Bayesian statistical inverse problems for a coupled Fokker-Planck-Darcy system
Abstract
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.
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BibTeX
@misc{pavliotis2026bayesian,
title = {{Bayesian statistical inverse problems for a coupled Fokker-Planck-Darcy system}},
author = {Grigorios A. Pavliotis and Andrew M. Stuart and Andrea Zanoni},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28242},
eprint = {2609.28242},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28242},
}
RIS
TY - UNPB
TI - Bayesian statistical inverse problems for a coupled Fokker-Planck-Darcy system
AU - Grigorios A. Pavliotis
AU - Andrew M. Stuart
AU - Andrea Zanoni
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28242
UR - https://arxiv.org/abs/2609.28242
AB - 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.
ER -