Visualizing Stellarator Objective Functions

In stellarator optimization, using proxies such as quasi-symmetry (QS) instead of direct neoclassical transport calculations can improve computational speed and smooth out the objective function landscape. Gradually increasing the mode numbers is another proven technique. Additionally, there is ongoing debate as to whether starting from a "cold start" or a "warm start" is more beneficial for stellarator optimization. This work presents a method to visualize the objective function landscape to gain a deeper understanding of these empirical observations. The method involves choosing two random orthogonal directions from a given equilibrium and perturbing them to visualize how the objective function varies. The comparative efficacy of various proxies such as $\Gamma_c$, QS, and $\varepsilon_{eff}$ is explored, finding that $\Gamma_c$ is noisier and less smooth than QS or $\varepsilon_{eff}$, making it a less effective objective. Different boundary representations are also compared to assess their impact on the optimization process. The findings provide valuable insights into the optimization techniques and their effectiveness, contributing to more effective stellarator design strategies