Bayesian soft X-ray tomography using non-stationary Gaussian Processes

Abstract

In this study, a Bayesian based non-stationary Gaussian Process method for the inference of soft X-ray emissivity distribution along with its associated uncertainties, has been developed. For the investigation of equilibrium condition and fast magnetohydrodynamic (MHD) behaviors in nuclear fusion plasmas, it is of importance to infer, especially in the plasma center, spatially resolved soft X-ray profiles from a limited number of noisy line integral measurements. For this ill-posed inversion problem, Bayesian probability theory can provide a posterior probability distribution over all possible solutions under given model assumptions. Specifically, the use of a non-stationary Gaussian Process to model the emission allows the model to adapt to the varying length scales of the underlying diffusion process. In contrast to other conventional methods, the prior regularization is realized in a probability form which enhances the capability of uncertainty analysis, in consequence, scientists who concern the reliability of their results will benefit from it. Under the assumption of normally distributed noise, the posterior distribution evaluated at a discrete number of points becomes a multivariate normal distribution whose mean and covariance are analytically available, making inversions and calculation of uncertainty fast. Additionally, the hyper-parameters embedded in the model assumption can be optimized through a Bayesian Occam"s Razor formalism and thereby automatically adjust the model complexity. This method is shown to produce convincing reconstructions and good agreements with independently calculated results from the Maximum Entropy (MaxEnt) and Equilibrium-Based Iterative Tomography Algorithm (EBITA) methods.

Inference of temperature and density profiles via forward modeling of an x-ray imaging crystal spectrometer within the Minerva Bayesian analysis framework

“…Compared to other common representations as e.g. basis function approaches 39,40 , Gaussian processes combines parametrization and regularization of profiles and also adjusts the model complexity through just one entity, the length scale parameter 22,38,41 . The model can be upgraded (see e.g.…”

“…The model can be upgraded (see e.g. Li et al 22 ) by allowing non stationary Gaussian processes, taking into account different length scales across the plasma radius. For the present temperature and density profile inference, this is however not required.…”

“…In the W7-X stellarator, the plasma cross-sections are not circular. Thus, more sophisticated inversion methods are required [20][21][22][23][24][25][26][27] . Some general challenges in unfolding line integrated spectral information are: A: Performing a simultaneous tomographic inversion of all plasma parameter profiles having direct impact on measured spectra.…”

Bayesian electron density inference from JET lithium beam emission spectra using Gaussian processes

“…(19). Due to the small deuterium pressure inside the tokamak during the beaminto-gas experiments, a strong beam attenuation is not expected.…”

“…Gaussian processes were introduced in the fusion community in [17] and are implemented as a standard representation of profile quantities in the Minerva framework [12]. It has been used for current tomography [17,18], soft x-ray tomography [19], and representing profile quantities [17,20,21]. The covariance function of a Gaussian process is defined as a parametrised function whose parameters, so called hyperparameters, determine aspects of the function such as overall scale and length scale.…”