Computation and Control pp 145-154 | Cite as

# Prediction Bands for ILL-Posed Problems

Chapter

## Abstract

Consider the linear ill-posed operator equation where The ill-posedness of the problem (see [1]) means that a small perturbation in the data

$$Kx = z$$

(1.1)

*K*:*X*→*Y*is an operator between two Hilbert spaces with the inner product denoted by$$\left\langle {x,y} \right\rangle ,\,x,y\, \in \mathcal{X}\,or\,x,y\, \in \,\mathcal{Y}\,$$

*z*may result in large changes in the solution to (1.1). This discontinuous dependence of the solution on the data requires regularization in order to approximately solve the ill-posed equation.## Keywords

Cumulative Distribution Function True Solution Singular System Truncation Level Spectral Filter
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

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*The Theory of Tikhonov Regulation for Fred-holm Equations of the First Kind*, Pitman Boston, 1984.Google Scholar - [2]C. W. Groetsch, C. R. Vogel, “Asymptotic Theory of Filtering for Linear Operator Equations with Discrete Noisy Data”,
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© Birkhäuser Boston 1989