  We present a specific-purpose globalized and preconditioned Newton-CG solver
to minimize a metric-aware curved high-order mesh distortion. The solver is
specially devised to optimize curved high-order meshes for high polynomial
degrees with a target metric featuring non-uniform sizing, high stretching
ratios, and curved alignment -- exactly the features that stiffen the
optimization problem. To this end, we consider two ingredients: a
specific-purpose globalization and a specific-purpose
Jacobi-$\text{iLDL}^{\text{T}}(0)$ preconditioning with varying accuracy and
curvature tolerances (dynamic forcing terms) for the CG method. These
improvements are critical in stiff problems because, without them, the large
number of non-linear and linear iterations makes curved optimization
impractical. Finally, to analyze the performance of our method, the results
compare the specific-purpose solver with standard optimization methods. For
this, we measure the matrix-vector products indicating the solver computational
cost and the line-search iterations indicating the total amount of objective
function evaluations. When we combine the globalization and the linear solver
ingredients, we conclude that the specific-purpose Newton-CG solver reduces the
total number of matrix-vector products by one order of magnitude. Moreover, the
number of non-linear and line-search iterations is mainly smaller but of
similar magnitude.
