Stochastic Convergence Analysis for Large-Scale Linear Discrete Ill-posed Problems
For researchers in inverse problems and regularization, this work offers theoretical guarantees and practical parameter selection for large-scale ill-posed problems with random noise.
The paper provides stochastic convergence analysis for weighted Tikhonov regularization in large-scale linear discrete ill-posed problems, deriving error bounds and an a priori parameter-choice rule. Numerical experiments show the predicted parameter is nearly optimal and the adaptive method is effective.
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.