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A new discrepancy principle (2005)
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| Venue: | J. Math. Anal. Appl |
| Citations: | 5 - 4 self |
BibTeX
@ARTICLE{Ramm05anew,
author = {A. G. Ramm},
title = {A new discrepancy principle},
journal = {J. Math. Anal. Appl},
year = {2005},
volume = {310},
pages = {342345}
}
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Abstract
The aim of this note is to prove a new discrepancy principle. The advantage of the new discrepancy principle compared with the known one consists of solving a minimization problem (see problem (2) below) approximately, rather than exactly, and in the proof of a stability result. To explain this in more detail, let us recall the usual discrepancy principle, which can be stated as follows. Consider an operator eqution Au = f, (1) where A: H → H is a bounded linear operator on a Hilbert space H, and assume that the range R(A) is not closed, so that problem (1) is ill-posed. Assume that f = Ay where y is the minimal-norm solution to (1), and that noisy data fδ are given, such that ||fδ − f| | ≤ δ. One wants to construct a stable approximation to y, given fδ. The variational regularization method for solving this problem consists of solving the minimization problem F (u): = ||Au − fδ| | 2 + ɛ||u| | 2 = min. (2) It is well known that problem (2) has a solution and this solution is unique (see e.g. [1]). Let uδ,ɛ solve (2). Consider the equation for finding ɛ = ɛ(δ):
