Anomaly Detection in Random Heterogeneous Media

1. Front Matter

   Pages 1-18

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2. Probabilistic interpretation of EIT

   1. Front Matter

      Pages 9-9

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   2. Mathematical setting

      • Martin Simon

      Pages 11-22

   3. Feynman-Kac formulae

      • Martin Simon

      Pages 23-51

3. Anomaly detection in heterogeneous media

   1. Front Matter

      Pages 53-53

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   2. Stochastic homogenization: Theory and numerics

      • Martin Simon

      Pages 55-89

   3. Statistical inversion

      • Martin Simon

      Pages 91-110

4. Back Matter

   Pages 111-150

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This monograph is concerned with the analysis and numerical solution of a stochastic inverse anomaly detection problem in electrical impedance tomography (EIT). Martin Simon studies the problem of detecting a parameterized anomaly in an isotropic, stationary and ergodic conductivity random field whose realizations are rapidly oscillating. For this purpose, he derives Feynman-Kac formulae to rigorously justify stochastic homogenization in the case of the underlying stochastic boundary value problem. The author combines techniques from the theory of partial differential equations and functional analysis with probabilistic ideas, paving the way to new mathematical theorems which may be fruitfully used in the treatment of the problem at hand. Moreover, the author proposes an efficient numerical method in the framework of Bayesian inversion for the practical solution of the stochastic inverse anomaly detection problem. 

• Calderón’s inverse conductivity problem
• EIT
• applied probability theory
• electrical impedance tomography
• random media
• statistical inverse problems
• stochastic analysis
• partial differential equations

• Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Mainz, Germany

  Martin Simon

Martin Simon has worked as a researcher at the Institute of Mathematics at the University of Mainz from 2008 to 2014. During this period he had several research stays at the University of Helsinki. He has recently joined an asset management company as a financial mathematician. 

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