JournalszaaVol. 15, No. 2pp. 475–493

On Optimal Regularization Methods for the Backward Heat Equation

  • Ulrich Tautenhahn

    University of Applied Sciences, Zittau, Germany
  • T. Schröter

    Technische Universität Chemnitz, Germany
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Abstract

In this paper we consider different regularization methods for solving the heat equation u1+Au=0(0t<T)u_1 + Au = 0 (0 ≤ t < T) backward in time, where A:HHA : H \to H is a linear (unbounded) operator in a Hilbert space HH with norm \| \cdot \| and zδz^{\delta} are the available (noisy) data for u(T)u(T) with zδu(T)δ\| z^{\delta} - u(T)\| ≤ \delta. Assuming u0E\|u{0}\| ≤ E we consider different regularized solutions qαδ(t)q^{\delta}_{\alpha} (t) for u(t)u(t) and discuss the question how to choose the regularization parameter α=α(δ,E,t)\alpha = \alpha (\delta, E, t) in order to obtain optimal estimates supqαδ(t)u(t)E1tTδtT\| q^{\delta}_{\alpha} (t) - u(t)\| ≤ E^{1 – \frac{t}{T} \delta \frac{t}{T}} where the supremum is taken over zδH,u(0)Ez^{\delta} \in H, \|u(0)\| ≤ E and zδu(T)δ\|z^{\delta} - u(T)\| ≤ \delta.

Cite this article

Ulrich Tautenhahn, T. Schröter, On Optimal Regularization Methods for the Backward Heat Equation. Z. Anal. Anwend. 15 (1996), no. 2, pp. 475–493

DOI 10.4171/ZAA/711