Estimation of linear functionals from indirect noisy data without knowledge of the noise level

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.

[1]  On the analysis of distance functions for linear ill-posed problems with an application to the integration operator in L 2 , 2007 .

[2]  B. Hofmann.  Approximate source conditions in Tikhonov–Phillips regularization and consequences for inverse problems with multiplication operators , 2006 .

[3]  F. Natterer.  Error bounds for tikhonov regularization in hilbert scales , 1984 .

[4]  Willi Freeden,et al.  Handbook of geomathematics , 2010 .

[5]  Peter Maass,et al.  A mollifier method for linear operator equations of the first kind , 1990 .

[6]  D. Donoho.  Statistical Estimation and Optimal Recovery , 1994 .

[7]  J. Kusche,et al.  Regularization of gravity field estimation from satellite gravity gradients , 2002 .

[8]  Bernd Hofmann,et al.  Analysis of Profile Functions for General Linear Regularization Methods , 2007, SIAM J. Numer. Anal..

[9]  S. Svensson.  Solution of the altimetry-gravimetry problem , 1983 .

[10]  Gernot Plank,et al.  Assessment of three numerical solution strategies for gravity field recovery from GOCE satellite gravity gradiometry implemented on a parallel platform , 2002 .

[11]  Bernd Hofmann,et al.  Convergence rates for Tikhonov regularization from different kinds of smoothness conditions , 2006 .

[12]  Alfred K. Louis.  Approximate inverse for linear and some nonlinear problems , 1995 .

[13]  Peter Mathé,et al.  Regularization of some linear ill-posed problems with discretized random noisy data , 2006, Math. Comput..

[14]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[15]  Peter Mathé,et al.  Direct Estimation of Linear Functionals from Indirect Noisy Observations , 2002, J. Complex..

[16]  Willi Freeden,et al.  Spherical Tikhonov regularization wavelets in satellite gravity gradiometry with random noise , 2001 .

[17]  F. Bauer,et al.  Local Solutions to Inverse Problems in Geodesy , 2006 .

[18]  Peiliang Xu,et al.  A Simulation Study of Smoothness Methods In Recovery of Regional Gravity Fields , 1994 .

[19]  Functional Methods for Gravity Field Approximation , 1986 .

[20]  R. S. Anderssen,et al.  The Linear Functional Strategy for Improperly Posed Problems , 1986 .

[21]  A. Tikhonov.  On the stability of inverse problems , 1943 .

[22]  Michael A. Golberg,et al.  A method of adjoints for solving some Ill-posed equations of the first kind , 1979 .

[23]  Peiliang Xu.  Truncated SVD methods for discrete linear ill-posed problems , 1998 .