Scalable Iterative Data-Adaptive RKHS Regularization

Abstract.

We present iDARR, a scalable iterative data-adaptive RKHS regularization method for solving ill-posed linear inverse problems. This method searches for solutions in subspaces where the true solution can be identified, with the data-adaptive reproducing kernel Hilbert space (RKHS) penalizing the spaces of small singular values. At the core of the method is a new generalized Golub–Kahan bidiagonalization procedure that recursively constructs orthonormal bases for a sequence of RKHS-restricted Krylov subspaces. The method is scalable, with a complexity of [Formula: see text] for [Formula: see text]-by-[Formula: see text] matrices, where [Formula: see text] denotes the number of iterations. Numerical tests on the Fredholm integral equation and two-dimensional image deblurring demonstrate that it outperforms the widely used [Formula: see text] and [Formula: see text] norms, consistently producing stable and accurate solutions that converge when the noise level decreases.

Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/feilumath/iDARR and in the supplementary materials ( iDARR-main.zip [1.28MB]). [Formula: see text]