TY - JOUR

T1 - Uncertainty quantification in three-dimensional magnetohydrodynamic equilibrium reconstruction via surrogate-assisted Bayesian inference

AU - W7-X Team

AU - Köberl, Robert

AU - von Toussaint, Udo

AU - Bungartz, Hans Joachim

AU - Schilling, Jonathan

AU - Albert, Christopher G.

N1 - Publisher Copyright:
© 2023 The Authors. Contributions to Plasma Physics published by Wiley-VCH GmbH.

PY - 2023/6/1

Y1 - 2023/6/1

N2 - 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.

AB - 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.

KW - equilibrium reconstruction

KW - magnetohydrodynamics

KW - polynomial chaos expansion

KW - uncertainty quantification

UR - http://www.scopus.com/inward/record.url?scp=85159083578&partnerID=8YFLogxK

U2 - 10.1002/ctpp.202200173

DO - 10.1002/ctpp.202200173

M3 - Article

AN - SCOPUS:85159083578

SN - 0863-1042

VL - 63

JO - Contributions to Plasma Physics

JF - Contributions to Plasma Physics

IS - 5-6

M1 - e202200173

ER -