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Bayesian Nonparametric Inference in Elliptic PDEs: Convergence Rates and Implementation

Matteo Giordano

TL;DR

The paper addresses recovering a spatially varying diffusivity $f$ in a high-dimensional elliptic PDE from noisy PDE-solutions using Bayesian nonparametric methods with Gaussian priors. It proves posterior consistency and explicit convergence rates for priors based on the Dirichlet--Laplacian eigenbasis and Matérn kernels, including an optimal prediction-rate $\epsilon_n = n^{-(\alpha+1)/(2\alpha+2+d)}$ and a deriveable $L^2$-risk rate for $f$. The authors implement two discretization schemes—Dirichlet--Laplacian sieve priors with pCN-MCMC and Matérn priors on a finite element mesh—and demonstrate through numerical experiments that the posterior concentrates around the true diffusivity with accurate PDE reconstructions and reasonable computation times. This work provides both rigorous theoretical guarantees and practical, scalable algorithms for PDE-constrained inverse problems with uncertainty quantification, enabling robust inference of diffusion fields from observed PDE data.

Abstract

Parameter identification problems in partial differential equations (PDEs) consist in determining one or more functional coefficient in a PDE. In this article, the Bayesian nonparametric approach to such problems is considered. Focusing on the representative example of inferring the diffusivity function in an elliptic PDE from noisy observations of the PDE solution, the performance of Bayesian procedures based on Gaussian process priors is investigated. Building on recent developments in the literature, we derive novel asymptotic theoretical guarantees that establish posterior consistency and convergence rates for methodologically attractive Gaussian series priors based on the Dirichlet-Laplacian eigenbasis. An implementation of the associated posterior-based inference is provided and illustrated via a numerical simulation study, where excellent agreement with the theory is obtained.

Bayesian Nonparametric Inference in Elliptic PDEs: Convergence Rates and Implementation

TL;DR

The paper addresses recovering a spatially varying diffusivity in a high-dimensional elliptic PDE from noisy PDE-solutions using Bayesian nonparametric methods with Gaussian priors. It proves posterior consistency and explicit convergence rates for priors based on the Dirichlet--Laplacian eigenbasis and Matérn kernels, including an optimal prediction-rate and a deriveable -risk rate for . The authors implement two discretization schemes—Dirichlet--Laplacian sieve priors with pCN-MCMC and Matérn priors on a finite element mesh—and demonstrate through numerical experiments that the posterior concentrates around the true diffusivity with accurate PDE reconstructions and reasonable computation times. This work provides both rigorous theoretical guarantees and practical, scalable algorithms for PDE-constrained inverse problems with uncertainty quantification, enabling robust inference of diffusion fields from observed PDE data.

Abstract

Parameter identification problems in partial differential equations (PDEs) consist in determining one or more functional coefficient in a PDE. In this article, the Bayesian nonparametric approach to such problems is considered. Focusing on the representative example of inferring the diffusivity function in an elliptic PDE from noisy observations of the PDE solution, the performance of Bayesian procedures based on Gaussian process priors is investigated. Building on recent developments in the literature, we derive novel asymptotic theoretical guarantees that establish posterior consistency and convergence rates for methodologically attractive Gaussian series priors based on the Dirichlet-Laplacian eigenbasis. An implementation of the associated posterior-based inference is provided and illustrated via a numerical simulation study, where excellent agreement with the theory is obtained.
Paper Structure (14 sections, 1 theorem, 55 equations, 6 figures, 2 tables)

This paper contains 14 sections, 1 theorem, 55 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Materials and Methods
  3. Likelihood, Prior and Posterior
  4. Convergence Rates
  5. Examples of Gaussian Priors
  6. Results
  7. Results with Truncated Gaussian Series Priors
  8. Results with the Matérn Process Prior
  9. Discussion
  10. Further Numerical Results
  11. Proof of Theorem \ref{['Theo:Main']}
  12. Step I: posterior contraction rates in prediction risk.
  13. Step II: remaining claims.
  14. Acknowledgement.

Key Result

Theorem 1

For fixed positive integers $\alpha,\beta\in\mathbb{N}$ such that $\alpha>\beta+d/2$, consider the scaled prior $\Pi_n$ in Eq:Prior, where $\Pi_{W_n}$ is a centred Gaussian Borel probability measure supported on a measurable linear subspace of $C^\beta(\mathcal{O})$, with RKHS $\mathcal{H}_{W_n}\sub Further assume that For fixed $F_0\in H^\alpha_0(\mathcal{O})$, suppose that there exists a sequen

Figures (6)

  • Figure 1: (Left): An example of diffusivity function $f$ with four circular regions of higher conductivity. (Right): $n=4500$ noisy observations from the corresponding PDE solution $G(f)$.
  • Figure 2: Left to right: posterior mean estimates $\bar{f}_n$ of the diffusivity function $f$ for increasing sample sizes $n= 100, 250, 1000$.
  • Figure 3: Left to right: The first, second, and fiftieth Dirichlet--Laplacian eigenfunctions $e_0, \ e_1$ and $e_{49}$, computed via finite element methods.
  • Figure 4: Dirichlet--Laplacian eigenvalues $\lambda_j$ in the range $[0,500]$, computed via finite element methods.
  • Figure 5: (Left): acceptance rate over the first 10,000 pCN samples. The rate stabilises around $30\%$ after the initial burn-in time (first 5000 iterates). (Right): in blue, the log-likelihood $l_n(\Phi\circ\vartheta_h)$ of the first 3000 iterates; in red, the log-likelihood $l_n(f_0)$ of the true diffusion coefficient $f_0$.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Example 2: Dirichlet--Laplacian eigenbasis
  • Example 3: Matérn covariance kernel