A globalized and preconditioned Newton-CG solver for metric-aware curved high-order mesh optimization
Guillermo Aparicio-Estrems, Abel Gargallo-Peiró, Xevi Roca
TL;DR
This work tackles the stiffness inherent in metric-aware curved high-order mesh optimization by introducing a dedicated globalized and preconditioned Newton-CG solver. The method combines a novel line-search globalization with a step-length predictor and dynamic forcing terms for both the residual and curvature, plus a switching preconditioner strategy that toggles between Jacobi and $\text{iLDL}^{\mathrm{T}}(0)$ guided by curvature and conditioning. Additional innovations include an MDF-based permutation and metric-aware node ordering to stabilize the incomplete factorizations, and a restricted Newton projection to form efficient, low-dimensional Newton directions. Across 2D and 3D test cases and an $h$-adaptive mesh application, the specific-purpose solver reduces the total number of matrix-vector products by about an order of magnitude and lowers non-linear and line-search iterations compared to standard solvers, enabling robust optimization for high polynomial degrees and complex target metrics. These advances promise substantial computational savings for curved high-order mesh optimization and suggest avenues for GPU acceleration and broader metric-driven adaptations.
Abstract
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.