Identification of conductivity by minimising a gradient co-linearity mismatch norm

Let ${\mit\Omega}\subset I\hspace{-3pt}R^2$ be bounded by two heteroclinic orbits, ${\mit\Gamma}_1, {\mit\Gamma}_2$ of the $\nabla u$-flow. Then $\nabla\cdot (c \nabla u)=0$ in ${\mit\Omega}$ implies $c\equiv 0$ in $\bar{\mit\Omega}$ [\textsc{Chicone} and \textsc{Gerlach}, 1987]. Let $u\in {\cal C}^2 ({\mit\Omega})\cap{\cal C}^0 (\bar{\mit\Omega})$ be known. The (unique) conductivity $\hat a$, which complies with $\nabla\cdot(\hat a\nabla u)=f$, can be identified by minimising with respect to $b$ the norm of $\nabla\tilde a [b] \times\nabla u - \nabla b\times\nabla p$ under constraints, where $\nabla\cdot(b\nabla p)=f$ and $\tilde a[b]\partial_j u := b\partial_j p$, $j=$ 1 \textbf{or} 2. This is an attempt at justifying the ``comparison model'' algorithm [\textsc{Scarascia} and \textsc{Ponzini}, 1972], which has seen successful practical applications to inverse hydrogeology ever since.