Identification of nonlinear elliptic equations (Q1911771)

The authors study the identification of the nonlinearities \(a\) and \(b\) in the following boundary value problems: \[ -\text{div}(a(\nabla y))\ni f\quad\text{in }\Omega,\quad y=0\quad\text{on }\partial\Omega \] and \[ -\Delta y+b(y)\ni f\quad\text{in }\Omega,\quad y=0\quad\text{on }\partial\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with Lipschitzian boundary, and \(f\in L^2(\Omega)\). If \(n=1\), \(\Omega=(0,1)\) is considered. The functions \(a\) and \(b\) are chosen from certain classes of monotone functions from \(\mathbb{R}^n\) to \(\mathbb{R}^n\) (resp. \(\mathbb{R}^1\) to \(\mathbb{R}^1\)). An optimization theoretic approach and an algorithm for the estimation of state-dependent coefficients \(a\) and \(b\) are presented. The identification problem is formulated as a nonlinear least-squares problem, which is approximated and analyzed in the framework of convex analysis. The algorithm is based on a nonlinear control formulation of the modified least-squares approach.