Least squares polynomial chaos expansion: A review of sampling strategies
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A new methodology is proposed for generating realizations of a random vector with values in a finite-dimensional Euclidean space that are statistically consistent with a dataset of observations of this vector. The probability distribution of this random vector, while a priori not known, is presumed to be concentrated on an unknown subset of the Euclidean space. A random matrix is introduced whose columns are independent copies of the random vector and for which the number of columns is the number of data points in the dataset. The approach is based on the use of (i) the multidimensional kernel-density estimation method for estimating the probability distribution of the random matrix, (ii) a MCMC method for generating realizations for the random matrix, (iii) the diffusion-maps approach for discovering and characterizing the geometry and the structure of the dataset, and (iv) a reduced-order representation of the random matrix, which is constructed using the diffusion-maps vectors associated with the first eigenvalues of the transition matrix relative to the given dataset. The convergence aspects of the proposed methodology are analyzed and a numerical validation is explored through three applications of increasing complexity. The proposed method is found to be robust to noise levels and data complexity as well as to the intrinsic dimension of data and the size of experimental datasets. Both the methodology and the underlying mathematical framework presented in this paper contribute new capabilities and perspectives at the interface of uncertainty quantification, statistical data analysis, stochastic modeling and associated statistical inverse problems.
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Abstract The International Monitoring System (IMS) is part of the verification regime for the Comprehensive Nuclear-Test-Ban-Treaty Organization (CTBTO). At entry-into-force, half of the 80 radionuclide stations will be able to measure concentrations of several radioactive xenon isotopes produced in nuclear explosions, and then the full network may be populated with xenon monitoring afterward (Bowyer et al., 2013). Fission-based production of 99Mo for medical purposes also releases radioxenon isotopes to the atmosphere (Saey, 2009). One of the ways to mitigate the effect of emissions from medical isotope production is the use of stack monitoring data, if it were available, so that the effect of radioactive xenon emissions could be subtracted from the effect from a presumed nuclear explosion, when detected at an IMS station location. To date, no studies have addressed the impacts the time resolution or data accuracy of stack monitoring data have on predicted concentrations at an IMS station location. Recently, participants from seven nations used atmospheric transport modeling to predict the time-history of 133Xe concentration measurements at an IMS station in Germany using stack monitoring data from a medical isotope production facility in Belgium. Participants received only stack monitoring data and used the atmospheric transport model and meteorological data of their choice. Some of the models predicted the highest measured concentrations quite well (a high composite statistical model comparison rank or a small mean square error with the measured values). The results suggest release data on a 15 min time spacing is best. The model comparison rank and ensemble analysis suggests that combining multiple models may provide more accurate predicted concentrations than any single model. Further research is needed to identify optimal methods for selecting ensemble members and those methods may depend on the specific transport problem. None of the submissions based only on the stack monitoring data predicted the small measured concentrations very well. The one submission that best predicted small concentrations also included releases from nuclear power plants. Modeling of sources by other nuclear facilities with smaller releases than medical isotope production facilities may be important in discriminating those releases from releases from a nuclear explosion.
Pacific Northwest National Laboratory (PNNL) is developing a prototype spectral x-ray radiography (SpecX) non-destructive assay system for noninvasive material mass quantification of uranium-bearing powders. The SpecX system uses a custom material estimation algorithm, which requires a high-fidelity pixel-by-pixel detector response model for accurate quantification. We performed a detailed characterization and built a model of a High-Energy X-ray Imaging Technology (HEXITEC) CdTe pixelated spectral detector. Here we present the detector characterization which led to that model, including a determination of the pixel-wise energy calibration, the detection efficiency, the detector uniformity, the detector stability, and a comparison of methods to remove charge-shared events.
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