APPROXIMATION OF BAYESIAN INVERSE PROBLEMS FOR PDES

Inverse problems are often ill posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is based on an approach to regularization, employing a Bayesian...

Full description

Saved in:
Bibliographic Details
Published in:SIAM journal on numerical analysis 2010-01, Vol.48 (1), p.322-345
Main Authors: COTTER, S. L., DASHTI, M., STUART, A. M.
Format: Article
Language:English
Subjects:
Airy equation
Approximation
Banach space
Bayesian analysis
Covariance
Data assimilation
Estimates
Forward problem
Inverse problems
Lagrangian function
Markov analysis
Mathematical analysis
Mathematical functions
Mathematical models
Navier Stokes equation
Navier-Stokes equations
Partial differential equations
Stability
Studies
Velocity distribution
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Inverse problems are often ill posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is based on an approach to regularization, employing a Bayesian formulation of the problem, which leads to a notion of well posedness for inverse problems, at the level of probability measures. The stability which results from this well posedness may be used as the basis for quantifying the approximation, in finite dimensional spaces, of inverse problems for functions. This paper contains a theory which utilizes this stability property to estimate the distance between the true and approximate posterior distributions, in the Hellinger metric, in terms of error estimates for approximation of the underlying forward problem. This is potentially useful as it allows for the transfer of estimates from the numerical analysis of forward problems into estimates for the solution of the related inverse problem. It is noteworthy that, when the prior is a Gaussian random field model, controlling differences in the Hellinger metric leads to control on the differences between expected values of polynomially bounded functions and operators, including the mean and covariance operator. The ideas are applied to some non-Gaussian inverse problems where the goal is determination of the initial condition for the Stokes or Navier-Stokes equation from Lagrangian and Eulerian observations, respectively.
ISSN:0036-1429
1095-7170
DOI:10.1137/090770734