Pitfalls of Projection: A study of Newton-type solvers for incremental potentials.
CoRR(2023)
摘要
Nonlinear systems arising from time integrators like Backward Euler can
sometimes be reformulated as optimization problems, known as incremental
potentials. We show through a comprehensive experimental analysis that the
widely used Projected Newton method, which relies on unconditional semidefinite
projection of Hessian contributions, typically exhibits a reduced convergence
rate compared to classical Newton's method. We demonstrate how factors like
resolution, element order, projection method, material model and boundary
handling impact convergence of Projected Newton and Newton.
Drawing on these findings, we propose the hybrid method Project-on-Demand
Newton, which projects only conditionally, and show that it enjoys both the
robustness of Projected Newton and convergence rate of Newton. We additionally
introduce Kinetic Newton, a regularization-based method that takes advantage of
the structure of incremental potentials and avoids projection altogether. We
compare the four solvers on hyperelasticity and contact problems.
We also present a nuanced discussion of convergence criteria, and propose a
new acceleration-based criterion that avoids problems associated with existing
residual norm criteria and is easier to interpret. We finally address a
fundamental limitation of the Armijo backtracking line search that occasionally
blocks convergence, especially for stiff problems. We propose a novel
parameter-free, robust line search technique to eliminate this issue.
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