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Elliptic Bayesian Inverse Problems on Metric Graphs

David Bolin, Wenwen Li, Daniel Sanz-Alonso

Abstract

This paper studies the formulation, well-posedness, and numerical solution of Bayesian inverse problems on metric graphs, in which the edges represent one-dimensional wires connecting vertices. We focus on the inverse problem of recovering the diffusion coefficient of a (fractional) elliptic equation on a metric graph from noisy measurements of the solution. Well-posedness hinges on both stability of the forward model and an appropriate choice of prior. We establish the stability of elliptic and fractional elliptic forward models using recent regularity theory for differential equations on metric graphs. For the prior, we leverage modern Gaussian Whittle--Matérn process models on metric graphs with sufficiently smooth sample paths. Numerical results demonstrate accurate reconstruction and effective uncertainty quantification.

Elliptic Bayesian Inverse Problems on Metric Graphs

Abstract

This paper studies the formulation, well-posedness, and numerical solution of Bayesian inverse problems on metric graphs, in which the edges represent one-dimensional wires connecting vertices. We focus on the inverse problem of recovering the diffusion coefficient of a (fractional) elliptic equation on a metric graph from noisy measurements of the solution. Well-posedness hinges on both stability of the forward model and an appropriate choice of prior. We establish the stability of elliptic and fractional elliptic forward models using recent regularity theory for differential equations on metric graphs. For the prior, we leverage modern Gaussian Whittle--Matérn process models on metric graphs with sufficiently smooth sample paths. Numerical results demonstrate accurate reconstruction and effective uncertainty quantification.

Paper Structure

This paper contains 31 sections, 9 theorems, 132 equations, 7 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Metric Graphs and Function Spaces
  4. Quantum Graphs
  5. Inverse Problem Formulation
  6. Prior Distribution
  7. Likelihood Function
  8. Posterior Distribution
  9. Well-posedness Theory
  10. Main Results
  11. Stability of the Forward Map
  12. Elliptic Problem
  13. Fractional Elliptic Problem
  14. Numerical Experiments
  15. Implementation
  16. ...and 16 more sections

Key Result

Proposition 3.1

Let $u\in H^1(\Gamma)$ and set $a:=e^{u}$. Assume $\kappa\in L^\infty(\Gamma)$ with $\operatorname*{ess\,inf}_{x\in\Gamma}\kappa(x)\ge \kappa_{\min}>0$. Let $\mathcal{L}_u:=\kappa^2-\nabla\!\cdot(a\nabla)$ be endowed with standard Kirchhoff vertex conditions. Then the following hold:

Figures (7)

  • Figure 1: Elliptic problem ($\beta =1).$ Comparison of ground truth, posterior mean, and MAP estimate for the parameter (left column) and for the corresponding PDE solution (right column).
  • Figure 2: Elliptic case ($\beta =1)$. Top row: posterior marginal standard deviation computed from MCMC samples for the parameter (left), defined as in Equation \ref{['eq:post_sd']}, and for the corresponding PDE solution (right). Bottom row: difference between the truth and the posterior mean for the parameter (left) and the PDE solution (right).
  • Figure 3: Fractional elliptic problem ($\beta = 3/2).$ Comparison of ground truth, posterior mean, and MAP estimate for the parameter (left column) and for the corresponding PDE solution (right column).
  • Figure 4: Fractional elliptic problem ($\beta = 3/2)$. Top row: posterior marginal standard deviation computed from MCMC samples for the parameter $u$ (left) and for the corresponding PDE solution $p$ (right). Bottom row: difference between the truth and the posterior mean for the parameter (left) and the PDE solution (right).
  • Figure 5: Boxplots of reconstruction errors over repeated trials. Top row: RMSE for the log-conductivity estimator $\widehat{u}$ and the corresponding plug-in state estimator $\widehat{p}=p(\widehat{u})$. Bottom row: range-normalized error for $u$ and relative $L^2(\Gamma)$ errors (mass-matrix based) for the conductivity $\widehat{a}=\exp(\widehat{u})$ and the state $\widehat{p}$.
  • ...and 2 more figures

Theorems & Definitions (22)

  • Proposition 3.1: Properties of $\mathcal{L}_u$
  • proof
  • Proposition 3.3: Regularity of Prior Gaussian Process (See bolin2024regularity)
  • Corollary 3.4: Forward Map on Prior Support
  • proof
  • Remark 3.5
  • Theorem 4.1: Stability of the Forward Map
  • Corollary 4.2: Application to the Gaussian Prior $\mu_0$
  • proof
  • Remark 4.3
  • ...and 12 more