We address functional uncertainty quantification for ill-posed inverse problems where it is possible to evaluate a possibly rank-deficient forward model, the observation noise distribution is known, and there are known parameter constraints. We present four constraint-aware confidence intervals extending the work of Batlle et al. (2023) by making the intervals both computationally feasible and less conservative. Our approach first shrinks the potentially unbounded constraint set compact in a data-adaptive way, obtains samples of the relevant test statistic inside this set to estimate a quantile function, and then uses these computed quantities to produce the intervals. Our data-adaptive bounding approach is based on the approach by Berger and Boos (1994), and involves defining a subset of the constraint set where the true parameter exists with high probability. This probabilistic guarantee is then incorporated into the final coverage guarantee in the form of an uncertainty budget. We then propose custom sampling algorithms to efficiently sample from this subset, even when the parameter space is high-dimensional. Optimization-based interval methods formulate confidence interval computation as two endpoint optimizations, where the optimization constraints can be set to achieve different types of interval calibration while seamlessly incorporating parameter constraints. However, choosing valid optimization constraints has been elusive. We show that all four proposed intervals achieve nominal coverage for a particular functional both theoretically and in practice, with numerical examples demonstrating superior performance of our intervals over the OSB interval in terms of both coverage and expected length. In particular, we show the superior performance in a realistic unfolding simulation from high-energy physics that is severely ill-posed and involves a rank-deficient forward model.

翻译：本文针对病态逆问题中的泛函不确定性量化展开研究，其中前向模型（可能秩亏）可被评估，观测噪声分布已知，且参数约束条件明确。我们在Batlle等人（2023）研究基础上，提出了四种约束感知的置信区间构造方法，通过使区间计算在计算上可行且降低保守性来扩展原有工作。我们的方法首先以数据自适应方式将可能无界的约束集压缩为紧集，在该集合内对相关检验统计量进行采样以估计分位数函数，进而利用这些计算量生成置信区间。数据自适应定界方法基于Berger和Boos（1994）提出的思路，通过定义约束集的一个子集（真实参数以高概率存在于该子集内）实现。这一概率保证随后以不确定性预算的形式整合到最终的覆盖度保证中。我们进一步提出了定制化采样算法，即使参数空间为高维情形，也能高效地从该子集中采样。基于优化的区间方法将置信区间计算建模为两个端点优化问题，其中优化约束的设置可实现不同类型的区间校准，并能无缝整合参数约束。然而，如何选择有效的优化约束始终是未解难题。我们从理论和实践两个层面证明，所提出的四种区间对特定泛函均能达到名义覆盖度，数值算例显示我们的区间在覆盖度和期望长度方面均优于OSB区间。特别地，我们在高能物理领域一个严重病态且涉及秩亏前向模型的现实性反卷积模拟中，展示了所提方法的优越性能。