On Optimal Regularization Methods for the Backward Heat Equation

Abstract

In this paper we consider different regularization methods for solving the heat equation u + Au = 0 (0 < i < T) backward in time, where A : H-, H is a linear (unbounded) operator in a Hubert space H with norm and z 6 are the available (noisy) data for u(T) with 11 z6-u(T)ii < 5. Assuming 11 u(0)11 < E we consider different regularized solutions q(t) for u(t) and discuss the question how to choose the regularization parameter = cs(5,E,t) in order to obtain optimal estimates sup q(t)-u(t)11 < E'+'&+ where the supremum is taken over z 6 E H, ll u (0)11 < E and 11 z6-u(T)II < 5.

A simple regularization method for the ill-posed evolution equation

“…So, the right-hand side of (16) is logarithmic stability estimate. This logarithmic order is also given in [2], [9], [14], [11], [21], [22].…”

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

“…Problems of this type have been under consideration, e. g., in the papers [16,50,61,62,72,74] and are one of the classical ill-posed problems with various engineering applications, see, e.g., [2,5,48] and the references cited there.…”

On an initial inverse problem in nonlinear heat equation associated with time-dependent coefficient

“…(1) a) In Tautenhahn and Schröter [13], the authors regularized the homogeneous problem (f = 0) and showed that the best possible estimate of the worst case error is given by…”

“…Tautenhahn and Schröter [13] established an optimal error estimate for (2). Liu in [6] introduced a group preserving scheme.…”