Optimal Stable Solution of Cauchy Problems for Elliptic Equations

Abstract

We consider ill-posed Cauchy problems for elliptic partial differential equations $u_{tt}-Lu=0(0<t\le T,x\in \Omega \subset {\mathbb{R}}^n)$ 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 $\Vert \cdot \Vert$ 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 $\Vert y–y^{\delta }\Vert \le \delta$. 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.