Newton's Method is widely used to find the solution of complex non-linear simulation problems in Computer Graphics. To guarantee a descent direction, it is common practice to clamp the negative eigenvalues of each element Hessian prior to assembly - a strategy known as Projected Newton (PN) - but this perturbation often hinders convergence. In this work, we observe that projecting only a small subset of element Hessians is sufficient to secure a descent direction. Building on this insight, we introduce Progressively Projected Newton (PPN), a novel variant of Newton's Method that uses the current iterate residual to cheaply determine the subset of element Hessians to project. The global Hessian thus remains closer to its original form, reducing both the number of Newton iterations and the amount of required eigen-decompositions. We compare PPN with PN and Project-on-Demand Newton (PDN) in a comprehensive set of experiments covering contact-free and contact-rich deformables (including large stiffness and mass ratios), co-dimensional, and rigid-body simulations, and a range of time step sizes, tolerances and resolutions. PPN consistently performs fewer than 10% of the projections required by PN or PDN and, in the vast majority of cases, converges in fewer Newton iterations, which makes PPN the fastest solver in our benchmark. The most notable exceptions are simulations with very large time steps and quasistatics, where PN remains a better choice.

翻译：牛顿法被广泛应用于计算机图形学中复杂非线性仿真问题的求解。为保证下降方向，通常的做法是在组装前对每个单元海森矩阵的负特征值进行截断——这一策略被称为投影牛顿法——但此扰动常阻碍收敛。本文观察到，仅投影少量单元海森矩阵即可确保下降方向。基于这一见解，我们提出渐进投影牛顿法，这是牛顿法的一种新型变体，它利用当前迭代残差廉价地确定需要投影的单元海森矩阵子集。由此，全局海森矩阵更接近其原始形式，从而减少牛顿迭代次数与所需特征分解次数。我们通过涵盖无接触及密接弹性体（包括大刚度比与质量比）、共维仿真及刚体仿真，以及多种时间步长、容差与分辨率的一系列综合实验，将PPN与PN及按需投影牛顿法进行了对比。PPN所需的投影操作始终少于PN或PDN的10%，且在绝大多数情况下以更少的牛顿迭代次数收敛，这使得PPN成为我们基准测试中求解速度最快的算法。最显著的例外是采用极大时间步长及准静态仿真的场景，此时PN仍是更优选择。