Approximation errors in nonstationary inverse problems

Huttunen, Janne M. J.; Kaipio, Jari P.

doi:10.3934/ipi.2007.1.77

Cited by 52 publications

(69 citation statements)

References 12 publications

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“…In the framework of Bayesian inversion, the model error can be incorporated as an additional additive noise (Huttunen and Kaipio, 2007). Assuming independence, the probability density function of the sum of the two noises is given by:…”

Section: Accelerating Mcmc Methodsmentioning

confidence: 99%

Accelerated Markov chain Monte Carlo sampling in electrical capacitance tomography

Watzenig

Neumayer

Fox

2011

COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering

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Purpose -The purpose of this paper is to establish a cheap but accurate approximation of the forward map in electrical capacitance tomography in order to approach robust real-time inversion in the framework of Bayesian statistics based on Markov chain Monte Carlo (MCMC) sampling. Design/methodology/approach -Existing formulations and methods to reduce the order of the forward model with focus on electrical tomography are reviewed and compared. In this work, the problem of fast and robust estimation of shape and position of non-conducting inclusions in an otherwise uniform background is considered. The boundary of the inclusion is represented implicitly using an appropriate interpolation strategy based on radial basis functions. The inverse problem is formulated as Bayesian inference, with MCMC sampling used to efficiently explore the posterior distribution. An affine approximation to the forward map built over the state space is introduced to significantly reduce the reconstruction time, while maintaining spatial accuracy. It is shown that the proposed approximation is unbiased and the variance of the introduced additional model error is even smaller than the measurement error of the tomography instrumentation. Numerical examples are presented, avoiding all inverse crimes. Findings -Provides a consistent formulation of the affine approximation with application to imaging of binary mixtures in electrical tomography using MCMC sampling with Metropolis-Hastings-Green dynamics. Practical implications -The proposed cheap approximation indicates that accurate real-time inversion of capacitance data using statistical inversion is possible. Originality/value -The proposed approach demonstrates that a tolerably small increase in posterior uncertainty of relevant parameters, e.g. inclusion area and contour shape, is traded for a huge reduction in computing time without introducing bias in estimates. Furthermore, the proposed framework -approximated forward map combined with statistical inversion -can be applied to all kinds of soft-field tomography problems.

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Section: Accelerating Mcmc Methodsmentioning

confidence: 99%

Accelerated Markov chain Monte Carlo sampling in electrical capacitance tomography

Watzenig

Neumayer

Fox

2011

COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering

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“…Here, the temporally and spatially varying concentration of some substance is typically the physical entity of primary interest. In these problems, the stochasticity of the models is typically due to unknown boundary data for the CD model or the fluid dynamics model (Sepp€ anen, Vauhkonen, Somersalo, & Kaipio, 2001;Sepp€ anen, Vauhkonen, Vauhkonen, Somersalo, & Kaipio, 2001a;Voutilainen et al, 2007), unknown velocity field (Lipponen, Sepp€ anen, & Kaipio, 2010a,b;Sepp€ anen et al, 2001a), and highly approximate reduced-order models for the primary unknowns Huttunen, Kaipio, & Somersalo, 2007;Lipponen, Sepp€ anen, & Kaipio, 2011).…”

Section: Estimation Of Primary Variables and Flows Using The State Spmentioning

confidence: 99%

Process tomography and estimation of velocity fields

Kaipio¹,

Seppänen²,

Vauhkonen³

et al. 2015

Industrial Tomography

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“…The approximation error approach was extended to nonstationary inverse problems in [32] nonstationary (heat transfer) problems were considered, and in [33] and [34], in which nonlinear problems and state space identification problems were considered, respectively. a modification in which the approximation error statistics can be updated with accumulating information was proposed in [35] and an application to hydrogeophysical monitoring in [36].…”

Section: Introductionmentioning

confidence: 99%

Bayesian approximation error approach in full-wave ultrasound tomography

Koponen

Huttunen

Tarvainen

et al. 2014

IEEE Trans. Ultrason., Ferroelect., Freq. Contr.

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In ultrasound tomography, the spatial distribution of the speed of sound in a region of interest is reconstructed based on transient measurements made around the object. The computation of the forward problem (the full-wave solution) within the object is a computationally intensive task and can often be prohibitive for practical purposes. The purpose of this paper is to investigate the feasibility of using approximate forward solvers and the partial recovery from the related errors by employing the Bayesian approximation error approach. In addition to discretization error, we also investigate whether the approach can be used to reduce the reconstruction errors that are due to the adoption of approximate absorbing boundary models. We carry out two numerical studies in which the objective is to reduce the computational times to around 3% of the time that would be required by a numerically accurate forward solver. The results show that the Bayesian approximation error approach improves the reconstructions.

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Approximation errors in nonstationary inverse problems

Cited by 52 publications

References 12 publications

Accelerated Markov chain Monte Carlo sampling in electrical capacitance tomography

Accelerated Markov chain Monte Carlo sampling in electrical capacitance tomography

Process tomography and estimation of velocity fields

Bayesian approximation error approach in full-wave ultrasound tomography

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