Regularized parameter identification in elliptic boundary value problems (Q1118994)

In the one-dimensional two-point boundary value problem \(-(au_ x)_ x=f\), \(x\in (0,1)\), with Dirichlet, Neumann, or mixed boundary conditions at \(x=0\) and \(x=1\), the diffusion coefficient \(a=a(x)\) is identified from measured u-values \(z=z(x)\) using a Tikhonov regularization approach: \(\| u-z\|^ 2+\epsilon \| a-b\|^ 2\to \min,\) where norms are in \(L^ 2(0,1)\), b is a suitable estimate for a, and \(\epsilon\) is a small regularization parameter. For solving the discretized and continuous nonlinear minimization problems the Gauss-Newton method is applied. An effective solution procedure is described which requires the solution of a 2nd order problem and a 4th order problem in each time step.