Goal-Oriented Error Estimation

Energy-norm estimators control the error everywhere in $\Omega$. EIT, however, only uses a few numbers: the electrode voltages. Goal-oriented estimation controls the error of a quantity of interest $\mathcal{Q}(u)$ directly.

Dual-weighted residual. Let $a(u,v)=\ell (v)$ be the forward problem, $u_h$ its Galerkin solution, and $\mathcal{Q}$ linear. Let $z$ solve the dual (adjoint) problem $a(v,z)=\mathcal{Q}(v)$ for all $v$. Then

$$\mathcal{Q}(u)-\mathcal{Q}(u_h)=a(u-u_h,z)=\ell (z)-a(u_h,z)=r(z-z_h)$$

for any $z_h$ in the finite element space, by Galerkin orthogonality. $r$ is the residual functional. Splitting $r(z-z_h)$ into cell contributions gives indicators ${\eta }_K={\rho }_K(u_h){\omega }_K(z)$: the local residual of the forward solution weighted by the local approximation error of the dual solution. Cells are refined only where both are large, i.e. where errors are made and matter for $\mathcal{Q}$.

For EIT the duals are already there. The measured voltage $V_m=Q_m\mathbf{u}$ is a linear functional of the state, and its dual problem is the adjoint problem of the Adjoint State Method with a unit misfit on measurement $m$. These are the adjoint fields that build the Jacobian rows (see Conductivity Tensor). One set of dual solves serves all current patterns. For the data misfit itself, the dual is the adjoint state $\lambda$ of the gradient computation. The DWR estimator then controls the error in the objective, and the same framework yields error estimators for the reconstructed conductivity (Becker and Vexler).

Practical points. The weight $z-z_h$ must be approximated, for example by a higher-order recovery of $z_h$ or by solving the dual problem on a finer space. The estimator is not guaranteed to be an upper bound, but it is usually much sharper for $\mathcal{Q}$ than energy-norm estimators.

A simple product indicator. By Cauchy–Schwarz, $|a(u-u_h,z-z_h)|\le {\sum }_K\Vert u-u_h{\Vert }_{a,K}\Vert z-z_h{\Vert }_{a,K}$. Estimating both local energy errors with the Zienkiewicz-Zhu Estimator gives the indicator ${\eta }_K={\eta }_K(u){\eta }_K(z)$. Summed over patterns and measurements, it needs no residual evaluation. It refines only where the primal error is large and the measurements are sensitive, and it vanishes when either solution is resolved exactly.

In ModularEIT.jl: goal_oriented_indicator.

References

1. R. Becker, R. Rannacher (2001). An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 10, 1–102. doi:10.1017/S0962492901000010
2. 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
3. M. Ainsworth, J. T. Oden (2000). A Posteriori Error Estimation in Finite Element Analysis. Wiley. doi:10.1002/9781118032824