Uncertainty Quantification for Landweber Iteration with Randomized Normal-Operator Approximation
随机正规算子近似下Landweber迭代的不确定性量化AI 解读 ↓
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随机正规算子近似下Landweber迭代的不确定性量化
- 研究问题
- 在线性病态逆问题中，如何同时量化观测噪声、正则化偏差与随机算子近似带来的重建不确定性，并降低计算成本？
- 方法与思路
- 作者分析噪声经Landweber重建映射的传播，结合正则化偏差界构造置信区间，并用随机奇异值分解近似正规算子。
- 材料陈述的贡献
- 摘要称方法将额外随机化误差纳入不确定性量化，数值实验显示在降低计算成本的同时保留准确重建与可靠估计。
- 阅读关注点
- 核对噪声传播、偏差界与随机近似误差的组合方式，以及置信区间覆盖依据、迭代停止方式和数值实验比较条件。
- 阅读边界
- 这是arXiv预印本，不能据此声称已同行评审或期刊接收；摘要未给出覆盖率条件、误差界细节及计算收益数值。
原文摘要
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.