Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography

Roininen, Lassi; Huttunen, Janne M. J.; Lasanen, Sari

doi:10.3934/ipi.2014.8.561

Cited by 116 publications

(152 citation statements)

References 16 publications

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“…In particular, due to the definition of κ (see (2.1)), the magnitude of the error depends on the ratio δ/ρ, and, asymptotically, the error decreases exponentially as this ratio increases. This explains the observations in [40] and [47], namely, that the error in the covariance is negligible if the distance of ∂D ext from ∂D is greater than the correlation length ρ, i.e. δ/ρ > 2.…”

Section: Theorem 32 (Main Result)mentioning

confidence: 74%

“…In this way, the boundary effects coming from the domain truncation are negligible in D if the window is large enough. In [39,47] an empirical rule for the window size is suggested, where it is observed that the window boundary should be at least as far away from D as the correlation length of the Matérn field. However, a precise error analysis has not been carried out to date.…”

mentioning

confidence: 99%

“…Moreover, they give rise to analytic expressions for the covariance error. Robin boundary conditions for the SPDE on the bounded domain have been considered in [17,47]. Robin conditions involve a coefficient which has to be tuned.…”

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confidence: 99%

See 2 more Smart Citations

Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields

Khristenko¹,

Scarabosio²,

Swierczynski³

et al. 2019

SIAM/ASA J. Uncertainty Quantification

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We consider the generation of samples of a mean-zero Gaussian random field with Matérn covariance function. Every sample requires the solution of a differential equation with Gaussian white noise forcing, formulated on a bounded computational domain. This introduces unwanted boundary effects since the stochastic partial differential equation is originally posed on the whole R d , without boundary conditions. We use a window technique, whereby one embeds the computational domain into a larger domain, and postulates convenient boundary conditions on the extended domain. To mitigate the pollution from the artificial boundary it has been suggested in numerical studies to choose a window size that is at least as large as the correlation length of the Matérn field. We provide a rigorous analysis for the error in the covariance introduced by the window technique, for homogeneous Dirichlet, homogeneous Neumann, and periodic boundary conditions. We show that the error decays exponentially in the window size, independently of the type of boundary condition. We conduct numerical experiments in 1D and 2D space, confirming our theoretical results.

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Section: Theorem 32 (Main Result)mentioning

confidence: 74%

mentioning

confidence: 99%

See 1 more Smart Citation

Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields

Khristenko¹,

Scarabosio²,

Swierczynski³

et al. 2019

SIAM/ASA J. Uncertainty Quantification

View full text Add to dashboard Cite

show abstract

“…In addition, we take the prior on a(x) to be a Gaussian measure, µ a = N (a * , C a ) on L 2 (Ω) with C a defined similarly as C β in Section 2. To be precise, we use a [31]. Far right: The weighted approach of the current paper as discussed in Section 2, with κ β = 0 (homogeneous Neumann).…”

Section: Problem Setupmentioning

confidence: 99%

Estimation of the Robin coefficient field in a Poisson problem with uncertain conductivity field

2018

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We consider the reconstruction of a heterogeneous coefficient field in a Robin boundary condition on an inaccessible part of the boundary in a Poisson problem with an uncertain (or unknown) inhomogeneous conductivity field in the interior of the domain. To account for model errors that stem from the uncertainty in the conductivity coefficient, we treat the unknown conductivity as a nuisance parameter and carry out approximative premarginalization over it, and invert for the Robin coefficient field only. We approximate the related modelling errors via the Bayesian approximation error (BAE) approach. The uncertainty analysis presented here relies on a local linearization of the parameter-to-observable map at the maximum a posteriori (MAP) estimates, which leads to a normal (Gaussian) approximation of the parameter posterior density. To compute the MAP point we apply an inexact Newton conjugate gradient approach based on the adjoint methodology. The construction of the covariance is made tractable by invoking a low-rank approximation of the data misfit component of the Hessian. Two numerical experiments are considered: one where the prior covariance on the conductivity is isotropic, and one where the prior covariance on the conductivity is anisotropic. Results are compared to those based on standard error models, with particular emphasis on the feasibility of the posterior uncertainty estimates. We show that the BAE approach is a feasible one in the sense that the predicted posterior uncertainty is consistent with the actual estimation errors, while neglecting the related modelling error yields infeasible estimates for the Robin coefficient. In addition, we demonstrate that the BAE approach is approximately as computationally expensive (measured in the number of PDE solves) as the conventional error approach.

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“…each other. For example, Gaussian priors with Matérn covariance operator C = C α,β,γ = β(I + γ 2 ∆) −α [36,12] are singular for different values of α > 0 or β > 0. We refer to [12] for a further discussion and for a particular subclass of equivalent Gaussian priors with Matérn covariance.…”

Section: Robustness In Hellinger Distancementioning

confidence: 99%

On the local Lipschitz stability of Bayesian inverse problems

Sprungk

2020

Inverse Problems

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In this note we consider the robustness of posterior measures occuring in Bayesian inference w.r.t. perturbations of the prior measure and the log-likelihood function. This extends the well-posedness analysis of Bayesian inverse problems. In particular, we prove a general local Lipschitz continuous dependence of the posterior on the prior and the log-likelihood w.r.t. various common distances of probability measures. These include the Hellinger and Wasserstein distance and the Kullback-Leibler divergence. We only assume the boundedness of the likelihoods and measure their perturbations in an L p -norm w.r.t. the prior. The obtained robustness yields under mild assumptions the well-posedness of Bayesian inverse problems, in particular, a well-posedness w.r.t. the Wasserstein distance which is missing in the existing literature. Moreover, our results indicate an increasing sensitivity of Bayesian inference as the posterior becomes more concentrated, e.g., due to more or more accurate data. This confirms and extends previous observations made in the sensitivity analysis of Bayesian inference.

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Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography

Cited by 116 publications

References 16 publications

Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields

Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields

Estimation of the Robin coefficient field in a Poisson problem with uncertain conductivity field

On the local Lipschitz stability of Bayesian inverse problems

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