Variational Regularization

Because the Calderón Problem is severely ill-posed (see Stability of the Calderón Problem), EIT reconstructions trade exact data fit for stability. They minimise a variational objective

$$\underset{\sigma \in \Sigma }{\min }\underset{\text{data fidelity}}{\underbrace{d(\mathcal{F}(\sigma ),y^{\delta })}}+\beta \underset{\text{regulariser}}{\underbrace{\mathcal{R}(\sigma )}},$$

where

• $\mathcal{F}$ is the Forward Map (for example voltages for given Current Patterns) and $y^{\delta }$ the noisy data;
• $d$ is a distance or divergence (see Data Fidelity Terms), most often $\frac{1}{2}\Vert \mathcal{F}(\sigma )-y^{\delta }{\Vert }_2^2$;
• $\mathcal{R}$ encodes prior knowledge: smoothness (Tikhonov Regularization), piecewise constancy (Total Variation), low rank or spectral truncation (Truncated SVD Regularization), or a learned prior (Learned Regularization);
• $\beta >0$ is the regularisation parameter (see Choosing the Regularization Parameter);
• $\Sigma$ is the admissible set, for example ${\sigma }_{\min }\le \sigma \le {\sigma }_{\max }$ (see Box Constraints on Conductivity).

What makes it a regularisation method. For linear problems, and under conditions for nonlinear ones, the minimisers converge to a true solution as the noise level $\delta \to 0$, provided $\beta =\beta (\delta )$ is chosen suitably ($\beta \to 0$ and ${\delta }^2/\beta \to 0$). In the Bayesian view, the objective is the negative log-posterior and its minimiser is the MAP estimate (see Bayesian Inversion).

Regularisation can also be implicit: early stopping of iterative methods, restricting to few low-frequency current patterns, or a coarse discretisation (see Implicit Regularization).

The objective is minimised with gradient-based methods. Gradients come from the Adjoint State Method, and minimisation uses Gauss–Newton, L-BFGS-B or splitting schemes such as ADMM.

In ModularEIT.jl: RegularizedObjective, minimize.

References

1. H. W. Engl, M. Hanke, A. Neubauer (1996). Regularization of Inverse Problems. Kluwer. doi:10.1007/978-94-009-1740-8
2. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344
3. M. Benning, M. Burger (2018). Modern regularization methods for inverse problems. Acta Numerica 27, 1–111. doi:10.1017/S0962492918000016