EIT starts with Maxwell’s equations and ends, in the computer, with sparse symmetric linear systems. This article follows the chain of modelling and discretisation steps between the two.
1. The physical model
Under the Quasi-Static Approximation, Ohm’s Law in Continuum Form together with the Continuity Equation gives the Conductivity Equation
For alternating currents the conductivity becomes complex (see Complex Conductivity). How current enters through the electrodes is modelled by an electrode model, from the idealised continuum model to the Complete Electrode Model (see Electrode Models).
2. The analytic problem
Prescribing currents gives a Neumann Problem, prescribing voltages a Dirichlet Problem. Both are well posed in the right function spaces (see Sobolev and Trace Spaces). The Neumann problem determines the potential only up to a constant, which is fixed by a grounding condition. The Weak Formulation of the Conductivity Equation and the Lax-Milgram Theorem give existence and uniqueness. The Dirichlet and Thomson Principles characterise the solution by energy minimisation. The map from applied currents to measured voltages is the Neumann-to-Dirichlet Map. Sampled at the electrodes, it is the Forward Map.
3. Discretisation
The Galerkin Method restricts the weak form to a finite element space, e.g. Lagrange Finite Elements for the potential and piecewise constants for the conductivity. The integrals become matrices:
- the Weighted Stiffness Matrix , linear in σ, which the Conductivity Tensor makes explicit;
- the Mass Matrix, Stiffness Matrix and Boundary Mass and Stiffness Matrices for norms, projections and boundary terms;
- the injection and measurement operators of the Discrete Electrode Models.
The singular Neumann system needs its null space handled explicitly (see Null Space of the Neumann Problem, Grounding of the Potential). Dirichlet data are imposed by elimination (see Enforcing Dirichlet Conditions).
4. The linear systems
Every forward solve, every adjoint solve and every Jacobian evaluation solves
with a sparse, symmetric, positive semidefinite and one column per current pattern. Which solver fits depends on the mesh, the number of right-hand sides and how often σ changes (see Choosing a Linear Solver). How fine the mesh has to be, and where, is the subject of Meshes and Geometry.
References
- S. C. Brenner, L. R. Scott (2008). The Mathematical Theory of Finite Element Methods, 3rd ed. Springer. doi:10.1007/978-0-387-75934-0
- M. Cheney, D. Isaacson, J. C. Newell (1999). Electrical Impedance Tomography. SIAM Rev. 41(1), 85–101. doi:10.1137/S0036144598333613