Approximate marginalization of absorption and scattering in fluorescence diffuse optical tomography

Mozumder, Meghdoot; Tarvainen, Tanja; Arridge, Simon; Kaipio, Jari P.; D’Andrea, Cosimo; Kolehmainen, Ville

doi:10.3934/ipi.2016.10.227

Cited by 15 publications

(11 citation statements)

References 49 publications

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“…In the context of electrical impedance tomography (EIT), the BAE approach has been used to simultaneously premarginalize over the unknown domain shape and the contact impedances of the electrodes [20]. Furthermore, in [21], the approach was used to premarginalize over the distributed scattering coefficient in diffuse optical tomography (DOT) and, in [22], the method was used to premarginalize over both the scattering and absorption coefficients in the context of fluorescence diffuse optical tomography (fDOT). The BAE method has also been applied to X-ray tomography to premarginalize over distributed parameters outside a region of interest [23].…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 99%

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

2018

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“…On the other hand, the Gaussian approximation of the approximation error, which is guided by the BAE approach, might fail to adequately capture the distribution of the approximation error. However, as shown in various studies in the BAE literature, this Gaussian approximation is reasonable in broad classes of inverse problems; see, e.g., [24,26,31,32]. Additionally, in the present work we considered a greedy approach for tackling the binary OED optimization problems.…”

Section: Discussionmentioning

confidence: 92%

Optimal Design of Large-scale Bayesian Linear Inverse Problems Under Reducible Model Uncertainty: Good to Know What You Don't Know

Alexanderian¹,

Petra²,

Stadler³

et al. 2021

SIAM/ASA J. Uncertainty Quantification

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We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unfied framework. We build on the Bayesian approximation error (BAE) framework, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.

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“…The Bayesian approximation error approach was originally used to take into account the modeling errors induced by numerical model reduction Somersalo, 2006, 2007). It is still used as such today, but it is also applied to handle various approximation and modeling errors in wide variety of inverse problems (Lehikoinen et al, 2007;Nissinen et al, 2007Nissinen et al, , 2011Kaipio and Kolehmainen, 2013;Koponen et al, 2014;Lähivaara et al, 2015;Mozumder et al, 2016;Nicholson et al, 2018). Once we specify our approximate models, in the BAE, any errors induced by the use of simplified models, reducing the dimension of the parameter space, and/or model uncertainties are embedded into a single additive error term.…”

Section: Introductionmentioning

confidence: 99%

A Bayesian-based approach to improving acoustic Born waveform inversion of seismic data for viscoelastic media

Muhumuza,

Roininen,

Huttunen

et al. 2019

Preprint

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In seismic waveform inversion, the reconstruction of the subsurface properties is usually carried out using approximative wave propagation models to ensure computational efficiency. The viscoelastic nature of the subsurface is often unaccounted for, and two popular approximations-the acoustic and linearized Born inversion-are widely used. This leads to reconstruction errors since the approximations ignore realistic (physical) aspects of seismic wave propagation in the heterogeneous earth. In this study, we show that the Bayesian approximation error approach can be used to partially recover from errors, addressing elastic and viscous effects in acoustic Born inversion for viscoelastic media. The results of numerical examples indicate that neglecting the modeling errors induced by the approximations results in very poor recovery of the subsurface velocity fields.

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Approximate marginalization of absorption and scattering in fluorescence diffuse optical tomography

Cited by 15 publications

References 49 publications

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

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

Optimal Design of Large-scale Bayesian Linear Inverse Problems Under Reducible Model Uncertainty: Good to Know What You Don't Know

A Bayesian-based approach to improving acoustic Born waveform inversion of seismic data for viscoelastic media

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