- yes (6) (remove)
- Abel integral equations (1)
- Complexity (1)
- Complexity and performance of numerical algorithms (1)
- Gaussian random noise (1)
- Ill-Posed Problems (1)
- Improperly posed problems (1)
- Inverse problems in Banach spaces (1)
- Inverses Problem (1)
- Lavrentiev regularization for equations with monotone operators (1)
- Linear Integral Equations (1)
- On the efficient discretization of integral equations of the third kind (1998)
- We propose a new discretization scheme for solving ill-posed integral equations of the third kind. Combining this scheme with Morozov's discrepancy principle for Landweber iteration we show that for some classes of equations in such method a number of arithmetic operations of smaller order than in collocation method is required to appoximately solve an equation with the same accuracy.
- Error estimates for band-limited spherical regularization wavelets in some inverse problems of satellite geodesy (1999)
- In this paper we discuss a special class of regularization methods for solving the satellite gravity gradiometry problem in a spherical framework based on band-limited spherical regularization wavelets. Considering such wavelets as a reesult of a combination of some regularization methods with Galerkin discretization based on the spherical harmonic system we obtain the error estimates of regularized solutions as well as the estimates for regularization parameters and parameters of band-limitation.
- Brakhage's implicit iteration method and Information Complexity of equations with operators having closed range (1999)
- An a posteriori stopping rule connected with monitoringthe norm of second residual is introduced forBrakhage's implicit nonstationary iteration method, applied to ill-posed problems involving linear operatorswith closed range. It is also shown that for someclasses of equations with such operators the algorithmconsisting in combination of Brakhage's method withsome new discretization scheme is order optimal in the sense of Information Complexity.
- Morozov's discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite-dimensional subspaces (2000)
- On the adaptive selection of the parameter in regularization of ill-posed problems (2003)
- We study a possiblity to use the structure of the regularization error for a posteriori choice of the regularization parameter. As a result, a rather general form of a selection criterion is proposed, and its relation to the heuristical quasi-optimality principle of Tikhonov and Glasko (1964), and to an adaptation scheme proposed in a statistical context by Lepskii (1990), is discussed. The advantages of the proposed criterion are illustrated by using such examples as self-regularization of the trapezoidal rule for noisy Abel-type integral equations, Lavrentiev regularization for non-linear ill-posed problems and an inverse problem of the two-dimensional profile reconstruction.
- Regularization without Preliminary Knowledge of Smoothness and Error Behavior (2004)
- The mathematical formulation of many physical problems results in the task of inverting a compact operator. The only known sensible solution technique is regularization which poses a severe problem in itself. Classically one dealt with deterministic noise models and required both the knowledge of smoothness of the solution function and the overall error behavior. We will show that we can guarantee an asymptotically optimal regularization for a physically motivated noise model under no assumptions for the smoothness and rather weak assumptions on the noise behavior which can mostly obtained out of two input data sets. An application to the determination of the gravitational field out of satellite data will be shown.