Stochastic Convergence Analysis for Large-Scale Linear Discrete Ill-posed Problems

TL;DR

This paper provides a stochastic convergence analysis for large-scale linear ill-posed problems, deriving error bounds and adaptive strategies for regularization under different noise models.

Contribution

It introduces a new stochastic error analysis and adaptive parameter selection method for weighted Tikhonov regularization in large-scale ill-posed problems.

Findings

Derived stochastic error bounds for expectation and high-probability under specific noise models.

Proposed an a priori parameter-choice rule and an adaptive strategy for large-scale computation.

Numerical experiments confirm the effectiveness and near-optimality of the proposed methods.

Abstract

We study weighted Tikhonov regularization for large-scale linear discrete ill-posed problems with random noise. Under a polynomial upper-bound assumption on the generalized eigenvalues of the discrete forward operator, we derive stochastic error bounds for two noise models: expectation bounds for independent zero-mean bounded-variance noise, and high-probability bounds for independent sub-Gaussian noise. The analysis yields an a priori parameter-choice rule and suggests an adaptive strategy suitable for large-scale computation. Numerical experiments support the theory and show that the predicted parameter is nearly optimal and that the adaptive method is effective in practice.