The authors consider the estimation of the diffusion coefficient \(a\) in \(-\text{div}(a \text{grad} u)+cu=f\) in \(\Omega\subset\mathbb{R}^ n\) where \(u\), \(c\) and \(f\) are known, and a feasible boundary condition is satisfied by \(u\) on the boundary \(\partial\Omega\). The problem can be formulated as inverting the parameter-to-solution mapping \(a\to u(a)\) at \(z\), i.e., to solve \(u(a)=z\). The authors study the regularized versions of the problem, e.g. \(\min\{{1\over 2}\| u(a)-z\|^ 2+{\beta\over 2}\langle a,Pa\rangle\}\), \(a\in Q_{ad}\), \(\langle a,Pa\rangle\leq\gamma\), where \(P\) is a bounded linear selfadjoint nonnegative operator. A model function technique is proposed to iteratively determine optimal values of regularization parameters \(\beta>0\) and/or \(\gamma>0\), and to estimate the error in the data if it is not known a priori. Numerical examples are given in case \(n=1\).