耦合Fokker-Planck-Darcy系统的贝叶斯统计反问题

中文摘要 AI 辅助

我们研究了非参数统计反问题，即从Fokker-Planck解的离散噪声观测中恢复耦合Fokker-Planck-Darcy系统中的空间相关介电常数。我们考虑在有界域上的耦合抛物-椭圆系统，其中Fokker-Planck方程采用物理上自然的无通量边界条件，Darcy方程采用齐次Dirichlet边界条件。该反问题是间接的，因为未知系数仅通过椭圆方程进入，且仅通过其对抛物方程中密度的影响被观测到。我们首先发展了正向问题的分析理论，为统计分析所需，建立了适定性和在介电常数允许集上一致先验界。然后，我们推导了反问题的两个稳定性估计，即Lipschitz型正向估计和广义后向估计。在对数平移介电常数上放置重标度高斯过程先验，我们证明了后验以观测数量的显式多项式速率收缩到真值，后验均值以相同速率收敛。一维和二维的数值实验，使用预条件Crank-Nicolson和集合卡尔曼滤波算法，补充了我们的理论结果。

英文摘要

We study the nonparametric statistical inverse problem of recovering the space-dependent permittivity in a coupled Fokker-Planck-Darcy system from discrete, noisy observations of the Fokker-Planck solution. We consider the coupled parabolic-elliptic system on a bounded domain with the physically natural no-flux boundary condition for the Fokker-Planck equation and homogeneous Dirichlet boundary conditions for the Darcy equation. The inverse problem is indirect, since the unknown coefficient enters only through the elliptic equation and is observed only through its effect on the density in the parabolic equation. We first develop the analytical theory of the forward problem required for the statistical analysis, establishing well-posedness and a priori bounds uniform over the admissible set of permittivities. We then derive two stability estimates for the inverse problem, namely a Lipschitz-type forward estimate and a generalized backward estimate. Placing a rescaled Gaussian process prior on the log-shifted permittivity, we show that the posterior contracts around the truth at an explicit polynomial rate in the number of observations, with the posterior mean converging at the same rate. Numerical experiments in one and two dimensions, using preconditioned Crank-Nicolson and ensemble Kalman filter algorithms, complement our theoretical results.

发表机构

• Imperial College London（帝国理工学院）
• California Institute of Technology（加州理工学院）
• Scuola Normale Superiore（比萨高等师范学院）

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