Full Length Paper Series B

Mathematical Programming

, Volume 134, Issue 1, pp 101-125

Robust inversion, dimensionality reduction, and randomized sampling

  • Aleksandr AravkinAffiliated withDepartment of Earth and Ocean Sciences, University of British Columbia
  • , Michael P. FriedlanderAffiliated withDepartment of Computer Science, University of British Columbia Email author 
  • , Felix J. HerrmannAffiliated withDepartment of Earth and Ocean Sciences, University of British Columbia
  • , Tristan van LeeuwenAffiliated withDepartment of Earth and Ocean Sciences, University of British Columbia

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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