Deflation Techniques for Stellarator Equilibrium and Optimization

TL;DR

This paper introduces deflation as a practical method to explore the highly nonconvex landscape of stellarator equilibrium and optimization. By embedding deflation operators into nonlinear solves and optimization problems, the authors demonstrate the discovery of multiple equilibria and distinct local minima without relying on numerous diverse initial guesses. The approach yields families of equilibria with shared core properties or cores and helically perturbed states, and identifies multiple high quality coil configurations, highlighting both the method's effectiveness and its limitations. The work suggests avenues for methodological refinements and uses as a general tool for exploring complex stellarator design spaces with minimal additional implementation effort.

Abstract

Stellarator optimization is a multi-objective, non-convex problem characterized by a complex objective landscape containing many local minima. The solution resulting from a single optimization is highly sensitive to factors such as the initial guess, objective weights, and the optimization method employed. However, merely varying these factors does not guarantee that a physically distinct minimum will be found; optimizations often fail to converge to good minima or simply return to the same or very similar local minima despite large-scale parameter scans. This paper presents a novel application of deflation methods to effectively explore this landscape. By modifying the objective function to penalize and "deflate" away already-found solutions, this technique encourages the optimizer towards attractive, distinct new minima while using a single initial guess and optimization setup. We provide a primer on deflation for nonlinear systems and non-convex optimization before applying it to non-axisymmetric equilibrium and stellarator optimization problems. Key results include the discovery of families of global equilibria with similar core characteristics and the convergence to helical core equilibria without prescient initial guesses. Furthermore, we demonstrate that augmenting stage-one stellarator and stage-two coil optimization with deflation constraints readily produces multiple high-quality, distinct solutions, establishing the method's efficacy and ease of use.

Figures (11)

• Figure 1: Boundaries at two different toroidal angles for the 25 deflated first-order NAE-constrained equilibria.
• Figure 2: Rotational transform profiles and QS error for the 25 deflated first-order NAE-constrained equilibria.
• Figure 3: A comparison of two of the 25 deflated first-order NAE-constrained equilibria. (Left) The flux surfaces of the two equilibria at the $\phi=0$ cross-section, compared to the original NAE surfaces. (Right top) The rotational transform profile for the two solutions, with both matching well near-axis. (Right Bottom) The Boozer QS error metric for the two solutions, with both displaying the expected quadratic scaling of the QS error with $\rho$.
• Figure 4: The two branches of equilibrium found for the helical core state presented in cooper_tokamak_2010. (Left) The axisymmetric equilibrium state, found when solving from an axisymmetric initial condition. (Right) The helical core equilibrium, found when solving from the deflated equilibrium solve as an the initial state. The solved equilibrium's axis is shown as a black dot, while the deflated solve solution's is a red cross, showing that the deflated equilibrium problem encouraged an axis perturbation in the correct direction, without requiring any prior insight.
• Figure 5: Boundaries at two different toroidal angles for the 18 optimized QH equilibria found through deflation which passed the filters.