Stability and Fast Solvers for Ill-Conditioned Linear Inverse Problems via Regularization and Preconditioning
DOI:
https://doi.org/10.55578/jdso.2605.005Keywords:
Ill-conditioned inverse problems, Regularization methods, preconditioning techniques, Krylov subspace solvers, Numerical stability and efficiencyAbstract
Ill-conditioned linear inverse problems are pervasive in scientific computing and engineering, yet their inherent instability and slow iterative convergence remain major computational bottlenecks. To address these challenges, we propose a unified computational framework that couples Tikhonov regularization with preconditioned Krylov subspace solvers. By reformulating the inverse problem through regularized normal equations, the proposed approach systematically suppresses noise amplification while preserving essential solution features. To further accelerate convergence, we introduce problem-adapted preconditioners that cluster the spectrum of the regularized system, enabling preconditioned conjugate gradient (PCG) methods to converge in significantly less iteration. Theoretical insights into conditioning, spectral filtering, and preconditioner design are synthesized to demonstrate how regularization and preconditioning operate synergistically. A numerical study on a nearly rank-deficient system validates the framework, highlighting both numerical stability and rapid convergence. The results confirm that the integrated strategy offers a robust, scalable, and computationally efficient pathway for solving large-scale ill-conditioned inverse problems.
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No data were used to support the findings of this study. Therefore, data sharing is not applicable to this article.
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Copyright (c) 2026 Adane Akate Ayalew, Emőke Imre (Author)
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