Estimation of linear functionals from indirect noisy data without knowledge of the noise level
- PDF / 185,056 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 75 Downloads / 190 Views
Estimation of linear functionals from indirect noisy data without knowledge of the noise level Sergei V. Pereverzev · Bernd Hofmann
Received: 16 April 2010 / Accepted: 14 May 2010 / Published online: 22 June 2010 © Springer-Verlag 2010
Abstract In this paper we discuss how one can avoid the use of information about levels of data noise and operator perturbations in the regularization of ill-posed linear operator equations. We present an approach that allows an estimation of linear functionals on the solutions with the best possible order of the accuracy uniformly over classes of solutions and admissible functionals. Proposed approach is based on the concept of distance functions and employs a deterministic version of the balancing principle. We argue that this approach can be of interest in satellite geodesy. Keywords Ill-posed problems · Data-functional strategy · Distance functions · Satellite gravity gradiometry Mathematics Subject Classification (2000)
47A52 · 65J20 · 86A22 · 92F05
1 Introduction The question addressed in the present paper is: “How can the value of a bounded linear functional be estimated from indirect noisy observation without any knowledge of noise?” Suppose we observe data yξ of the form yξ = Ax + ξ,
(1.1)
S. V. Pereverzev (B) Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Science, Altenbergstrasse 69, 4040 Linz, Austria e-mail: [email protected] B. Hofmann Department of Mathematics, Chemnitz University of Technology, Chemnitz 09107, Germany e-mail: [email protected]
123
122
Int J Geomath (2010) 1:121–131
where A is an injective linear operator acting between Hilbert spaces X and Y . Moreover, x is an element of a convex subset M ⊂ X and ξ is a noise vector. We are interested in estimating the value of the linear functional F(x), and wish to do this in such a way as to minimize the error occurring at the worst x ∈ M. To motivate our interest in estimating F(x), note that in many geoscientific applications the quantities of interest correspond to the values of bounded linear functionals defined on the solution of some observation equation. For example, satellite gravimetry mission GOCE (Gravity field and steady-state Ocean Circulation Explorer), launched in March 2009 by ESA, aims to develop a model of the Earth’s gravity field from satellite observations. In mathematical terms, the observation model leads to an equation of the form (1.1) (see, e.g. Freeden and Pereverzev 2001), where x is the unknown gravity potential at the Earth’s surface. The GOCE mission strives for a high-accuracy, high-resolution model of the Earth’s static gravity field, represented by spherical harmonic coefficients of the Earth’s gravity potential x. Each such coefficient is the value of a corresponding linear functional. Therefore, for the mission success it is desirable to have a reliable strategy for estimating the values of linear functionals. At this point it is worth to remind that the observation equations (1.1) for the estimati
Data Loading...
