Abstract: Modern stellarator design requires numerical optimization to navigate the high-dimensional spaces

used to describe their geometry. Physical insight into the self-adjointness properties of the underlying

equations enables advanced optimization methods through the efficient calculation of sensitivity information.

The first applications of the adjoint method to stellarator design are reviewed. An adjoint

drift-kinetic equation is derived based on the self-adjointness property of the Fokker-Planck collision

operator [1]. This adjoint method allows one to understand the sensitivity of neoclassical quantities,

such as the radial collisional transport and bootstrap current, to perturbations of the magnetic field

strength. The well-known self-adjointness property of the MHD force operator is generalized to include

perturbations of the rotational transform and the currents outside the confinement region [2-3]. This

adjoint method enables evaluation of the sensitivity of equilibrium properties to perturbations of coil

shapes or the plasma boundary. Adjoint methods have also been developed to reduce stellarator coil

complexity [4], eliminate magnetic islands [5], and obtain quasisymmetric vacuum fields. Applications

of these adjoint methods for sensitivity analysis and optimization are reviewed [6].

[1] E. J. Paul, I. G. Abel, M. Landreman, and W. Dorland, “An adjoint method for neoclassical

stellarator optimization,” Journal of Plasma Physics 85, 795850501 (2019).

[2] T. Antonsen, Jr., E. J. Paul, and M. Landreman, “Adjoint approach to calculating shape gradients

for 3D magnetic confinement equilibria,” Journal of Plasma Physics 85, 905850207 (2019).

[3] E. J. Paul, T. Antonsen, Jr., M. Landreman, and W. A. Cooper, “Adjoint approach to calculating

shape gradients for 3D magnetic confinement equilibria,” Journal of Plasma Physics 86, 905860103

(2020).

[4] E. J. Paul, M. Landreman, A. Bader, and W. Dorland, “An adjoint method for gradient-based

optimization of stellarator coil shapes,” Nuclear Fusion 58, 076015 (2018).

[5] A. Geraldini, M. Landreman, and E. J. Paul, “An adjoint method for determining the sensitivity

of island size to magnetic field variations,” Journal of Plasma Physics 87, 905870302 (2021).

[6] E. J. Paul, M. Landreman, and T. M. Antonsen, “Gradient-based optimization of 3D MHD equilibria,”

Journal of Plasma Physics 87, 905870214 (2021).

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