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Bayesian statistical inverse problems for a coupled Fokker-Planck-Darcy system

Grigorios A. Pavliotis, Andrew M. Stuart, Andrea Zanoni

arXiv:2609.28242v1math.STcs.NAmath.APmath.NAstat.TH

TL;DR

The paper addresses recovery of a space-dependent permittivity from discrete, noisy observations of the Fokker–Planck density, rather than direct observations of the Darcy solution. It develops forward analysis, inverse stability estimates, and a Bayesian approach, showing posterior contraction at an explicit rate with the posterior mean attaining the same rate.

  • Problem

    The problem is to recover permittivity from density observations even though the coefficient enters through the elliptic equation and the Darcy solution is not observed directly.

    Source: §1 · p. 1

  • Method

    The analysis establishes forward well-posedness and stability estimates, then uses a rescaled Gaussian process prior to study posterior contraction.

    Source: §1 · p. 3, §3.1 · p. 6, §4 · p. 14

  • Results

    The posterior contracts around the true permittivity at an explicit, computable rate, and the posterior mean attains the same rate.

    Source: §1 · p. 4

  • Takeaways & Limitations

    One- and two-dimensional numerical experiments confirm the qualitative behavior predicted by the theory.

    Source: §6 · p. 27

    The regularity and observation assumptions are restricted to times bounded away from the initial time because generic initial data fail the required compatibility conditions.

    Source: §4 · p. 16

Abstract

from arXiv · show

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.

1 Introduction

The paper studies recovery of a space-dependent permittivity from noisy density observations, where the coefficient affects the observed density only indirectly through the coupled system. Under no-flux and Dirichlet boundary conditions, it develops forward and inverse-problem analyses and establishes posterior contraction for a Bayesian approach.

1 Introduction

  • The inverse problem recovers permittivity from density measurements rather than direct observations of the Darcy solution.The unknown coefficient enters through the elliptic equation and is observed indirectly through its effect on the Fokker–Planck solution.Source: §1 · pp. 1–2
  • A rescaled Gaussian process prior supports posterior contraction by combining observed-quantity contraction with stability of the inverse map.This places the analysis in the nonlinear statistical inverse-problem framework of, using a stability estimate based on [28, Lemma 2.2].Source: §1 · p. 2
  • Green-function reduction to the McKean–Vlasov setting is obstructed by Green-function singularity, sparse diagonal measurements, and unsuitable periodic boundaries.The related torus analysis in therefore does not directly resolve the present setting.Source: §1 · p. 2
  • No-flux density and homogeneous Dirichlet potential boundary conditions make the problem physically relevant but complicate forward well-posedness and regularity.The observation window is bounded away from t = 0 because compatibility conditions cannot be imposed a priori when the initial potential depends on the initial density.Source: §1 · p. 3
  • No-flux/Dirichlet boundaries yield strictly positive elliptic forcing that supports a stability estimate, whereas periodic boundaries require zero-mean forcing and sign changes.The authors report that extending their stability analysis to periodic boundaries is unclear, although numerical experiments suggest recovery may still be possible.Source: §1 · p. 3
  • The authors establish forward well-posedness and uniform bounds over an appropriate set of variable permittivities.These bounds include upper and lower bounds on the density.Source: §1 · p. 3
  • The inverse analysis proves Lipschitz-type forward stability and generalized backward stability estimates suitable for Bayesian contraction analysis.The generalized estimate has the form required by the framework in [28].Source: §1 · pp. 3–4
  • The posterior contracts around the true permittivity at an explicit, computable rate, and the posterior mean attains the same rate.The rate depends on a regularity parameter of the forward-problem solution.Source: §1 · p. 4

2 Problem setting

The forward problem couples Fokker–Planck density evolution with a Darcy potential under no-flux and homogeneous Dirichlet boundary conditions. The inverse problem uses noisy density observations to recover the permittivity indirectly.

2.1 The forward problem

  • The model is posed on a bounded smooth domain in dimensions 1–3, with no-flux conditions for the density and homogeneous Dirichlet conditions for the potential.The forward problem has an initial condition and assumes smooth permittivity and initial data.Source: §2.1 · p. 4

2.2 The inverse problem

  • The data are N noisy density measurements at independently and uniformly sampled space-time points in a positive-length observation window, with independent Gaussian noise of known level.The observation window may be arbitrarily small provided it has positive length.Source: §2.2 · p. 5
  • The target α is inferred indirectly because it enters only through the elliptic equation, while observations measure the density u.The admissible permittivities have regularity parameter ℓ≥4, required by the backward stability estimate's solution-regularity conditions.Source: §2.2 · p. 5

3 Properties of the coupled system

The paper establishes unique solvability of the coupled system over a uniform short-time interval, then derives bounds and regularity estimates needed for inverse analysis. The density is positive and normalized, and its bounds yield further regularity for the Darcy potential.

