Uncertainty quantification in three‐dimensional magnetohydrodynamic equilibrium reconstruction via surrogate‐assisted Bayesian inference

Abstract

In three‐dimensional (3D) equilibrium, reconstruction defining parameters of an ideal magneto‐hydrodynamic equilibrium are inferred from a set of plasma diagnostic measurements. For the reconstructed parameters, various forms of uncertainty estimates exist within common 3D reconstruction frameworks. These estimates often assume a Gaussian posterior distribution. The validity of this assumption is not obvious in such highly nonlinear inverse problems, and therefore the accuracy of the estimates cannot be guaranteed. In this work, we formulate the problem of 3D equilibrium reconstruction in a Bayesian sense and explore the posterior distribution of reconstruction parameters via Markov chain Monte Carlo (MCMC) sampling. The target reconstruction parameters, that is, shape and scaling factors of the pressure and toroidal current‐density profiles as well as the total toroidal flux, are taken from a reduced subspace of Wendelstein 7‐X equilibrium configurations. Since the corresponding forward model evaluations are computationally demanding, we replace the forward model via a polynomial chaos expansion surrogate. We compare the posterior distribution obtained via MCMC sampling to Laplace's approximation, which assumes a Gaussian posterior. We find that the two approaches provide similar results in regimes where the uncertainty on the plasma diagnostic signals is low. However, discrepancies between the posteriors are observed in cases of higher diagnostic signal uncertainty. Therefore, we provide further validation of the commonly used Laplace's approximation as a method of uncertainty quantification in 3D equilibrium reconstruction and showcase some of its limitations.

Magnetohydrodynamic Equilibrium Reconstruction with Consistent Uncertainties

“…We further constrain the pressure profile to be monotonic and 0 at the plasma boundary and enforce these constraints with exponentiation and integration, as well as a renormalization. For the current profile GP, a smooth rational quadratic kernel [17] is used, since detailed information on the current profile shape can be hard to capture with magnetic diagnostics alone [11] and the toroidal current is expected to be comparatively low for standard W7-X equilibria. Again, exponentiation and integration ensure monotonicity, but in contrast to the pressure profile, the current is 0 at the magnetic axis, and the scaling factor can be positive or negative.…”

“…This makes 3D equilibrium reconstruction a fixed-point problem that is traditionally solved iteratively in a least-squares sense [3,6,7]. While the least-squares approach can result in suitable reconstructed parameters and uncertainty estimates [8], evaluations of the forward model are computationally demanding (a single equilibrium reconstruction can take up to several hours [9,10]), and common error estimates might fail to capture relevant uncertainties when high levels of noise are present for the equilibrium diagnostics [11]. Existing Bayesian frameworks [12] are also limited by the high computational costs of the forward model.…”

Special issue: Machine learning methods in plasma physics

“…Emulators are often used to perform sensitivity studies or inferences that are otherwise numerically challenging using the underlying simulator. The paper of Köberl et al [ 7 ] demonstrates such an application, analyzing the uncertainty of a 3D magnetohydrodynamic equilibrium reconstruction using an emulator based on polynomial chaos expansions. While other machine‐learning methodologies are useful, neural networks are the most common emulators due to the availability of open‐source libraries and generic applicability.…”