3,099 research outputs found
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
dimension , 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-Lo\`eve 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-Lo\`eve 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-Lo\`eve truncation.Comment: 19 pages, 0 figures, submitted to SIAM Journal on Numerical Analysi
Sparse Deterministic Approximation of Bayesian Inverse Problems
We present a parametric deterministic formulation of Bayesian inverse
problems with input parameter from infinite dimensional, separable Banach
spaces. In this formulation, the forward problems are parametric, deterministic
elliptic partial differential equations, and the inverse problem is to
determine the unknown, parametric deterministic coefficients from noisy
observations comprising linear functionals of the solution.
We prove a generalized polynomial chaos representation of the posterior
density with respect to the prior measure, given noisy observational data. We
analyze the sparsity of the posterior density in terms of the summability of
the input data's coefficient sequence. To this end, we estimate the
fluctuations in the prior. We exhibit sufficient conditions on the prior model
in order for approximations of the posterior density to converge at a given
algebraic rate, in terms of the number of unknowns appearing in the
parameteric representation of the prior measure. Similar sparsity and
approximation results are also exhibited for the solution and covariance of the
elliptic partial differential equation under the posterior. These results then
form the basis for efficient uncertainty quantification, in the presence of
data with noise
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