Surrogate-based multilevel Monte Carlo methods for uncertainty quantification in the Grad-Shafranov free boundary problem

Abstract

We explore a hybrid technique to quantify the variability in the numerical solutions to a free boundary problem associated with magnetic equilibrium in axisymmetric fusion reactors amidst parameter uncertainties. The method aims at reducing computational costs by integrating a surrogate model into a multilevel Monte Carlo method. The resulting surrogate-enhanced multilevel Monte Carlo methods reduce the cost of simulation by factors as large as $10^4$ compared to standard Monte Carlo simulations involving direct numerical solutions of the associated Grad-Shafranov partial differential equation. Accuracy assessments also show that surrogate-based sampling closely aligns with the results of direct computation, confirming its effectiveness in capturing the behavior of plasma boundary and geometric descriptors.

Figures (4)

• Figure 1: Schematic of the cross section of a tokamak. The solid black line represents the wall of the reactor. The grey rectangles represent the coils, each located in a region denoted as $\Omega_{c_i}$, with the index $i$ running over the total number of coils present in the device. The violet region $\Omega_p$ represents the region occupied by the plasma. Due to the problem's (anti)symmetry, it is enough for the analysis to focus on the right half side of the diagram.
• Figure 2: Left: Mean CPU time per realization vs. number of spatial grid points $M_\ell$ for both direct and surrogate computations. The computing time was obtained by averaging over 100 realizations. Right: Offline costs of construction of both single spatial level and multi spatial level surrogates.
• Figure 3: Left: Estimated sampling CPU time vs. tolerance $\epsilon$. Right: value of $V_\ell$ vs. number of spatial grid points $M_\ell$.
• Figure 4: The overlayed plasma boundaries of 50 random realizations are displayed as violet curves. The solid violet line is the plasma boundary of the expected poloidal flux generated with tolerance $\epsilon=4\times 10^{-4}$. On the leftmost panel, the inner and outer walls of the reactor are displayed in solid black and dark red respectively, while the green square highlights the zoomed--in in the remaining images. The dark green dots are the x-points of the expected solution. All simulations were performed using the discretization level $\ell=5$ on geometry-conforming uniform meshes.

Theorems & Definitions (4)

• Remark 1: Efficiency of multi-level sampling
• Theorem 1
• Lemma 1
• Remark 2