Analysis of an Adaptive Safeguarded Newton-Anderson Algorithm of Depth One with Applications to Fluid Problems
<|reference_start|>Analysis of an Adaptive Safeguarded Newton-Anderson Algorithm of Depth One with Applications to Fluid Problems: The purpose of this paper is to develop a practical strategy to accelerate Newton's method in the vicinity of singular points. We present an adaptive safeguarding scheme with a tunable parameter, which we call adaptive gamma-safeguarding, that one can use in tandem with Anderson acceleration to improve the performance of Newton's method when solving problems at or near singular points. The key features of adaptive gamma-safeguarding are that it converges locally for singular problems, and it can detect nonsingular problems automatically, in which case the Newton-Anderson iterates are scaled towards a standard Newton step. The result is a flexible algorithm that performs well for singular and nonsingular problems, and can recover convergence from both standard Newton and Newton-Anderson with the right parameter choice. This leads to faster local convergence compared to both Newton's method, and Newton- Anderson without safeguarding, with effectively no additional computational cost. We demonstrate three strategies one can use when implementing Newton-Anderson and gamma-safeguarded Newton-Anderson to solve parameter-dependent problems near singular points. For our benchmark problems, we take two parameter-dependent incompressible flow systems: flow in a channel and Rayleigh-Benard convection.<|reference_end|>
@article{dallas2024analysis,
    title={Analysis of an Adaptive Safeguarded Newton-Anderson Algorithm of Depth
  One with Applications to Fluid Problems},
    author={Matt Dallas, Sara Pollock, Leo G. Rebholz},
    journal={arXiv preprint arXiv:2402.09295},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.09295},
    primaryClass={math.NA cs.NA}
}