Sobolev error estimates and a priori parameter selection for semi-discrete Tikhonov regularization
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J. Krebs
Abstract
The goal of this paper is to investigate the Tikhonov–Phillips method for semi-discrete linear ill-posed problems in order to determine tight error bounds and to obtain a good parameter choice. We consider the equation Aƒ = g, where the operator A is known, noisy discrete data 
with 
can be observed and a solution ƒ* is sought. Assuming that ƒ* is an element of a Sobolev space, we use the well-known theory of optimal recovery in Hilbert spaces for the reconstruction process. We then provide L2-error estimates in terms of the data density and derive an a priori selection for the regularization parameter, which guarantees an optimal compromise between approximation and stability. Finally, we illustrate the parameter selection with a simple example.
© de Gruyter 2009
Articles in the same Issue
- An iterative regularization method for ill-posed Hammerstein type operator equation
- Sobolev error estimates and a priori parameter selection for semi-discrete Tikhonov regularization
- Complexity analysis of the iteratively regularized Gauss–Newton method with inner CG-iteration
- Regularization methods for a Cauchy problem for a parabolic equation in multiple dimensions
- A new version of quasi-boundary value method for a 1-D nonlinear ill-posed heat problem
Articles in the same Issue
- An iterative regularization method for ill-posed Hammerstein type operator equation
- Sobolev error estimates and a priori parameter selection for semi-discrete Tikhonov regularization
- Complexity analysis of the iteratively regularized Gauss–Newton method with inner CG-iteration
- Regularization methods for a Cauchy problem for a parabolic equation in multiple dimensions
- A new version of quasi-boundary value method for a 1-D nonlinear ill-posed heat problem
