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.
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.
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.
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.
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.
This work is funded by the FFG Project (Bridge 1) TomoFlow under the FFG Project Number 6833795 in cooperation with voestalpine Stahl GmbH.
Neumayer, M., Bretterklieber, T., Flatscher, M. and Puttinger, S. (2017), "PCA based state reduction for inverse problems using prior information", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 36 No. 5, pp. 1430-1441. https://doi.org/10.1108/COMPEL-02-2017-0090Download as .RIS
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