Stellarators are a class of device for the magnetic confinement of plasmas without toroidal symmetry. As the confining magnetic field is produced by clever shaping of external electro-magnetic coils rather than through internal plasma currents, stellarators enjoy enhanced stability properties over their two-dimensional counterpart, the tokamak. However, the design of a stellarator with acceptable confinement properties requires numerical optimization of the magnetic field in the non-convex, high-dimensional spaces describing their geometry. Another major challenge facing the stellarator program is the sensitive dependence of confinement properties on electro-magnetic coil shapes, necessitating the construction of the coils under tight tolerances. In this Thesis, we address these challenges with the application of adjoint methods and shape sensitivity analysis.

Adjoint methods enable the efficient computation of the gradient of a function that depends on the solution to a system of equations, such as linear or nonlinear PDEs. Rather than perform a finite-difference step with respect to each parameter, one additional adjoint PDE is solved to compute the derivative with respect to any parameter. This enables gradient-based optimization in high-dimensional spaces and efficient sensitivity analysis. We present the first applications of adjoint methods for stellarator shape optimization.

The first example we discuss is the optimization of coil shapes based on the generalization of a continuous current potential model. We optimize the geometry of the coil-winding surface using an adjoint-based method, producing coil shapes that can be more easily constructed. Understanding the sensitivity of coil metrics to perturbations of the winding surface allows us to gain intuition about features of configurations that enable simpler coils. We next consider solutions of the drift-kinetic equation, a kinetic model for collisional transport in curved magnetic fields. An adjoint drift-kinetic equation is derived based on the self-adjointness property of the Fokker-Planck collision operator. This adjoint method allows us to understand the sensitivity of neoclassical quantities, such as the radial collisional transport and self-driven plasma current, to perturbations of the magnetic field strength. Finally, we consider functions that depend on solutions of the magneto-hydrodynamic (MHD) equilibrium equations. We generalize the well-known self-adjointness property of the MHD force operator to include perturbations of the rotational transform and the currents outside the confinement region. This self-adjointness property is applied to develop an adjoint method for computing the derivatives of such functions with respect to perturbations of coil shapes or the plasma boundary. We present a method of solution for the adjoint equations based on a variational principle used in MHD stability analysis.