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Noise Models for Ill-Posed Problems

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Handbook of Geomathematics

Abstract

The standard view of noise in ill-posed problems is that it is either deterministic and small (strongly bounded noise) or random and large (not necessarily small). Following Eggerment, LaRiccia and Nashed (2009), a new noise model is investigated, wherein the noise is weakly bounded. Roughly speaking, this means that local averages of the noise are small. A precise definition is given in a Hilbert space setting, and Tikhonov regularization of ill-posed problems with weakly bounded noise is analysed. The analysis unifies the treatment of “classical” ill-posed problems with strongly bounded noise with that of ill-posed problems with weakly bounded noise. Regularization parameter selection is discussed, and an example on numerical differentiation is presented.

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Author information

Authors and Affiliations

  1. Food and Resource Economics, University of Delaware, Newark, DE, USA

    Paul N. Eggermont & Vincent LaRiccia

  2. Department of Mathematics, University of Central Florida, Orlando, FL, USA

    M. Zuhair Nashed

Authors
  1. Paul N. Eggermont
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  2. Vincent LaRiccia
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  3. M. Zuhair Nashed
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Corresponding author

Correspondence to Paul N. Eggermont .

Editor information

Editors and Affiliations

  1. Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    M. Zuhair Nashed

  3. Computational Mathematics, Technische Universität Braunschweig, Braunschweig, Germany

    Thomas Sonar

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Eggermont, P.N., LaRiccia, V., Nashed, M.Z. (2015). Noise Models for Ill-Posed Problems. In: Freeden, W., Nashed, M., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54551-1_24

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