3.1 Existence and uniqueness of solutions

  • A fixed-point argument for the map S, which sends a trial function to its Fokker–Planck solution, yields a unique solution of the coupled system.The proof uses Schauder's fixed-point theorem.Source: §3.1 · pp. 6, 9
  • The solution density is nonnegative and has unit L1 mass throughout the interval of existence.The mass conservation follows from the no-flux boundary condition.Source: §3.1 · p. 9
  • The analysis establishes existence only on a short interval uniform over the admissible set; global-in-time existence is left for future work.The paper notes that addressing global existence would require studying the system's free energy.Source: §3.1 · p. 9

3.2 Boundedness and regularity of solutions

  • The density is uniformly bounded in Lp for every p≥1, and these bounds yield higher regularity for the Darcy potential.The bounds are part of the a priori estimates developed for inverse-problem analysis.Source: §3.2 · pp. 9, 10, 12
  • The density also has a uniform positive lower bound, u(x,t)≥b e^-Λ2T, throughout the domain and time interval.The proof uses a reflected-diffusion argument on the bounded domain.Source: §3.2 · p. 12

4 Analysis of the inverse problem

This section analyzes Bayesian recovery of permittivity using a log-shifted Gaussian-process prior, developing stability estimates for posterior contraction and deriving a posterior-mean estimate.

4 Analysis of the inverse problem

  • The unknown permittivity is represented through a log-shifted parameter with a nonlinear forward map in which it enters only through the elliptic equation.A Gaussian-process prior is placed on the transformed function; its regularity holds almost surely, while the norm bound used for deterministic estimates is imposed realization by realization.Source: §4 · p. 14

4.1 Stability estimates

  • The forward estimate gives continuous L2 dependence of the Fokker–Planck solution on the coefficient, while the backward estimate controls coefficient differences by solution differences.Both estimates are uniform over the admissible coefficient set; the backward estimate requires the higher solution regularity in Assumption 4.2.Source: §4.1 · pp. 16–19
  • The backward estimate is qualitative in practice: its constant can be very large and degenerates as the observation window approaches time zero or becomes too short.The estimates rely on a window bounded away from the initial time because generic initial data fail the boundary compatibility conditions needed for regularity at the parabolic corner.Source: §4 · p. 16, §4.1 · p. 20

4.2 Posterior contraction rates

  • Under the stated assumptions, the analysis establishes posterior contraction for the permittivity and a matching contraction rate for its posterior mean.The contraction statements are formulated on bounded-realization sets where the deterministic estimates apply.Source: §4 · p. 14, §4.2 · pp. 21–22
  • Choosing R′ and hence R sufficiently large yields (34); the posterior-mean estimate (35) follows from (34), [20, Theorem 2.3.2], and (23).Source: §4.2 · p. 22

5 Numerical experiments

The numerical experiments examine one- and two-dimensional recovery of the permittivity from discrete density observations. In one dimension, reconstruction improves with more observations, while the two-dimensional tests compare pCN and EnKF and show that higher modes remain difficult to infer.

5 Numerical experiments

  • The experiments use a Matérn-like Gaussian prior, finite-element space discretization, implicit Euler time stepping, and pCN posterior sampling in one and two dimensions.The reported noise standard deviation is γ = 0.002; the experiments set σ = 1 and T = 1.Source: §5 · pp. 22–23

5.1 Dependence on the number of observations

  • As N increases from 25 to 800, reconstructions improve; MAE and MSE decay slightly faster than N^-µ, while posterior-mean L2 decay is more variable.The theoretical rate uses µ ≃ 0.153; Figure 2 is a qualitative validation, not a strict quantitative verification.Source: Figure 1 · p. 23, §5.1 · p. 24, Figure 2 · p. 24
  • In one dimension, the first three mode marginals narrow and their bias decreases as N grows, but N = 800 still does not reliably capture modes four and five.Higher modes initially resemble the prior and begin changing shape as the observation count increases.Source: §5.1 · p. 24, Figure 3 · p. 25

5.2 Two-dimensional example

  • At N = 500 in two dimensions, pCN and EnKF recover the parameter field’s main features; EnKF approximates true coefficients slightly more accurately, while modes 11–15 remain poorly informed.Posterior distributions contract around the true coefficients for modes 1–5 and stay closer to the prior for modes 11–15.Source: §5.2 · p. 26, Figure 4 · p. 26, Figure 5 · p. 27

6 Conclusion

The paper establishes analytical and statistical results for recovering permittivity in the coupled Fokker–Planck–Darcy system, with numerical experiments in one and two dimensions showing qualitative agreement with theory. Open problems include improving the rate exponent, extending solvability results, and establishing contraction theory for EnKF.

6 Conclusion

  • The paper establishes well-posedness and uniform solution bounds, derives forward and backward stability estimates, and proves posterior contraction and posterior-mean convergence rates.One- and two-dimensional experiments confirm the qualitative behavior predicted by the theory.Source: §6 · p. 27
  • The regularity argument yields κ < 3, while κ = ℓ−1 is expected; proving the stronger bound would improve the contraction exponent, and global solvability questions remain open.The authors also note that the stability constant depends doubly exponentially on final time, making the resulting estimate qualitative rather than quantitative.Source: §6 · p. 28
  • Proposed extensions include two-species systems, minimax lower bounds, continuous-time observations, and contraction theory for EnKF, which currently has no such theory in this setting.The EnKF extension is motivated by its comparable performance to posterior sampling in the two-dimensional experiments.Source: §6 · p. 28
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