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Zeitschrift für Analysis und ihre Anwendungen

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Volume 15, Issue 2, 1996, pp. 475–493
DOI: 10.4171/ZAA/711

Published online: 1996-06-30

On Optimal Regularization Methods for the Backward Heat Equation

Ulrich Tautenhahn and T. Schröter[1]

(1) Technische Universität Chemnitz, Germany

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

Keywords: Backward heat equation, optimal parameter choice, optimal error bounds, regularization methods

Tautenhahn Ulrich, Schröter T.: On Optimal Regularization Methods for the Backward Heat Equation. Z. Anal. Anwend. 15 (1996), 475-493. doi: 10.4171/ZAA/711