发表机构
University of Manchester; Case Western Reserve University（曼彻斯特大学; 凯斯西储大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对线性逆问题中的Landweber迭代，结合随机奇异值分解的低秩近似与广义fiducial推断，提出一种量化重建不确定性的方法，在降低计算成本的同时提供准确重建和可靠置信区间。
AI 中文摘要
迭代方法在求解线性不适定逆问题中极为流行。尽管此类方法的确定性收敛（更精确地说，半收敛）已被广泛研究，但由于观测数据中的随机噪声导致的重建结果不确定性量化问题却受到的关注较少。在本工作中，我们以Radon变换作为激励性示例，研究线性逆问题中Landweber迭代的不确定性量化。我们通过分析测量数据中的不确定性如何通过重建映射传播，对Landweber重建进行统计解释。这一视角与广义fiducial推断密切相关，其中关于参数的不确定性是通过反转观测数据与未知（固定）量之间的关系而引入的。我们将由此产生的随机不确定性与正则化偏差的理论界相结合，以构建重建解的置信区间。评估所得不确定性估计需要对正算子进行重复运算，这在大规模问题中可能变得昂贵。为降低这一成本，我们使用随机奇异值分解获得与Radon变换相关的正规算子的低秩近似，并将该近似纳入不确定性量化中。特别是，这要求将由于使用此类随机技术而引入的额外随机化误差包含在我们的不确定性量化方法中。数值结果表明，所提出的方法在显著降低计算成本的同时，提供了准确的重建和可靠的不确定性估计。
英文摘要
Iterative methods are extremely popular for solving linear ill-posed inverse problems. While deterministic convergence, or more precisely, semi-convergence of such methods is widely studied, the problem of quantifying uncertainty in the resulting reconstructions due to random noise in the observed data has received far less attention. In this work, we study uncertainty quantification for the Landweber iteration in linear inverse problems using the Radon transform as a motivating example. We interpret the Landweber reconstruction statistically by analyzing how uncertainty in the measured data is propagated through the reconstruction map. This perspective is closely related to generalized fiducial inference, where uncertainty about the parameter is induced by inverting the relation between the observed data and the unknown (fixed) quantity. We combine the resulting stochastic uncertainty with a theoretical bound on the regularization bias to construct confidence intervals for the reconstructed solution. The evaluation of the resulting uncertainty estimates require repeated operations with the forward operator that can become expensive in large-scale problems. To reduce this cost, we use randomized singular value decomposition to obtain a low-rank approximation of the normal operator associated with the Radon transform and incorporate this approximation in quantifying uncertainty. {In particular, this requires that the additional randomization error introduced due to the use of such randomized techniques be included in our approach for uncertainty quantification.} Numerical results show that the proposed method provides accurate reconstructions and reliable uncertainty estimates at substantially reduced computational cost.
