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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| Veröffentlicht in: | SIAM journal on numerical analysis Jg. 49; H. 5/6; S. 2524 - 2542 |
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| Sprache: | Englisch |
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Society for Industrial and Applied Mathematics
01.01.2011
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| Abstract | 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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| AbstractList | 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. 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 $d \le 3$, 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 Lipschitz 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. [PUBLICATION ABSTRACT] |
| Author | DASHTI, M. STUART, A. M. |
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| Cites_doi | 10.1029/96WR00160 10.1088/0266-5611/25/11/115008 10.1137/060663660 10.1142/S0219530511001728 10.1016/S0045-7825(98)00100-5 10.1002/zamm.200800095 10.1016/j.jcp.2003.09.015 10.1137/090770734 10.1016/j.jcp.2006.01.048 10.1017/S0962492910000061 10.1137/S0036142902418680 10.1016/j.cma.2004.04.008 10.1137/050645142 10.3934/ipi.2009.3.87 10.1137/070680540 10.1016/j.cma.2008.08.019 |
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| Keywords | 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 |
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| SubjectTerms | 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 |
| Title | UNCERTAINTY QUANTIFICATION AND WEAK APPROXIMATION OF AN ELLIPTIC INVERSE PROBLEM |
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