Solving large-scale Bayesian inverse problems presents significant challenges, particularly when the exact (discretized) forward operator is unavailable. These challenges often arise in image processing tasks due to unknown defects in the forward process that may result in varying degrees of inexactness in the forward model. Moreover, for many large-scale problems, computing the square root or inverse of the prior covariance matrix is infeasible such as when the covariance kernel is defined on irregular grids or is accessible only through matrix-vector products. This paper introduces an efficient approach by developing an inexact generalized Golub-Kahan decomposition that can incorporate varying degrees of inexactness in the forward model to solve large-scale generalized Tikhonov regularized problems. Further, a hybrid iterative projection scheme is developed to automatically select Tikhonov regularization parameters. Numerical experiments on simulated tomography reconstructions demonstrate the stability and effectiveness of this novel hybrid approach.

翻译：求解大规模贝叶斯反问题面临显著挑战，尤其是在精确（离散化）前向算子不可得的情况下。这些挑战常出现在图像处理任务中，源于前向过程中的未知缺陷可能导致前向模型存在不同程度的不精确性。此外，对于许多大规模问题，计算先验协方差矩阵的平方根或逆矩阵并不可行，例如当协方差核定义在不规则网格上或仅能通过矩阵-向量乘积访问时。本文提出一种高效方法，通过构建一种能融合前向模型不同程度不精确性的不精确广义Golub-Kahan分解，以求解大规模广义Tikhonov正则化问题。进一步，我们发展了一种混合迭代投影方案来自动选择Tikhonov正则化参数。模拟断层扫描重建的数值实验验证了该新型混合方法的稳定性与有效性。