A convergent reconstruction method for an elliptic operator in potential form (Q1909984)

The author considers the equation \(-\Delta u+ qu= 0\) in the unit square in \(\mathbb{R}^2\) with boundary conditions given for \(u(0, y)\), \(u(1, y)\), \(u_y(x, 0)\), \(u_y(x, 1)\) for \(x, y\in (0, 1)\). The potential \(q\) is assumed to be a function of the variable \(x\) alone. The problem is to reconstruct \(q\) from the measurement of the data \(u(x, 1)\), \(x\in (0, 1)\), which are assumed to lie in the Hölder space \(C^\alpha[0, 1]\). The method developed in the paper is to show, that \(q\) is the fixed point of a certain contraction in a ball in \(C^\alpha[0, 1]\), if suitable smallness conditions on the inhomogeneous boundary data are imposed. Some numerical results are presented.