Box Constraints on Conductivity

Physical conductivities are positive and bounded:

$$\Sigma =\{\sigma \in L^{\infty }(\Omega ):{\sigma }_{\min }\le \sigma (x)\le {\sigma }_{\max }\text{a.e.}\},0<{\sigma }_{\min }\le {\sigma }_{\max }<\infty .$$

Why the bounds matter.

• Lower bound: it makes the forward problem uniformly elliptic and coercive (Lax-Milgram Theorem). Perfect insulators ($\sigma =0$) would disconnect the domain.
• Upper bound: it keeps the problem uniformly elliptic, gives continuity of the bilinear form, and excludes perfect conductors.
• Regularisation: bounds rule out oscillating minimising sequences and restore compactness in combination with a regulariser (see Kohn-Vogelius Functional and Implicit Regularization).
• Prior knowledge: typical biological tissue lies roughly between $10^{-2}$ and $2$ S/m (bone and fat low, blood and cerebrospinal fluid high). The best metallic conductor, silver, is about $6.3\times 10^7$ S/m. After normalising to a reference, simulations often use $\sigma \in [{\sigma }_{\min },1]$ with a small ${\sigma }_{\min }>0$.

Enforcing them.

• Bound-constrained optimisers: L-BFGS-B, or projected gradient steps $\sigma \leftarrow P_{\Sigma }(\sigma -\tau \nabla \Phi )$ with $P_{\Sigma }$ = clipping.
• As the regulariser ${\iota }_{\Sigma }$ in ADMM, whose prox is clipping.
• Reparametrisation: $\sigma ={\sigma }_{\min }+e^s$, or a sigmoid, turns the problem into an unconstrained one. This changes the geometry of the problem and the gradient ($\partial \sigma /\partial s$ chain factor).

In ModularEIT.jl: minimize.

References

1. L. Borcea (2002). Electrical impedance tomography. Inverse Problems 18(6), R99–R136. doi:10.1088/0266-5611/18/6/201
2. C. Gabriel, S. Gabriel, E. Corthout (1996). The dielectric properties of biological tissues: I. Literature survey. Phys. Med. Biol. 41(11), 2231–2249. doi:10.1088/0031-9155/41/11/001
3. R. H. Byrd, P. Lu, J. Nocedal, C. Zhu (1995). A Limited Memory Algorithm for Bound Constrained Optimization. SIAM J. Sci. Comput. 16(5), 1190–1208. doi:10.1137/0916069