用于数据一致反演的Copula变换
Copula Transformations for Data-Consistent Inversion
- University of Colorado Denver（科罗拉多大学丹佛分校）
- Colorado State University（科罗拉多州立大学）
- Aalto University（阿尔托大学）
- Sandia National Labs（桑迪亚国家实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究利用Copula理论建立iDCI与原始联合DCI解的关系，推导了Copula变换后的iDCI解并证明其可恢复原始DCI解，通过数值示例验证了相关策略的有效性。
AI中文摘要：
数据一致反演（DCI）构建概率测度，其推前分布与观测数据一致；迭代数据一致反演（iDCI）通过依次施加多个推前约束，将该框架扩展至广义随机反问题。尽管iDCI避免了对高维联合密度的直接近似，但其与原始联合DCI解的关系仍不明确。本研究通过Copula理论建立该关系，利用Sklar定理将DCI更新分解为独立的边际变换和相依变换，证明iDCI算法收敛后剩余的偏差完全由与观测及预测联合分布相关的Copula表征。该表征催生了Copula变换后的iDCI解，且证明精确的Copula变换可恢复原始DCI解。进一步在参考测度收敛序列和逐步丰富的可行集下，建立近似Copula变换的收敛结果。数值示例展示了感兴趣量映射诱导的几何如何决定Copula变换的重要性，说明了固定采样预算下提升计算精度的自适应参考测度细化策略，并通过异构异步获取的实验展示了广义随机反问题的逐步细化过程。
英文摘要：
Data-consistent inversion (DCI) constructs probability measures whose push-forward distributions agree with observed data, while iterative data-consistent inversion (iDCI) extends this framework to generalized stochastic inverse problems by enforcing multiple push-forward constraints sequentially. Although iDCI avoids the direct approximation of high-dimensional joint densities, its relationship to the original joint DCI solution has remained unclear. In this work, we establish this relationship through copula theory. Using Sklar's theorem, we derive a factorization of the DCI update into separate marginal and dependence transformations and show that the discrepancy remaining after convergence of the iDCI algorithm is entirely characterized by the copulas associated with the observed and predicted joint distributions. This characterization motivates a copula-transformed iDCI solution, and we prove that an exact copula transformation recovers the original DCI solution. We further establish convergence results for approximate copula transformations under converging sequences of reference measures and progressively enriched feasible sets. Numerical examples demonstrate how the geometry induced by the quantity-of-interest map governs the importance of the copula transformation, illustrate an adaptive reference-measure refinement strategy for improving computational accuracy under a fixed sampling budget, and demonstrate the progressive refinement of generalized stochastic inverse problems through heterogeneous, asynchronously acquired experiments.
