Adaptive Meshing in EIT

Adaptive mesh refinement repeats SOLVE → ESTIMATE → MARK → REFINE (see A Posteriori Error Estimation and Adaptive Meshing). In EIT it serves two different purposes, and they call for different indicators.

1. Accurate forward predictions

The discretisation error of the predicted voltages is a model error. If it is larger than the measurement noise, the reconstruction fits the discretisation error and produces artefacts (see Inverse Crime and Noise Models for EIT Data). The error concentrates where the solution is not smooth:

• Electrode edges. Under the Shunt Model, the current density is singular like $r^{-1/2}$ at the electrode edges. Under the Complete Electrode Model, a small contact impedance $z$ leaves a thin boundary layer with steep gradients instead. Point electrodes have logarithmic singularities.
• Conductivity interfaces. At a jump of $\sigma$ the normal current $\sigma {\partial }_{\nu }u$ is continuous, so ${\partial }_{\nu }u$ jumps. The potential has a kink there that piecewise polynomials resolve only with small cells.
• Near the boundary. High-frequency current patterns decay quickly into the interior (see Current Patterns), so they need fine cells close to the boundary.
• Reentrant corners, if the domain has them. Convex corners, such as those of a square image domain, are harmless.

One mesh has to serve all current patterns, so the indicators of all patterns are summed. Suitable indicators:

• the residual indicator: element residuals, current-density jumps across facets, and the boundary residuals of the electrode model;
• the Zienkiewicz-Zhu Estimator, which recovers a smooth current density and measures the distance to it;
• Goal-Oriented Error Estimation, which weights the residuals by adjoint fields of the measurements and targets exactly the voltage error.

2. Resolving the conductivity

When $\sigma$ lives on the same mesh as $u$, refinement also adds conductivity unknowns. For the reconstruction, cells are needed where $\sigma$ has features: inclusion boundaries show up as large jumps of a piecewise constant iterate, i.e. large contributions to its Total Variation. Refining there and coarsening flat regions gives a feature-adapted parametrisation. Two cautions apply:

• Resolution is limited by the data. EIT is severely ill-posed (see Stability of the Calderón Problem). Interior details far from the electrodes cannot be resolved by any mesh. Refining there only adds unknowns that the regulariser must control.
• Feature-driven refinement is blind to the forward error. It does not refine at the electrodes. Combining both indicators, or using separate meshes for $u$ (fine) and $\sigma$ (coarse), keeps the model error small.

A convergent adaptive algorithm for the regularised EIT problem refines with an estimator that combines the state, the adjoint and the conductivity (Jin, Xu, Zou).

Consequences for the algorithms

• Gradients. On a graded mesh the coefficient gradient $\partial J/\partial {\sigma }_a$ scales with the cell size, so steepest descent favours large cells. The $L^2$ gradient $M_{\sigma }^{-1}\partial J/\partial \sigma$ is mesh independent (see Conductivity Tensor and Gradient Representation and the Riesz Map).
• Transfer. After refinement or coarsening, the current iterate is transferred by L2 Projection. For piecewise constants, children inherit the value of their parent, and a coarsened parent gets the mean of its children.
• The experiment must not change with the mesh. Electrodes, current patterns and the grounding must be defined geometrically (for example electrode positions as length-weighted centroids, electrodes as boundary segments carried through refinement), not through node or facet counts, which change under boundary refinement. Otherwise every mesh simulates a slightly different measurement, and the discretisation error seems to stagnate.
• Discretisation-dependent functionals. Regularisers must be evaluated with the mesh geometry (cell areas, facet lengths), not per coefficient, or their weight changes under refinement.

Refining quadrilaterals and triangles

Quadtree refinement of quadrilaterals (octrees in 3D) splits a cell into four children and creates Hanging Nodes. It fits pixel meshes well and keeps the cells shape regular. Triangles can be refined conformingly by Newest Vertex Bisection or red-green refinement, without hanging nodes. Marking is usually done by Dörfler Marking.

In ModularEIT.jl: AdaptiveMesh, refine_mesh!, goal_oriented_indicator.

References

1. M. Molinari, B. H. Blott, S. J. Cox, G. J. Daniell (2002). Optimal imaging with adaptive mesh refinement in electrical impedance tomography. Physiol. Meas. 23(1), 121–128. doi:10.1088/0967-3334/23/1/311
2. B. Jin, Y. Xu, J. Zou (2017). A convergent adaptive finite element method for electrical impedance tomography. IMA J. Numer. Anal. 37(3), 1520–1550. doi:10.1093/imanum/drw045
3. R. Becker, B. Vexler (2004). A posteriori error estimation for finite element discretization of parameter identification problems. Numer. Math. 96(3), 435–459. doi:10.1007/s00211-003-0482-9
4. M. Ainsworth, J. T. Oden (2000). A Posteriori Error Estimation in Finite Element Analysis. Wiley. doi:10.1002/9781118032824