Mathematical Programming

, Volume 134, Issue 1, pp 101–125 | Cite as

Robust inversion, dimensionality reduction, and randomized sampling

  • Aleksandr Aravkin
  • Michael P. Friedlander
  • Felix J. Herrmann
  • Tristan van Leeuwen
Full Length Paper Series B

Abstract

We consider a class of inverse problems in which the forward model is the solution operator to linear ODEs or PDEs. This class admits several dimensionality-reduction techniques based on data averaging or sampling, which are especially useful for large-scale problems. We survey these approaches and their connection to stochastic optimization. The data-averaging approach is only viable, however, for a least-squares misfit, which is sensitive to outliers in the data and artifacts unexplained by the forward model. This motivates us to propose a robust formulation based on the Student’s t-distribution of the error. We demonstrate how the corresponding penalty function, together with the sampling approach, can obtain good results for a large-scale seismic inverse problem with 50 % corrupted data.

Keywords

Inverse problems Seismic inversion Stochastic optimization Robust estimation 

Mathematics Subject Classification

90C06 49N45 

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Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  • Aleksandr Aravkin
    • 1
  • Michael P. Friedlander
    • 2
  • Felix J. Herrmann
    • 1
  • Tristan van Leeuwen
    • 1
  1. 1.Department of Earth and Ocean SciencesUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada

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