Statistics and Computing

, Volume 25, Issue 4, pp 727–737

Sequential Monte Carlo methods for Bayesian elliptic inverse problems

  • Alexandros Beskos
  • Ajay Jasra
  • Ege A. Muzaffer
  • Andrew M. Stuart
Article

Abstract

In this article, we consider a Bayesian inverse problem associated to elliptic partial differential equations in two and three dimensions. This class of inverse problems is important in applications such as hydrology, but the complexity of the link function between unknown field and measurements can make it difficult to draw inference from the associated posterior. We prove that for this inverse problem a basic sequential Monte Carlo (SMC) method has a Monte Carlo rate of convergence with constants which are independent of the dimension of the discretization of the problem; indeed convergence of the SMC method is established in a function space setting. We also develop an enhancement of the SMC methods for inverse problems which were introduced in Kantas et al. (SIAM/ASA J Uncertain Quantif 2:464–489, 2014); the enhancement is designed to deal with the additional complexity of this elliptic inverse problem. The efficacy of the methodology and its desirable theoretical properties, are demonstrated for numerical examples in both two and three dimensions.

Keywords

Inverse problems Elliptic PDEs Groundwater flow  Adaptive SMC Markov chain Monte Carlo 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Alexandros Beskos
    • 1
  • Ajay Jasra
    • 2
  • Ege A. Muzaffer
    • 2
  • Andrew M. Stuart
    • 3
  1. 1.Department of Statistical ScienceUniversity College LondonLondonUK
  2. 2.Department of Statistics & Applied ProbabilityNational University of SingaporeSingaporeSingapore
  3. 3.Department of MathematicsUniversity of WarwickCoventryUK

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