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Sparsity-promoting hierarchical Bayesian model for EIT with a blocky target

Daniela Calvetti, Monica Pragliola, Erkki Somersalo

TL;DR

This work develops a sparsity-promoting hierarchical Bayesian framework for nonlinear electrical impedance tomography (EIT) that targets nearly piecewise-constant conductivities. By formulating a two-layer prior on the edge-incidence vector ζ = Lξ and applying the Iterative Alternating Sequential (IAS) algorithm, the authors achieve computationally efficient MAP estimates through adjoint-based linearization and a data-space approach detailed with Lanczos bidiagonalization. They analyze convexity properties, propose hyperparameter strategies anchored in data sensitivity, and demonstrate both MAP and quasi-MAP performances on simulated EIT data, highlighting substantial speedups with only modest differences in reconstruction quality. The findings provide practical guidance for fast, edge-preserving EIT reconstructions and contribute theoretical insights into the convergence landscape of IAS for nonlinear inverse problems.

Abstract

The electrical impedance tomography (EIT) problem of estimating the unknown conductivity distribution inside a domain from boundary current or voltage measurements requires the solution of a nonlinear inverse problem. Sparsity promoting hierarchical Bayesian models have been shown to be very effective in the recovery of almost piecewise constant solutions in linear inverse problems. We demonstrate that by exploiting linear algebraic considerations it is possible to organize the calculation for the Bayesian solution of the nonlinear EIT inverse problem via finite element methods with sparsity promoting priors in a computationally efficient manner. The proposed approach uses the Iterative Alternating Sequential (IAS) algorithm for the solution of the linearized problems. Within the IAS algorithm, a substantial reduction in computational complexity is attained by exploiting the low dimensionality of the data space and an adjoint formulation of the Tikhonov regularized solution that constitutes part of the iterative updating scheme. Numerical tests illustrate the computational efficiency of the proposed algorithm. The paper sheds light also on the convexity properties of the objective function of the maximum a posteriori (MAP) estimation problem.

Sparsity-promoting hierarchical Bayesian model for EIT with a blocky target

TL;DR

This work develops a sparsity-promoting hierarchical Bayesian framework for nonlinear electrical impedance tomography (EIT) that targets nearly piecewise-constant conductivities. By formulating a two-layer prior on the edge-incidence vector ζ = Lξ and applying the Iterative Alternating Sequential (IAS) algorithm, the authors achieve computationally efficient MAP estimates through adjoint-based linearization and a data-space approach detailed with Lanczos bidiagonalization. They analyze convexity properties, propose hyperparameter strategies anchored in data sensitivity, and demonstrate both MAP and quasi-MAP performances on simulated EIT data, highlighting substantial speedups with only modest differences in reconstruction quality. The findings provide practical guidance for fast, edge-preserving EIT reconstructions and contribute theoretical insights into the convergence landscape of IAS for nonlinear inverse problems.

Abstract

The electrical impedance tomography (EIT) problem of estimating the unknown conductivity distribution inside a domain from boundary current or voltage measurements requires the solution of a nonlinear inverse problem. Sparsity promoting hierarchical Bayesian models have been shown to be very effective in the recovery of almost piecewise constant solutions in linear inverse problems. We demonstrate that by exploiting linear algebraic considerations it is possible to organize the calculation for the Bayesian solution of the nonlinear EIT inverse problem via finite element methods with sparsity promoting priors in a computationally efficient manner. The proposed approach uses the Iterative Alternating Sequential (IAS) algorithm for the solution of the linearized problems. Within the IAS algorithm, a substantial reduction in computational complexity is attained by exploiting the low dimensionality of the data space and an adjoint formulation of the Tikhonov regularized solution that constitutes part of the iterative updating scheme. Numerical tests illustrate the computational efficiency of the proposed algorithm. The paper sheds light also on the convexity properties of the objective function of the maximum a posteriori (MAP) estimation problem.
Paper Structure (17 sections, 2 theorems, 119 equations, 8 figures)

