JournalszaaVol. 15, No. 4pp. 961–984

Optimal Stable Solution of Cauchy Problems for Elliptic Equations

  • Ulrich Tautenhahn

    University of Applied Sciences, Zittau, Germany
Optimal Stable Solution of Cauchy Problems for Elliptic Equations cover
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Abstract

We consider ill-posed Cauchy problems for elliptic partial differential equations uttLu=0(0<tT,xΩRn)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)HHL : D(L) \subset H \to H where HH denotes a Hilbert space with norm \| \cdot \| and inner product (,)(\cdot, \cdot). We assume that instead of exact data y=u(x,0)y = u(x,0) or y=ut(x,0)y = u_t(x,0) noisy data yδ=uδ(x,0)y^{\delta} = u^{\delta} (x,0) or yδ=u(x,0)y^{\delta} = u(x,0) are available, respectively, with yyδδ\|y – y^{\delta} \| ≤ \delta. Furthermore we assume certain smoothness conditions u(x,t)Mu(x,t) \in M with appropriate sets MM and answer the question concerning the best possible accuracy for identifying u(x,t)u(x,t) from the noisy data. For special sets MM the best possible accuracy depends either in a Hölder continuous way or in a logarithmic way on the noise level δ\delta. 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. 961–984

DOI 10.4171/ZAA/740