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

Anuj Abhishek, Thilo Strauss, Taufiquar Khan

TL;DR

This work develops a non-parametric Bayesian level-set framework for simultaneous reconstruction of two piecewise-constant coefficients in an elliptic PDE, motivated by Diffuse Optical Tomography. By representing each coefficient via level-set functions and placing Gaussian priors on these functions, the authors derive a rigorous infinite-dimensional Bayes formulation and prove well-posedness with Lipschitz stability of the posterior in Hellinger distance. The approach enables uncertainty quantification through MCMC (pCN) sampling, and the authors demonstrate two reconstruction strategies: sharp bi-level maps and continuous reconstructions with quantified credible regions. Numerical experiments on DOT-inspired phantoms validate robustness to noise and illustrate how posterior uncertainty concentrates along material boundaries, highlighting the practical impact for reliable two-parameter optical tomography imaging.

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.

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

TL;DR

This work develops a non-parametric Bayesian level-set framework for simultaneous reconstruction of two piecewise-constant coefficients in an elliptic PDE, motivated by Diffuse Optical Tomography. By representing each coefficient via level-set functions and placing Gaussian priors on these functions, the authors derive a rigorous infinite-dimensional Bayes formulation and prove well-posedness with Lipschitz stability of the posterior in Hellinger distance. The approach enables uncertainty quantification through MCMC (pCN) sampling, and the authors demonstrate two reconstruction strategies: sharp bi-level maps and continuous reconstructions with quantified credible regions. Numerical experiments on DOT-inspired phantoms validate robustness to noise and illustrate how posterior uncertainty concentrates along material boundaries, highlighting the practical impact for reliable two-parameter optical tomography imaging.

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.
Paper Structure (16 sections, 8 theorems, 31 equations, 5 figures, 2 tables)

This paper contains 16 sections, 8 theorems, 31 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A joint coefficient-identification problem
  3. Non-Parametric Bayesian Approach
  4. Related Research
  5. Organization
  6. Mathematical formulation of the problem
  7. Preliminary Results
  8. Modeling a level-set prior
  9. Likelihood and posterior update
  10. Numerical Simulations
  11. Markov Chain Monte Carlo
  12. Simulation results
  13. Method 1: Bi-Level Reconstruction
  14. Method 2: Continuous Reconstruction
  15. Noise-Study
  16. ...and 1 more sections

Key Result

Theorem 1.1

dashti17 Let $\Phi:X\times Y\to \mathbb{R}$ be $\nu_{0}$ measurable and let $Z_y$ defined as $\int_{X}\exp(-\Phi(v,y)) d\mu_{0}:=Z_y>0$ for $\mathbb{Q}_0$ a.s. $y$, then the conditional distribution of $v|y$ denoted by $\mu^y$ exists under $\nu$. Furthermore, $\mu^y\ll \mu_0$ and

Figures (5)

  • Figure 1: On the left is the fine mesh used to generate the data by solving the forward problem. On the right is the coarser finite element mesh used for solving the inverse problem.
  • Figure 2: Traceplots for parameter values as a function of MCMC draws after burn-in: Row 1 corresponds to the traceplots for the value of $u_1$ (diffusive level-set function) in some randomly chosen triangle in the reconstruction mesh, Row 2 corresponds to the traceplots for the value of $u_2$ (absorptive level-set function) in some randomly chosen triangle in the reconstruction mesh. Columns 1, 2 and 3 correspond respectively to the the three different phantom geometries as in Figure \ref{['mod1']}.
  • Figure 3: Reconstructions using Method 1 (Bi-Level Reconstruction): First and third columns have the plots of projections of true diffusion and absorption parameters on the FEM basis. In rows 1 and 2, the diffusive and absorptive regions coincide. In row 3, we have a phantom in which the diffusive and absorptive regions are at separate places. In column 2, we have the corresponding reconstructions of the diffusive parameter and in column 4, we have the reconstructions of the absorptive parameter. The data has been collected at $2\%$ relative noise.
  • Figure 4: Uncertainty quantification in Method 1 (Bi-Level Reconstruction): The rows correspond to the phantom geometries as in Figure \ref{['mod1']}. The first and third column in each row shows the lower Bayesian credible region ($15\%$) for the diffusive and absorptive regions respectively. The second and the fourth column show the upper credible region ($85\%$) for the diffusive and absorptive regions respectively.
  • Figure 5: Reconstructions using Method 2 (Continuous Reconstruction): The rows correspond to the phantom geometries as shown in Figure \ref{['mod1']}. In the first and fourth columns, we have the projections of true diffusive and absorptive phantoms on the FEM basis. In columns 2 and 5, we have the corresponding reconstructions of the diffusive and absorptive parameters, respectively. In the third and the sixth columns are plots of standard errors. From the standard error plots, it is clear that the greatest variance (and thus, the uncertainty) in posterior samples is around the boundaries of the diffusive/absorptive regions in the phantom. Data was corrupted with $2\%$ relative noise.

Theorems & Definitions (14)

  • Theorem 1.1
  • Proposition 1
  • Remark 1
  • Lemma 3.1
  • Remark 2
  • Theorem 3.2
  • Definition 3.3
  • Theorem 3.4
  • Remark 3
  • Proposition 2
  • ...and 4 more