This paper presents two new augmented flexible (AF)-Krylov subspace methods, AF-GMRES and AF-LSQR, to compute solutions of large-scale linear discrete ill-posed problems that can be modeled as the sum of two independent random variables, exhibiting smooth and sparse stochastic characteristics respectively. Following a Bayesian modelling approach, this corresponds to adding a covariance-weighted quadratic term and a sparsity enforcing $\ell_1$ term in the original least-squares minimization scheme. To handle the $\ell_1$ regularization term, the proposed approach constructs a sequence approximating quadratic problems that are partially solved using augmented flexible Krylov-Tikhonov methods. Compared to other traditional methods used to solve this minimization problem, such as those based on iteratively reweighted norm schemes, the new algorithms build a single (augmented, flexible) approximation (Krylov) subspace that encodes information about the different regularization terms through adaptable "preconditioning". The solution space is then expanded as soon as a new problem within the sequence is defined. This also allows for the regularization parameters to be chosen on-the-fly at each iteration. Compared to most recent work on generalized flexible Krylov methods, our methods offer theoretical assurance of convergence and a more stable numerical performance. The efficiency of the new methods is shown through a variety of experiments, including a synthetic image deblurring problem, a synthetic atmospheric transport problem, and fluorescence molecular tomography reconstructions using both synthetic and real-world experimental data.

翻译：本文提出两种新的增广柔性Krylov子空间方法——AF-GMRES和AF-LSQR，用于求解可建模为两个独立随机变量之和的大规模线性离散不适定问题，这两个变量分别呈现光滑和稀疏的随机特性。遵循贝叶斯建模方法，这相当于在原始最小二乘最小化框架中添加了一个协方差加权二次项和一个稀疏性约束的$\ell_1$项。为处理$\ell_1$正则化项，所提方法通过增广柔性Krylov-Tikhonov方法部分求解一系列逼近二次问题。与基于迭代重加权范数方案等传统方法相比，新算法构建了单一（增广、柔性）逼近（Krylov）子空间，通过自适应"预处理"编码不同正则化项的信息。随着序列中新问题的定义，解空间会同步扩展，同时允许在每次迭代过程中动态选择正则化参数。相较于近期关于广义柔性Krylov方法的研究，本文方法在理论上保证了收敛性并具有更稳定的数值表现。通过一系列实验验证了新方法的有效性，包括合成图像去模糊问题、合成大气传输问题以及基于合成数据和真实实验数据的荧光分子断层成像重建。