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Uncertainty quantification and weak approximation of an elliptic inverse problem

Masoumeh Dashti, Andrew M. Stuart

arXiv:1102.0143v1math.STstat.TH

TL;DR

The paper asks how to infer permeability from pressure observations when the unknown coefficient has a lognormal prior and the resulting posterior is infinite-dimensional. It establishes conditions for a well-defined, data-stable posterior and analyzes finite-dimensional approximation. The resulting theory gives weak-error control for posterior approximation and Hellinger continuity with respect to data.

  • Problem

    Inferring permeability from pressure observations requires a Bayesian posterior on function space, where the unknown diffusion coefficient is conditioned on data.

    Source: §1 · pp. 1–2

  • Method

    The paper combines Gaussian-prior regularity with boundedness and Lipschitz conditions for the observation operator, then studies truncated Karhunen–Loève approximation.

    Source: §1 · p. 3, §2.3 · p. 6

  • Results

    The posterior is well-defined and continuous in the Hellinger metric with respect to data, while finite-dimensional truncation has weak error bounded by Cψ(N).

    Source: §2.3 · p. 6, §4.1 · p. 11

  • Takeaways & Limitations

    The theory supports uncertainty quantification through posterior measures and weak approximation of functions of the log permeability.

    Source: §1 · p. 3, §2.3 · p. 6

    Observations are restricted to point evaluations with sufficient regularity or functionals on H1(D) when pressure lies in H1(D), with mean-zero Gaussian observational noise.

    Source: §1 · p. 2

Abstract

from arXiv · show

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-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.

1. Introduction.

The paper studies Bayesian recovery of a log permeability from noisy pressure observations, focusing on lognormal priors and finite-dimensional approximation. It develops conditions for a well-defined, data-stable posterior and analyzes approximation of that measure.

1. Introduction.

  • The inverse problem conditions an unknown diffusion coefficient on observations, yielding a more complicated measure than forward uncertainty analysis with a truncated random-field expansion.Lognormal priors are attractive when prior permeability bounds are unknown.Source: §1 · p. 1
  • The forward model is an elliptic Darcy equation in which pressure is the solution and exp(u) is the permeability; observations are noisy linear functionals of pressure.The functionals may be point evaluations or functionals on H1(D), subject to sufficient regularity.Source: §1 · p. 2
  • The Bayesian formulation maps log permeability through an observation operator and interprets the posterior on function space as a Radon–Nikodym derivative with respect to the prior.The paper derives conditions under which pressure observations can be treated as functions of the log permeability.Source: §1 · p. 2
  • The paper establishes function-space Bayes results and studies posterior stability under finite-dimensional prior approximation, alongside boundedness and Lipschitz continuity of the elliptic observation operator.These operator properties are shown for appropriate Banach spaces.Source: §1 · p. 3

2. Bayesian approach to inverse problems for functions.

The section develops Bayesian inverse-problem results on function spaces, including conditions for a well-defined, data-continuous posterior and its approximation. It applies these results to finite-dimensional Karhunen–Loève truncations and bounds errors in posterior expectations.

2.1. Well-defined and well-posed Bayesian inverse problems.

  • Under conditions on Φ and a Gaussian prior supported on X, the posterior is well-defined and continuous in the data.The section also recalls Hellinger continuity and posterior approximation results.Source: §2 · p. 3, §2.1 · p. 4

2. Bayesian approach to inverse problems for functions.

  • The framework separates analytic properties of Φ from choosing a Gaussian prior supported on X, and permits infinite-dimensional data or non-Gaussian observational error.The section describes these results as widely applicable beyond the specific inverse problem.Source: §2 · p. 3

2.2. Approximation of the posterior.

  • Approximating Φ controls the resulting posterior error when the likelihood approximations satisfy the stated uniform conditions.The section presents this result as a basis for finite-dimensional posterior approximation.Source: §2.2 · pp. 4–5

2.3. Approximating the posterior measure in a finite dimensional space.

  • The finite-dimensional approximation truncates the Karhunen–Loève basis and uses the resulting posterior on retained coefficients to estimate expectations.The target expectations may be of pressure fields or pressure covariances; the approximating expectations use G(uN, 0) under νN.Source: §2.3 · pp. 5–6
  • The weak error for expectations of suitable functions of the log permeability is bounded by Cψ(N) as the finite-dimensional truncation is refined.The result assumes the stated conditions on the Banach spaces, Gaussian prior, likelihood, and function being averaged.Source: §2.3 · p. 6
  • The finite-dimensional approximation results concern a broad class of Bayesian inverse problems with Gaussian priors; extension to Besov priors is deferred.The paper points to for the Besov priors and says that generalization will be presented elsewhere.Source: §2.3 · p. 7

3. Estimates on the observation operator.

The section establishes boundedness and Lipschitz properties of the observation operator for two types of pressure measurements. The required assumptions depend on whether observations are bounded linear functionals or pointwise evaluations.

