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Purpose Inverse problems are often marked by highly dimensional state vectors. The high dimension affects the quality of the estimation result as well as the computational complexity of the estimation problem. This paper aims to present a state reduction technique based on prior knowledge.Design/methodology/approach Ill-posed inverse problems require prior knowledge to find a stable solution. The prior distribution is constructed for the high-dimensional data space. The authors use the prior distribution to construct a reduced state description based on a lower-dimensional basis, which they derive from the prior distribution. The approach is tested for the inverse problem of electrical capacitance tomography.Findings Based on a singular value decomposition of a sample-based prior distribution, a reduced state model can be constructed, which is based on principal components of the prior distribution. The approximation error of the reduced basis is evaluated, showing good behavior with respect to the achievable data reduction. Owing to the structure, the reduced state representation can be applied within existing algorithms.Practical implications The full state description is a linear function of the reduced state description. The reduced basis can be used within any existing reconstruction algorithm. Increased noise robustness has been found for the application of the reduced state description in a back projection-type reconstruction algorithm.Originality/value The paper presents the construction of a prior-based state reduction technique. Several applications of the reduced state description are discussed, reaching from the use in deterministic reconstruction methods up to proposal generation for computational Bayesian inference, e.g. Markov chain Monte Carlo techniques. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]