For many applications involving a sequence of linear systems with slowly changing system matrices, subspace recycling, which exploits relationships among systems and reuses search space information, can achieve huge gains in iterations across the total number of linear system solves in the sequence. However, for general (i.e., non-identity) shifted systems with the shift value varying over a wide range, the properties of the linear systems vary widely as well, which makes recycling less effective. If such a sequence of systems is embedded in a nonlinear iteration, the problem is compounded, and special approaches are needed to use recycling effectively. In this paper, we develop new, more efficient, Krylov subspace recycling approaches for large-scale image reconstruction and restoration techniques that employ a nonlinear iteration to compute a suitable regularization matrix. For each new regularization matrix, we need to solve regularized linear systems, ${\bf A} + \gamma_\ell {\bf E}_k$, for a sequence of regularization parameters, $\gamma_\ell$, to find the optimally regularized solution that, in turn, will be used to update the regularization matrix. In this paper, we analyze system and solution characteristics to choose appropriate techniques to solve each system rapidly. Specifically, we use an inner-outer recycling approach with a larger, principal recycle space for each nonlinear step and smaller recycle spaces for each shift. We propose an efficient way to obtain good initial guesses from the principle recycle space and smaller shift-specific recycle spaces that lead to fast convergence. Our method is substantially reduces the total number of matrix-vector products that would arise in a naive approach. Our approach is more generally applicable to sequences of shifted systems where the matrices in the sum are positive semi-definite.

翻译：针对系统矩阵缓慢变化的线性系统序列问题，子空间回收技术通过挖掘系统间关联性并重用搜索空间信息，可显著减少整个序列中所有线性系统求解的迭代次数。然而对于一般性（非单位矩阵）且位移值大范围变化的移位移系统，不同线性系统的特性差异显著，导致回收效率降低。当此类系统序列嵌入非线性迭代时，问题更为复杂，需采用特殊策略实现有效回收。本文针对大规模图像重建与恢复技术，开发了新型高效Krylov子空间回收方法，该方法通过非线性迭代计算合适的正则化矩阵。对于每个新正则化矩阵，我们需要求解一系列正则化线性系统 ${\bf A} + \gamma_\ell {\bf E}_k$（对应不同正则化参数 $\gamma_\ell$），以获得最优正则化解进而更新正则化矩阵。本文通过分析系统与解的特性，选择适宜技术快速求解各系统：具体采用内外双回收策略，对每个非线性步骤构建较大的主回收空间，并对每个位移值构建较小的专用回收空间。我们提出高效方法，从主回收空间及位移专用回收空间中获取优质初始解，从而加速收敛。相较于朴素方法，本方法大幅减少了矩阵-向量乘积总量。该方案可推广至求和矩阵均为半正定的更一般性位移系统序列场景。