• The aim: to construct a reduced basis space.
• A space where we can find an approximation to a forward problem for any point in parameter space \(\mathscr{P}\).
• We build it by orthonormalising forward problem solutions at points \(\mathscr{P}_1, \mathscr{P}_2, ...\)
• These are called the reduced basis functions and make up our reduced basis space.

EIT formulation

\[\nabla\cdot(\sigma\nabla u)=0 \ \ \ \textrm{in }\Omega\\ \sigma\frac{\partial u}{\partial \textbf{n}} = \langle\sigma\nabla u,\textbf{n}\rangle=0 \ \ \ \textrm{on }\partial\Omega \backslash \cup^L_{l=1} e_l \\ u + z_l\sigma\frac{\partial u}{\partial \textbf{n}} = U_l \ \ \ \textrm{for } l=1,...,L \\ \int_{e_l}\sigma\frac{\partial u}{\partial \textbf{n}} d(\partial \Omega) = I_l \ \ \ \textrm{on } e_l \]

Variational Formulation

\[B((u,U),(v,V)) = \int_{\Omega} \sigma \nabla u \cdot \nabla v d \Omega \\ + \sum^L_{l=1} \frac{1}{z_l} \int_{e_l}(u-U_l)(v-V_l)d(\partial \Omega)\]

Ritz-Galerkin Discritization

\[\begin{gather} \begin{pmatrix} \boldsymbol A & - \boldsymbol B \\ - \boldsymbol B^T & \boldsymbol C\end{pmatrix} \begin{pmatrix}\boldsymbol \alpha \\ \boldsymbol \beta\end{pmatrix} = \begin{pmatrix}\boldsymbol 0 \\ \boldsymbol I \end{pmatrix} \label{CEM_sys} \end{gather}\]