  An adjoint method to calculate the gradient of island width in stellarators
is presented and applied to a set of magnetic field configurations. The
underlying method of calculation of the island width is that of Cary & Hanson
(1991) (with a minor modification), and requires that the residue of the island
centre be small. Therefore, the gradient of the residue is calculated in
addition. Both the island width and the gradient calculations are verified
using an analytical magnetic field configuration introduced in Reiman &
Greenside (1986). The method is also applied to the calculation of the shape
gradient of the width of a magnetic island in an NCSX vacuum configuration with
respect to positions on a coil. A gradient-based optimization is applied to a
magnetic field configuration studied in Hanson & Cary (1984) to minimize
stochasticity by adding perturbations to a pair of helical coils. Although only
vacuum magnetic fields and an analytical magnetic field model are considered in
this work, the adjoint calculation of the island width gradient could also be
applied to a magnetohydrodynamic (MHD) equilibrium if the derivative of the
magnetic field with respect to the equilibrium parameters was known. Using the
island width gradient calculation presented here, more general gradient-based
optimization methods can be applied to design stellarators with small magnetic
islands. Moreoever, the sensitivity of the island size may itself be optimized
to ensure that coil tolerances with respect to island size are kept as high as
possible.
