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.

翻译：我们提出了一种面向特定目标的全局化预条件牛顿-CG求解器，用于最小化度量感知的弯曲高阶网格畸变。该求解器专为优化高多项式阶数的弯曲高阶网格而设计，其目标度量具有非均匀尺度、高拉伸比和弯曲对齐特性——这些特征正是导致优化问题刚性的根源。为此，我们采用两种关键技术：面向特定目标的全局化策略，以及面向特定目标的雅可比-$\text{iLDL}^{\text{T}}(0)$预条件方法（结合动态强制项中的变精度与曲率容差）。这些改进对刚性问题的求解至关重要，因为若无这些措施，大量的非线性和线性迭代将使弯曲优化难以实现。最后，为评估方法性能，我们将该面向特定目标的求解器与标准优化方法进行对比，通过矩阵-向量乘积次数衡量求解器计算成本，通过线搜索迭代次数衡量目标函数总评估量。结果表明：当结合全局化策略与线性求解器改进后，该面向特定目标的牛顿-CG求解器将矩阵-向量乘积总数降低了一个数量级；同时，非线性迭代和线搜索迭代次数虽整体偏小，但量级基本相当。