UNCERTAINTY QUANTIFICATION AND WEAK APPROXIMATION OF AN ELLIPTIC INVERSE PROBLEM

We consider the inverse problem of determining the permeability from the pressure in a Darcy model of flow in a porous medium. Mathematically the problem is to find the diffusion coefficient for a linear uniformly elliptic partial differential equation in divergence form, in a bounded domain in dime...

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Bibliographic Details
Published in:SIAM journal on numerical analysis Vol. 49; no. 5/6; pp. 2524 - 2542
Main Authors: DASHTI, M., STUART, A. M.
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
Subjects:
Approximation
Banach space
Banach spaces
Covariance
Diffusion coefficient
Elliptic differential equations
Exact sciences and technology
Inverse problems
Mathematical analysis
Mathematical functions
Mathematical theorems
Mathematics
Numerical analysis
Numerical analysis in abstract spaces
Numerical analysis. Scientific computation
Partial differential equations
Partial differential equations, boundary value problems
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Permeability
Quantification
Random variables
Sciences and techniques of general use
Truncation
uncertainty quantification
Approximation
Error estimation
Initial value problem
Well posed problem
Banach space
Partial differential equation
Inverse problem
Truncation
35J99
Weak convergence
Bayesian approach
elliptic inverse problem
Numerical analysis
Porous medium flow
Elliptic equation
Boundary value problem
Linear equation
inverse problems
Lipschitz function
62G99
ISSN:0036-1429, 1095-7170
Online Access:Get full text
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Summary:We consider the inverse problem of determining the permeability from the pressure in a Darcy model of flow in a porous medium. Mathematically the problem is to find the diffusion coefficient for a linear uniformly elliptic partial differential equation in divergence form, in a bounded domain in dimension d < 3, from measurements of the solution in the interior. We adopt a Bayesian approach to the problem. We place a prior random field measure on the log permeability, specified through the Karhunen-Loève expansion of its draws. We consider Gaussian measures constructed this way, and study the regularity of functions drawn from them. We also study the Lipschitz properties of the observation operator mapping the log permeability to the observations. Combining these regularity and continuity estimates, we show that the posterior measure is well defined on a suitable Banach space. Furthermore the posterior measure is shown to be Lipschitz with respect to the data in the Hellinger metric, giving rise to a form of well posedness of the inverse problem. Determining the posterior measure, given the data, solves the problem of uncertainty quantification for this inverse problem. In practice the posterior measure must be approximated in a finite dimensional space. We quantify the errors incurred by employing a truncated Karhunen-Loève expansion to represent this meausure. In particular we study weak convergence of a general class of locally Lipscfritz functions of the log permeability, and apply this general theory to estimate errors in the posterior mean of the pressure and the pressure covariance, under refinement of the finite-dimensional Karhunen-Loève truncation.
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ISSN:0036-1429
1095-7170
DOI:10.1137/100814664