inverse Problems

Abstract

Many inverse problems that appear in applications can be modeled as an operator equation. In practice, most of these problems are ill-posed, and computing solutions to such problems in an efficient manner is challenging and has been of greatest interest among researchers in the recent past. While many approaches are developed within infinite-dimensional Hilbert space settings, practical applications often require solutions in finite-dimensional spaces, and we need to discretize the problem. In this manuscript, we study a novel discretization scheme along with a class of regularization techniques for solving linear ill-posed problems and obtain the optimal order error estimates under an a priori parameter choice strategy. We illustrate the computational efficacy of the proposed scheme through numerical examples, and the results demonstrate that the proposed scheme is more economical due to the amount of discrete information needed to solve the problem is significantly lower than the traditional finite-dimensional approach.