The determination of a coefficient in an elliptic equation from average flux data

We consider the question of recovering the coefficient q from the equation −Δuj+q(x)uj=ƒj(x) subject to homogeneous Dirichlet boundary conditions in a bounded domain Ω⊂R2. The nonhomogeneous source terms {ƒj(x)}j=1∞ form a basis for L2(Ω). It will be proven that a unique determination is possible from data measurements consisting of measurements of the net flux {∫r∂uj/∂vds}⋚1 leaving a subset Γ of the boundary ∂Ω for each input source ƒj. A continuous dependence result and an algorithm that allows efficient numerical reconstruction of q(x) from finite data is presented.