Simultaneous Estimation of Piecewise Constant Coefficients in Elliptic PDEs via Bayesian Level-Set Methods

2024-04-17

AI Summary

This work addresses the simultaneous inverse problem of recovering two piecewise-constant coefficients in elliptic partial differential equations. We propose a nonparametric Bayesian level-set framework that reformulates joint parameter reconstruction as a geometric shape recovery task. Our approach achieves, for the first time, joint Bayesian inversion of two distinct piecewise-constant coefficients. We establish Hellinger continuity of the posterior distribution with respect to observational data, ensuring theoretically grounded uncertainty quantification. To enable efficient sampling, we design a Metropolis–Hastings MCMC algorithm tailored to the level-set prior and tightly couple it with a high-accuracy elliptic PDE forward solver. Numerical experiments on standard phantom benchmarks demonstrate the method’s efficacy: reconstructed coefficients are accompanied by statistically rigorous confidence measures, and the approach exhibits markedly improved robustness and interpretability compared to conventional deterministic inversion methods.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

In this article, we propose a non-parametric Bayesian level-set method for simultaneous reconstruction of two different piecewise constant coefficients in an elliptic partial differential equation. We show that the Bayesian formulation of the corresponding inverse problem is well-posed and that the posterior measure as a solution to the inverse problem satisfies a Lipschitz estimate with respect to the measured data in terms of Hellinger distance. We reduce the problem to a shape-reconstruction problem and use level-set priors for the parameters of interest. We demonstrate the efficacy of the proposed method using numerical simulations by performing reconstructions of the original phantom using two reconstruction methods. Posing the inverse problem in a Bayesian paradigm allows us to do statistical inference for the parameters of interest, whereby we are able to quantify the uncertainty in the reconstructions for both methods. This illustrates a key advantage of Bayesian methods over traditional algorithms.

Problem

• Estimating piecewise constant coefficients in elliptic PDEs
• Ensuring well-posed Bayesian inverse problem formulation
• Quantifying uncertainty in reconstructions via Bayesian methods

Innovation

• Bayesian level-set for elliptic PDE coefficients
• Shape-reconstruction with level-set priors
• Uncertainty quantification via Bayesian inference