Stability Analysis and Efficient Algorithms for Nearly Singular Multidimensional Linear Inverse Problems

Multidimensional linear inverse problems pop up in various fields of science and engineering, from medical imaging to geophysical exploration. One of the main challenges is their inherent ill-posedness, which often shows up in practice as system matrices that are nearly singular or severely ill-conditioned. In these situations, direct solution methods can become highly sensitive to measurement noise and numerical fluctuations, resulting in unstable and physically meaningless reconstructions. This article delves into the stability analysis of these issues using condition numbers, singular value decomposition, and the Picard condition. It also discusses traditional regularization techniques, particularly Tikhonov regularization. We will take a closer look at how preconditioned iterative solvers can help tackle large-scale problems and provide a detailed MATLAB-based example on 2D image deblurring. This will illustrate how ill-conditioning manifests in real-world scenarios and how regularization and preconditioning work together to stabilize the solution. The interplay between robust regularization and advanced numerical methods is crucial for achieving stable, accurate, and computationally feasible solutions to large-scale, nearly singular inverse problems.