AI 中文总结
针对移动边界达西流的对数渗透率反演，提出逐观测表示公式的变分框架，大幅降低计算成本，1D精度接近MCMC，2D效果优于大规模集合卡尔曼反演。
AI 中文摘要
受树脂传递模塑工艺的启发，我们开发了一种无穷维贝叶斯框架，用于从压力观测值中恢复单相移动边界达西流中空间变化的对数渗透率。主要贡献是一种逐观测的表示公式，将参数到观测映射的敏感性与移动边界状态和伴随系统关联起来。这产生了一种降阶的Levenberg--Marquardt方法，用于在高斯先验的Cameron--Martin空间上进行最大后验（MAP）估计，以及线性化MAP（LMAP）近似的显式有限秩协方差。一旦计算出表示函数，LM更新和协方差组装就简化为观测空间中的线性代数运算，与离散化维度无关。在1D情况下，显式移动边界解用于建立适定性和MAP估计量的存在性，并推导闭式Fréchet导数和表示函数。在2D情况下，我们引入了一种耦合压力、界面速度和域演化的极弱混合公式。形式化的单侧方向形状线性化产生了相应的线性化状态和伴随系统，通过独立的并行伴随问题计算逐观测的表示函数。数值实验表明，1D LMAP近似与参考MCMC后验非常吻合，但仅需要O(10^1)次前向模拟，而非O(10^5--10^6)次。在2D情况下，该方法生成的重构和不确定性估计与集成规模较大的集合卡尔曼反演相当，但计算成本显著更低。该工作流在多种几何形状和实验配置中得到验证，无需重新构建反演方法。
英文摘要
We develop an infinite-dimensional Bayesian framework for recovering spatially varying log-permeability from pressure observations in single-phase Darcy flow with a moving boundary, motivated by resin transfer moulding. The main contribution is an observation-wise representer formulation linking sensitivities of the parameter-to-observable map to the moving-boundary state and adjoint systems. This yields a reduced Levenberg--Marquardt method for maximum a posteriori (MAP) estimation on the Cameron--Martin space of a Gaussian prior and an explicit finite-rank covariance for the linearised MAP (LMAP) approximation. Once the representers are computed, the LM update and covariance assembly reduce to linear algebra in observation space, independently of the discretisation dimension. In 1D, the explicit moving-boundary solution is used to establish posterior well-posedness and existence of MAP estimators, and to derive closed-form Fréchet derivatives and representers. In 2D, we introduce a very weak mixed formulation coupling pressure, interface velocity and domain evolution. A formal one-sided directional shape linearisation yields the corresponding linearised state and adjoint systems, with observation-wise representers computed through independent, parallel adjoint problems. Numerical experiments show that the 1D LMAP approximation closely matches a reference MCMC posterior while requiring \(\mathcal{O}(10^1)\) rather than \(\mathcal{O}(10^5\text{--}10^6)\) forward simulations. In 2D, the method produces reconstructions and uncertainty estimates comparable to ensemble Kalman inversion with large ensembles, at substantially lower computational cost. The workflow is demonstrated across several geometries and experimental configurations without reformulating the inversion methodology.
