We consider ill-posed Cauchy problems for elliptic partial differential equations with linear densely defined self-adjoint and positive definite operators where denotes a Hilbert space with norm and inner product . We assume that instead of exact data or noisy data or are available, respectively, with . Furthermore we assume certain smoothness conditions with appropriate sets and answer the question concerning the best possible accuracy for identifying from the noisy data. For special sets the best possible accuracy depends either in a Hölder continuous way or in a logarithmic way on the noise level . Furthermore, we discuss special regularization methods which realize this best possible accuracy.
Cite this article
Ulrich Tautenhahn, Optimal Stable Solution of Cauchy Problems for Elliptic Equations. Z. Anal. Anwend. 15 (1996), no. 4, pp. 961–984DOI 10.4171/ZAA/740