Estimation of discontinuous parameters of elliptic partial differential equations by regularization for surface representations (Q2747524)

In 1998, \textit{A. Neubauer} and \textit{O. Scherzer} [SIAM J. Appl. Math. 58, 1891-1900 (1998; Zbl 0916.65128)] developed a new variant of regularization for problems with discontinuous solutions, regularization for curve representations. Kindermann and Neubauer have generalized the method to parameter estimation problems in the two-dimensional case. They call their method regularization for surface representations. The discontinuous functions of two variables are replaced by the continuous graphs using an appropriate parametrization. So the usual Tikhonov regularization for nonlinear ill-posed problems in Hilbert spaces becomes applicable. The authors consider the problem of estimating the parameter \(\gamma\) in the elliptic problem \(\text{div} (\gamma\nabla u)=f\) in \(\Omega=(0,1)^2\), \(u=0\) on \(\partial\Omega\) from measurements of \(u\). For solutions having a surface representation the general theory of Tikhonov regularization formulated by \textit{H. W. Engl}, \textit{M. Hanke} and \textit{A. Neubauer} [Regularization of inverse problems, Kluwer Academic Publishers (1996; Zbl 0859.65054)] ensures the convergence of regularized parameters \(\gamma_\alpha\) to the exact solution \(\gamma^+\) in the obvious noisy data context. The solution errors are measured in \(L^r(\Omega)\)-Norms for \(1\leq r<\infty\). Moreover, finite-dimensional approximations are analyzed and illustrative numerical results are presented.