This paper contains 17 sections, 2 theorems, 119 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Computational model
  3. Forward model
  4. Inverse problem
  5. Discrete increments
  6. Bayesian sparsity-promoting model
  7. MAP estimate by alternating iterations
  8. Insights from the IAS algorithm for linear problems
  9. Hyperparameter selection
  10. Computational details of the IAS optimization
  11. Updating of the variance
  12. Updating the increment vector
  13. Lanczos bidiagonalization process
  14. MAP and quasi-MAP estimates
  15. Analysis of the convergence
  16. ...and 2 more sections

Key Result

Lemma 7.1

The matrix ${\mathsf D} =\left[D_{\mu\nu}\right]\in{\mathbb R}^{n\times n}$ is symmetric positive semidefinite. Moreover, if $\overline b = {\mathbb E}(b)$ is the noiseless voltage vector that corresponds to an underlying true conductivity then the matrix $\overline {\mathsf C} = \left[\overline C_{\mu\nu}\right]$ is symmetric positive semidefinite.

Figures (8)

  • Figure 1: On the left, the tessellation used for solving the inverse problem is shown. The disc with blue background corresponds to the compact domain $D$ containing the support of $\delta\sigma$, while in the collar domain with no shading, the conductivity is assumed to be equal to a known constant $\sigma_0 = 1$. To capture the singularities at the edges of the electrodes indicated by red, a significant refining of the mesh near the boundary is necessary: The total number of elements in the mesh is 5 816, while the number of elements in $D$, determining the number of degrees of freedom in the problem, is 1 940. in the middle and on the right, the meshes used to generate the data. In the middle, the background value is equal to $\sigma_0 = 1$, while in the pink inclusion, the conductivity is set to $\sigma = 4.2$. On the right, the background conductivity is $\sigma_0=1$, and in the rectangular inclusion, $\sigma = 3.5$, while in the circular inclusion, the value is $\sigma = 0.4$. In the mesh on the right, the meshing near the electrodes is finer than in the mesh on the left to ensure that the reconstruction results are not mesh-dependent.
  • Figure 2: Left panel shows the cumulative times needed to solve the linearized least squares problems of the IAS algorithm. The times do not include the computation of the Jacobian, which takes about one second per re-evaluation. Here, "Lanczos, no basis" refers to the Lanczos bidiagonalization without keeping the basis of the Krylov subspace in memory, while "Lanczos with basis" saves the basis. "Direct, normal eqs" refers to a direct solver of the system (\ref{['normal']}), and "Direct, adjoint" to the direct solver of the adjoint system (\ref{['adjoint']}). The plot shows that for midsize problems in which the matrices are available and can be kept in the cache, the direct solvers are competitive, the adjoint formulation being the fastest. The calculations were performed using Apple M2 Ultra processor in a Mac Studio computer. The right panel shows the number of Lanczos steps needed to solve the linear system (\ref{['adjoint']}) at required accuracy of having the norm of the residual below the threshold of $10^{-8}$.
  • Figure 3: The iterative solutions estimating the conductivity $\sigma$ after an odd number of IAS iterations with the hyperparameter model with $r=1$ corresponding to the gamma distribution prior model. Each non-linear least squares solution involves two linearization steps. Observe that after the eighth iteration (lower row), the estimated conductivity remains practically unaltered.
  • Figure 4: The left panel shows the cumulative times needed to solve the linearized least squares problems of the IAS algorithm using the direct method of the adjoint map, compared to the approximate CGLS solution with early stopping at discrepancy. The right panel shows the number of CGLS steps (equivalent to the number of Lanczos steps) needed before the stopping condition is met.
  • Figure 5: Comparison of the MAP estimate (left) and the qMAP estimate (center) obtained by replacing the exact solution of the approximation replacing the Tikhonov-regularized least squares problem by the least squares problem solved by using CGLS with an early stopping at the discrepancy. On the right, the difference of the two estimates is shown.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Lemma 7.1
  • Theorem 7.2