A Globalized and Preconditioned Newton-CG Solver for Metric-Aware Curved High-Order Mesh Optimization

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-${\text{iLDL}}^{\text{T}}\left(0\right)$ 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. First, to enhance the global convergence of the non-linear solver, the globalization strategy modifies Newton’s direction to a feasible step. In particular, our specific-purpose globalization strategy memorizes the length of the feasible step (step-length continuation) between the optimization iterations while ensuring sufficient decrease and progress. Second, to compute Newton’s direction in second-order optimization problems, we consider a conjugate-gradient iterative solver with specific-purpose preconditioning and dynamic forcing terms. To account for the metric stretching and alignment, the preconditioner uses specific orderings for the mesh nodes and the degrees of freedom. We also present a preconditioner switch between Jacobi and ${\text{iLDL}}^{\text{T}}\left(0\right)$ preconditioners to control the numerical ill-conditioning of the preconditioner. In addition, the dynamic forcing terms determine the required accuracy for the Newton direction approximation. Specifically, they control the residual tolerance and enforce sufficient positive curvature for the conjugate-gradients method. 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.