The Bayesian statistical framework provides a systematic approach to enhance the regularization model by incorporating prior information about the desired solution. For the Bayesian linear inverse problems with Gaussian noise and Gaussian prior, we propose a new iterative regularization algorithm that belongs to subspace projection regularization (SPR) methods. By treating the forward model matrix as a linear operator between the two underlying finite dimensional Hilbert spaces with new introduced inner products, we first introduce an iterative process that can generate a series of valid solution subspaces. The SPR method then projects the original problem onto these solution subspaces to get a series of low dimensional linear least squares problems, where an efficient procedure is developed to update the solutions of them to approximate the desired solution of the original problem. With the new designed early stopping rules, this iterative algorithm can obtain a regularized solution with a satisfied accuracy. Several theoretical results about the algorithm are established to reveal the regularization properties of it. We use both small-scale and large-scale inverse problems to test the proposed algorithm and demonstrate its robustness and efficiency. The most computationally intensive operations in the proposed algorithm only involve matrix-vector products, making it highly efficient for large-scale problems.

翻译：贝叶斯统计框架通过引入关于期望解的先验信息，提供了一种系统性的方法来增强正则化模型。针对高斯噪声和高斯先验下的贝叶斯线性逆问题，我们提出了一种新的迭代正则化算法，该算法属于子空间投影正则化（SPR）方法。通过将前向模型矩阵视为两个有限维希尔伯特空间之间的线性算子（引入新的内积），我们首先引入一个能生成一系列有效解子空间的迭代过程。随后，SPR方法将原始问题投影至这些解子空间上，得到一系列低维线性最小二乘问题，并开发了一种高效更新这些解的过程，以逼近原始问题的期望解。借助新设计的早停准则，该迭代算法能够获得具有满意精度的正则化解。我们建立了该算法的若干理论结果，揭示了其正则化性质。通过小规模和大规模逆问题对算法进行测试，验证了其鲁棒性和高效性。该算法中计算最密集的操作仅涉及矩阵-向量乘积，使其在处理大规模问题时极具效率。