On Optimal Regularization Methods for the Backward Heat Equation

Abstract

In this paper we consider different regularization methods for solving the heat equation $u_1+Au=0(0\le t<T)$ backward in time, where $A:H\to H$ is a linear (unbounded) operator in a Hilbert space $H$ with norm $\Vert \cdot \Vert$ and $z^{\delta }$ are the available (noisy) data for $u(T)$ with $\Vert z^{\delta }-u(T)\Vert \le \delta$. Assuming $\Vert u0\Vert \le E$ we consider different regularized solutions $q_{\alpha }^{\delta }(t)$ 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$\Vert q_{\alpha }^{\delta }(t)-u(t)\Vert \le E^{1–\frac{t}{T}\delta \frac{t}{T}}$ where the supremum is taken over $z^{\delta }\in H,\Vert u(0)\Vert \le E$ and $\Vert z^{\delta }-u(T)\Vert \le \delta$.