EFFICIENT COMPUTATIONAL ALGORITHMS FOR MAGNETIC EQUILIBRIUM IN A FUSION REACTOR

Abstract

In a magnetic confinement fusion reactor, like a Tokamak, hydrogen isotopes are injected into the chamber and heated to form a plasma. The hot plasma tends to expand, so it's crucial to confine it within the center of the reactor to prevent contact with the reactor walls. This confinement is achieved through magnetic fields generated by currents running through external coils surrounding the reactor. However, these currents may suffer from uncertainties, which could arise from factors like temperature fluctuations and material impurities. These variables introduce a level of unpredictability in the plasma's behavior and the overall stability of the fusion process.

This thesis aims to investigate the impact that stochasticity in the current intensities has on the confinement properties of the plasma and to estimate the expected behavior of the magnetic field. While the focus is on the variability in current intensities, the tools developed can be applied to other sources of uncertainty, such as the positioning of coils and the source term parameters. To quantify the variability in model predictions and to evaluate the statistical properties of solutions over a range of parameter values, traditional sampling methods like Monte Carlo, often require intensive and expensive nonlinear computations. To tackle this challenge, we propose three approaches.

Firstly, we focus on the development and application of a surrogate function, constructed via a stochastic collocation approach on a sparse grid in the parameter space. This surrogate function is employed to replace the nonlinear solution in Monte Carlo sampling processes. For our numerical experiments, we evaluate the efficiency and accuracy of the outcomes produced by the surrogate, in comparison with those obtained through direct nonlinear solutions. Our findings indicate that a carefully selected surrogate function reduces the sampling cost -- achieving acceleration factors ranging from 7 to over 30 -- while preserving the accuracy of the results.

The second part of the thesis explores the multilevel Monte Carlo approach, investigating its potential for cost savings compared to simple Monte Carlo. This method involves conducting the majority of computations on a sequence of coarser spatial grids compared to what a simple Monte Carlo simulation would typically use. We examine this approach with non-linear computation, using both uniformly refined meshes and adaptively refined grids guided by a discrete error estimator. Numerical experiments reveal substantial cost reductions achieved through multilevel methods, typically ranging from a factor of 60 to exceeding 200. Adaptive gridding results in more accurate computation of relevant geometric parameters.

In the last part of this work, we explore hybridmethods that integrate surrogates with multilevel Monte Carlo to further reduce the sampling cost. We establish the optimal construction and sampling costs for the surrogate-based multilevel Monte Carlo. Numerical results demonstrate that surrogate-based multilevel Monte Carlo remarkably reduces the computational burden, requiring only 0.1 to 14 seconds for a target relative mean square error ranging from $8\times 10^{-3}$ to $2\times10^{-4}$, reducing the cost of direct computation by factors of 50 to 300. In terms of accuracy, the surrogate-based sampling results exhibit close congruence with those obtained via direct computation, both in plasma boundary and geometric descriptors.

The primary contributions of our work entail the application of stochastic collocation techniques and multilevel Monte Carlo methods to analyze plasma behavior under uncertainties in current within fusion reactors. Furthermore, we establish the universal sampling cost for the surrogate-enhanced multilevel Monte Carlo approach. Our methodology presents a paradigm in how we approach and manage computational challenges in this field.