Iterated Tikhonov Regularization via Flexible Arnoldi Reduction

Abstract

We propose the Iterated Flexible Arnoldi–Tikhonov (IFAT) method for large-scale discrete ill-posed problems. IFAT embeds nonstationary Tikhonov updates into a flexible Arnoldi reduction whose basis is enriched with problem-aware directions, enabling regularization in subspaces that standard Arnoldi and classical projected Tikhonov schemes may fail to capture. Building on the Arnoldi-based preconditioner of Buccini–Onisk Reichel and recent projected/iterated frameworks with adaptive parameter choice, we (i) unify flexible sub space enrichment with discrepancy-principle parameter selection on the reduced problem, (ii) derive a residual update that requires no additional matrix–vector products, and (iii) establish monotone error decrease and stability under inherited spectral–equivalence assumptions. Extensive experiments in image deblurring, signal reconstruction, tomography, MRI, seismic deconvolution, and electrical impedance tomography demonstrate that IFAT achieves consistently higher reconstruction quality (PSNR/SSIM, SNR/CNR, EPI) with approximately 40–60% fewer outer iterations than IAT and classical Tikhonov. Additional comparisons with Landweber and damped Gauss–Newton show that IFAT converges substantially faster than gradient-based schemes and is less sensitive to noise amplification than second-order approaches. Against the best published baselines from BOR (2023) and PIT–GF (2025), IFAT attains lower or comparable relative reconstruction error at the same discrepancy breakout. These results indicate that flexible augmentation provides a robust and practically significant improvement over fixed-subspace methods for real-world inverse problems.