The EMS Publishing House is now EMS Press and has its new home at

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen

Full-Text PDF (1366 KB) | Metadata | Table of Contents | ZAA summary
Volume 15, Issue 4, 1996, pp. 961–984
DOI: 10.4171/ZAA/740

Published online: 1996-12-31

Optimal Stable Solution of Cauchy Problems for Elliptic Equations

Ulrich Tautenhahn

We consider ill-posed Cauchy problems for elliptic partial differential equations $u_{tt} - Lu = 0 (0 < t ≤ T, x \in \Omega \subset \mathbb R^n)$ with linear densely defined self-adjoint and positive definite operators$ L : D(L) \subset H \to H$ where $H$ denotes a Hilbert space with norm $\| \cdot \|$ and inner product $(\cdot, \cdot)$. We assume that instead of exact data $y = u(x,0)$ or $y = u_t(x,0)$ noisy data $y^{\delta} = u^{\delta} (x,0)$ or $y^{\delta} = u(x,0)$ are available, respectively, with $\|y – y^{\delta} \| ≤ \delta$. Furthermore we assume certain smoothness conditions $u(x,t) \in M$ with appropriate sets $M$ and answer the question concerning the best possible accuracy for identifying $u(x,t)$ from the noisy data. For special sets $M$ the best possible accuracy depends either in a Hölder continuous way or in a logarithmic way on the noise level $\delta$. Furthermore, we discuss special regularization methods which realize this best possible accuracy.

Keywords: Ill-posed problems, elliptic partial differential equations, Cauchy problems, optimal regularization methods, optimal error bounds

Tautenhahn Ulrich: Optimal Stable Solution of Cauchy Problems for Elliptic Equations. Z. Anal. Anwend. 15 (1996), 961-984. doi: 10.4171/ZAA/740