Statistical inverse problems: discretization, model reduction and inverse crimes

Authors:
Jari Kaipio;Erkki Somersalo
Affiliations:
Department of Applied Physics, University of Kuopio, Kuopio, Finland;Institute of Mathematics, Helsinki University of Technology, Finland
Venue:
Journal of Computational and Applied Mathematics - Special issue: Applied computational inverse problems
Year:
2007

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Abstract

The article discusses the discretization of linear inverse problems. When an inverse problem is formulated in terms of infinite-dimensional function spaces and then discretized for computational purposes, a discretization error appears. Since inverse problems are typically ill-posed, neglecting this error may have serious consequences to the quality of the reconstruction. The Bayesian paradigm provides tools to estimate the statistics of the discretization error that is made part of the measurement and modelling errors of the estimation problem. This approach also provides tools to reduce the dimensionality of inverse problems in a controlled manner. The ideas are demonstrated with a computed example.