Progressively Projected Newton's Method

TL;DR

Progressively Projected Newton (PPN) targets robust descent in implicit Newton-based time integration by selectively projecting a small subset of element Hessians based on the current residual $\mathbf{g}$, leaving the global Hessian $\mathbf{H}$ largely intact. It maintains a split Hessian $\mathbf{H} = \mathbf{H}_p + \mathbf{H}_u$ where $\mathcal{H}_p$ contains the projected elements; a residual-driven tolerance $\delta$ (updated by $\alpha$ and $\beta$) guides when to project more. The method incrementally updates the Hessian, using SPD solvers (PCG or LLT) and early termination if indefiniteness is detected, enabling fewer projections and faster convergence than PN or PDN across diverse deformable and rigid-body simulations. Empirical results show up to 90% fewer projections and up to 50% fewer Newton iterations, with speedups up to $\times 2.5$ over PN and $\times 1.5$ over PDN; exceptions occur at very large time steps or quasistatics where PN may win. Overall, PPN offers a robust, easy-to-integrate enhancement to existing PN pipelines, delivering substantial computational savings in graphics simulations.

Abstract

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.

Figures (10)

• Figure 1: Rolling sphere. A deformable sphere with initial horizontal velocity rolls on a flat surface. Different Newton step velocity tolerance are used: $10^{-1}m\per s$ (yellow), $10^{-2}m\per s$ (blue), $10^{-3}m\per s$ (green), $10^{-4}m\per s$ (red).
• Figure 2: Ablation test for the projection aggressiveness of PPN with an iterative PCG linear solver (top) and a direct LLT solver (bottom). Color scale is independent per table. The red box highlights the selected parametrization for the rest of this document.
• Figure 3: Runtime breakdown for the chosen parametrization of PPN in the "Press" scene using PCG (top) and LLT (bottom) linear solvers. The average number of Newton iterations (#N) and the percentage of projected element Hessians (ph) are shown at the right of each bar.
• Figure 4: Number of projected Hessians (top) and Newton iterations (bottom) in relation to PN for all solvers in different parametrizations of the "Press" scene: mesh resolution (columns), time step size (marker shape), solver type (color) and tolerance (x-axis).
• Figure 5: Quasistatic extrusion. An elastic box is stretched by twice its size in a quasistatic setting using different resolutions (rows) and Poisson's ratios (colums). Solvers use different colors, and eigenvalue filtering different shading: solid for clamping and hatched for mirroring. The number of Newton iterations is shown at the right of each bar.