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Volume 20, Issue 5
Newton-Anderson at Singular Points

Matt Dallas & Sara Pollock

DOI: 10.4208/ijnam2023-1029

Int. J. Numer. Anal. Mod., 20 (2023), pp. 667-692.

Published online: 2023-09

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  • Abstract

In this paper we develop convergence and acceleration theory for Anderson acceleration applied to Newton’s method for nonlinear systems in which the Jacobian is singular at a solution. For these problems, the standard Newton algorithm converges linearly in a region about the solution; and, it has been previously observed that Anderson acceleration can substantially improve convergence without additional a priori knowledge, and with little additional computation cost. We present an analysis of the Newton-Anderson algorithm in this context, and introduce a novel and theoretically supported safeguarding strategy. The convergence results are demonstrated with the Chandrasekhar H-equation and a variety of benchmark examples.

  • Keywords

Anderson acceleration, Newton’s method, safeguarding, singular problems.

  • AMS Subject Headings

65J15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-20-667, author = {Dallas , Matt and Pollock , Sara}, title = {Newton-Anderson at Singular Points}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2023}, volume = {20}, number = {5}, pages = {667--692}, abstract = {

In this paper we develop convergence and acceleration theory for Anderson acceleration applied to Newton’s method for nonlinear systems in which the Jacobian is singular at a solution. For these problems, the standard Newton algorithm converges linearly in a region about the solution; and, it has been previously observed that Anderson acceleration can substantially improve convergence without additional a priori knowledge, and with little additional computation cost. We present an analysis of the Newton-Anderson algorithm in this context, and introduce a novel and theoretically supported safeguarding strategy. The convergence results are demonstrated with the Chandrasekhar H-equation and a variety of benchmark examples.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1029}, url = {http://global-sci.org/intro/article_detail/ijnam/22007.html} }
TY - JOUR T1 - Newton-Anderson at Singular Points AU - Dallas , Matt AU - Pollock , Sara JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 667 EP - 692 PY - 2023 DA - 2023/09 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1029 UR - https://global-sci.org/intro/article_detail/ijnam/22007.html KW - Anderson acceleration, Newton’s method, safeguarding, singular problems. AB -

In this paper we develop convergence and acceleration theory for Anderson acceleration applied to Newton’s method for nonlinear systems in which the Jacobian is singular at a solution. For these problems, the standard Newton algorithm converges linearly in a region about the solution; and, it has been previously observed that Anderson acceleration can substantially improve convergence without additional a priori knowledge, and with little additional computation cost. We present an analysis of the Newton-Anderson algorithm in this context, and introduce a novel and theoretically supported safeguarding strategy. The convergence results are demonstrated with the Chandrasekhar H-equation and a variety of benchmark examples.

Matt Dallas & Sara Pollock. (2023). Newton-Anderson at Singular Points. International Journal of Numerical Analysis and Modeling. 20 (5). 667-692. doi:10.4208/ijnam2023-1029
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