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Bayesian inference for the fractional Calderón problem with a single measurement

Pu-Zhao Kow, Janne Nurminen, Jesse Railo

TL;DR

This work addresses the inverse problem of recovering a fractional Schrödinger potential from a single exterior measurement with additive Gaussian noise, formulated within a Bayesian framework. It introduces a reparametrized forward map and places priors on the transformed coefficient using both rescaled Gaussian priors and Gaussian sieve priors, enabling rigorous posterior contraction analysis. The main contributions are the derivation of logarithmic posterior contraction rates around the true coefficient, proofs of minimax optimality via instability results, refined forward and inverse problem estimates, and numerical demonstrations in one dimension. These results provide statistical guarantees and uncertainty quantification guidance for nonlocal Calderón-type problems with highly limited data. The paper also discusses Bernstein–von Mises considerations and includes practical MCMC-based illustrations of the Bayesian procedure.

Abstract

This paper investigates the consistency of a posterior distribution in the single-measurement fractional Calderón problem with additive Gaussian noise. We consider a Bayesian framework with rescaled and Gaussian sieve priors, using a collection of noisy, discrete observations taken from a suitable exterior domain. Our main result shows that the posterior distribution concentrates around the true parameter as the number of measurements increases. Furthermore, we establish tight convergence rates for the reconstruction error of the posterior mean. A central technical challenge is to obtain refined stability estimates for both the forward and inverse problems. In particular, the required forward estimates are delicate to obtain because the fractional elliptic problems do not enjoy as strong regularity theory as their classical counterparts.

Bayesian inference for the fractional Calderón problem with a single measurement

TL;DR

This work addresses the inverse problem of recovering a fractional Schrödinger potential from a single exterior measurement with additive Gaussian noise, formulated within a Bayesian framework. It introduces a reparametrized forward map and places priors on the transformed coefficient using both rescaled Gaussian priors and Gaussian sieve priors, enabling rigorous posterior contraction analysis. The main contributions are the derivation of logarithmic posterior contraction rates around the true coefficient, proofs of minimax optimality via instability results, refined forward and inverse problem estimates, and numerical demonstrations in one dimension. These results provide statistical guarantees and uncertainty quantification guidance for nonlocal Calderón-type problems with highly limited data. The paper also discusses Bernstein–von Mises considerations and includes practical MCMC-based illustrations of the Bayesian procedure.

Abstract

This paper investigates the consistency of a posterior distribution in the single-measurement fractional Calderón problem with additive Gaussian noise. We consider a Bayesian framework with rescaled and Gaussian sieve priors, using a collection of noisy, discrete observations taken from a suitable exterior domain. Our main result shows that the posterior distribution concentrates around the true parameter as the number of measurements increases. Furthermore, we establish tight convergence rates for the reconstruction error of the posterior mean. A central technical challenge is to obtain refined stability estimates for both the forward and inverse problems. In particular, the required forward estimates are delicate to obtain because the fractional elliptic problems do not enjoy as strong regularity theory as their classical counterparts.

Paper Structure

This paper contains 15 sections, 12 theorems, 99 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. Main results
  4. The statistical model
  5. Results with rescaled Gaussian priors
  6. Results with Gaussian sieve priors
  7. Tightness of the logarithmic contraction rate
  8. Remarks on the Bernstein-von Mises (BvM) theorem
  9. Estimates for the fractional Schrödinger equation
  10. Estimates for the forward problem
  11. Stability estimates for the inverse problem
  12. Statistical conclusions
  13. Posterior contraction rates
  14. Minimax optimality of the contraction rates
  15. Numerical demonstrations of the Bayesian procedure

Key Result

Theorem 2.3

Let $d\in\mathbb{N}$, let $\Omega$, $\mathcal{O}$ and $\mathcal{D}$ be nonempty bounded smooth domains in $\mathbb{R}^{d}$ with $\overline{\mathcal{O}}\subset\Omega$ and $\overline{\Omega}\cap\overline{\mathcal{D}}=\emptyset$. Let $0<s<1$, $0\not\equiv\phi\in C_{c}^{\infty}(\Omega_{e})$ and $\Phi$ b as $N\rightarrow\infty$, where $\delta_{N}=N^{-\frac{\alpha}{2\alpha+d}}$.

Figures (2)

  • Figure 5.1: Plot of $u_{f}|_{\Omega}$
  • Figure 5.2: MCMC iterations

Theorems & Definitions (13)

  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • ...and 3 more