A Preconditioned Krylov Subspace Method for Linear Inverse Problems with General-Form Tikhonov Regularization

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2607-A2633, August 2024.
Abstract. Tikhonov regularization is a widely used technique in solving inverse problems that can enforce prior properties on the desired solution. In this paper, we propose a Krylov subspace based iterative method for solving linear inverse problems with general-form Tikhonov regularization term [math], where [math] is a positive semidefinite matrix. An iterative process called the preconditioned Golub–Kahan bidiagonalization (pGKB) is designed, which implicitly utilizes a proper preconditioner to generate a series of solution subspaces with desirable properties encoded by the regularizer [math]. Based on the pGKB process, we propose an iterative regularization algorithm via projecting the original problem onto small dimensional solution subspaces. We analyze the regularization properties of this algorithm, including the incorporation of prior properties of the desired solution into the solution subspace and the semiconvergence behavior of the regularized solution. To overcome instabilities caused by semiconvergence, we further propose two pGKB based hybrid regularization algorithms. All the proposed algorithms are tested on both small-scale and large-scale linear inverse problems. Numerical results demonstrate that these iterative algorithms exhibit excellent performance, outperforming other state-of-the-art algorithms in some cases.