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 $x^TMx$, where $M$ is a positive semi-definite 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 $x^TMx$. Based on the pGKB process, we propose an iterative regularization algorithm via projecting the original problem onto small dimensional solution subspaces. We analyze regularization effect of this algorithm, including the incorporation of prior properties of the desired solution into the solution subspace and the semi-convergence behavior of regularized solution. To overcome instabilities caused by semi-convergence, 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.

翻译：Tikhonov正则化是求解反问题中广泛使用的技术，可对期望解施加先验属性。本文针对带有一般形式Tikhonov正则化项$x^TMx$（其中$M$为半正定矩阵）的线性反问题，提出一种基于Krylov子空间的迭代方法。我们设计了名为预处理Golub-Kahan双对角化（pGKB）的迭代过程，该过程隐式利用适当预处理算子生成一系列具有正则化器$x^TMx$所编码理想属性的解子空间。基于pGKB过程，我们通过将原问题投影到低维解子空间上，提出一种迭代正则化算法。我们分析该算法的正则化效应，包括将期望解的先验属性融入解子空间以及正则化解的半收敛行为。为克服半收敛引起的不稳定性，我们进一步提出两种基于pGKB的混合正则化算法。所有算法在小规模及大规模线性反问题上均进行测试。数值结果表明，这些迭代算法展现出优异性能，在某些情形下优于其他最先进算法。