Iterative Reconstruction Loop

Most practical EIT reconstructions minimise a regularised objective

$$\Phi (\sigma )=\sum _{i=1}^N{J}_i(u_i(\sigma ))+\beta \mathcal{R}(\sigma )$$

iteratively, where $u_i(\sigma )$ solves the forward problem for current pattern $g_i$. One iteration:

given σ_k
1. assemble L_σk                                    (weighted stiffness matrix)
2. solve  L_σk u_i = g_i             for all i      (state equations)
3. misfit  J(σ_k) = Σ_i d(u_i|Γ, f_i)
4. solve  L_σk λ_i = ∂_u d(u_i|Γ, f_i)  for all i   (adjoint equations)
5. gradient  ∇J = Σ_i −∇u_i·∇λ_i                    (functional derivative)
6. add regulariser  β ∇R(σ_k)
7. update σ_{k+1} = σ_k + τ_k p_k                   (optimiser: GN, L-BFGS-B, ...)
until stopping criterion

The search direction $p_k$ is a descent direction, for example $p_k=-(\nabla J+\beta \nabla \mathcal{R})$ for steepest descent. The step size ${\tau }_k>0$ comes from the optimiser (see Line Search).

Components and where they are described:

• steps 1–2: Weighted Stiffness Matrix, Block Krylov Methods;
• step 3: Data Fidelity Terms;
• steps 4–5: Adjoint State Method, Adjoint Equation, Functional Derivative of the Data Misfit;
• step 6: Tikhonov Regularization, Smoothed Total Variation, learned priors (Learned Regularization);
• step 7: Gauss-Newton Method, Levenberg-Marquardt Method, L-BFGS-B, or splitting with ADMM (see Nested ADMM Reconstruction);
• stopping: Stopping Criteria.

The per-pattern solves are independent, so the method parallelises naturally over patterns.

In ModularEIT.jl: minimize.

References

1. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich (2009). Optimization with PDE Constraints. Springer. doi:10.1007/978-1-4020-8839-1
2. W. R. B. Lionheart (2004). EIT reconstruction algorithms: pitfalls, challenges and recent developments. Physiol. Meas. 25(1), 125–142. doi:10.1088/0967-3334/25/1/021