3.1. Measurements from bounded linear functionals of p ∈H1(D) .

  • For d=2,3, Proposition 3.1 assumes bounded D with C1-regular boundary, f in H−1(D), g in L2(D), and φ extendable to H1(D); Corollary 3.2 considers G(u)=(l1(p),...,lK(p)) for bounded linear functionals lj on H1(D).Source: §3.1 · pp. 7–8

3.2. Pointwise measurements of p.

  • For pointwise pressure measurements, the solution map from C( D̄ ) to L∞(D) is locally Lipschitz continuous.The pointwise-measurement setting assumes f in Lr(D), g in L2r(D), and r > d/2, with the stated boundary and extension conditions.Source: §3.2 · pp. 8, 10
  • The pointwise-measurement estimates also establish Lipschitz continuity under additional regularity conditions on the permeability and boundary data.The cited result requires one log-permeability input in Ct(D) for t > 0 and φ extendable to W1,2r(D).Source: §3.2 · pp. 9–10

4. Properties of the posterior measure for the elliptic inverse problem.

The paper establishes posterior well-posedness for the elliptic inverse problem and analyzes finite-dimensional approximation using Fourier truncation. The approximation results address errors in pressure uncertainty summaries, while well-posedness is shown under specified prior regularity conditions.

4.1. Well-posedness of the posterior measure.

  • On the torus, the posterior is absolutely continuous with respect to the Gaussian prior and continuous in the Hellinger metric with respect to the data.The theorem assumes Assumption 3.6 and a prior N(0, (−∆)−s) with s > d/2.Source: §4.1 · pp. 10–11
  • For a general bounded domain, the stated prior regularity supports posterior well-posedness when s > d − 1.The passage specifies s > d − 1/2 for the prior to assign probability one to Ct(D), and states well-posedness provided s > d − 1.Source: §4.1 · p. 11

W N = span{ϕk1,...,kd, |k1| ≤N, . . . , |kd| ≤N}

  • Theorem 4.2 estimates weak-approximation errors in the posterior pressure mean and covariance under Fourier truncation of the Karhunen–Loève expansion.These quantities quantify uncertainty in pressure conditioned on the data.Source: W N = span{ϕk1,...,kd, |k1| ≤N, . . .… · p. 12
  • The approximation argument combines Lipschitz properties of pressure or pressure covariance with bounds on the Fourier-truncation projection.The proof uses X = Ct and X′ = X′′ = L∞ under the theorem’s assumptions.Source: W N = span{ϕk1,...,kd, |k1| ≤N, . . .… · pp. 12, 15
  • In two dimensions, the pressure approximation error is bounded by C ∥u∥Ct(D) N^-t (log N)^2 for x, y ∈ T2.The bound applies to the stated Fourier truncation estimate.Source: W N = span{ϕk1,...,kd, |k1| ≤N, . . .… · p. 15

5. Conclusion.

The paper treats Bayesian inference for the diffusion coefficient in a uniformly elliptic divergence-form PDE from noisy solution observations. It establishes posterior well-posedness and develops weak error estimates for finite-dimensional Karhunen–Loève truncation.

5. Conclusion.

  • The paper addresses recovery of a diffusion coefficient from noisy observations of the solution to a uniformly elliptic divergence-form PDE using a Bayesian approach.It applies prior results on posterior well-definedness and well-posedness to this inverse problem.Source: §5 · p. 15
  • A general theorem on weak approximation of the posterior under finite-dimensional Karhunen–Loève truncation is applied to derive Fourier-truncation error estimates.The estimates concern the posterior measure’s weak error.Source: §5 · p. 15
  • The paper identifies application to other inverse problems, rare-event analysis, improved convergence rates under stronger conditions on G, and non-Gaussian Besov-type priors as future work.Source: §5 · p. 15

Appendix A. Let D ⊂Rd be open and bounded and p ∈W 1,q

The appendix develops estimates for the pressure’s W 1,q norm on bounded domains, including boundary effects. It also considers how Hölder continuity of the diffusion coefficient affects the estimate’s dependence on its norm.

Appendix A. Let D ⊂Rd be open and bounded and p ∈W 1,q

  • The appendix’s estimates bound the pressure’s W 1,q norm while tracking dependence on the diffusion coefficient.The argument uses local interior and boundary results to obtain an estimate on a general bounded domain.Source: Appendix A · p. 15
  • For a Hölder-continuous diffusion coefficient, the appendix obtains a pressure estimate with polynomial dependence on its Ct(D) norm.The appendix states that this dependence is considered under the assumptions of Theorem A-3.Source: Appendix A · pp. 15, 18
  • The estimates apply for 2 < q < 6, with the stated general-domain theorem covering bounded C1 domains in dimensions 2 and 3.The theorem passage specifies a continuous diffusion coefficient.Source: Appendix A · p. 16
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