Volume 13, Issue 5
Anderson Acceleration of Nonlinear Solvers for the Stationary Gross-Pitaevskii Equation

Dominique Forbes, Leo G. Rebholz & Fei Xue

Adv. Appl. Math. Mech., 13 (2021), pp. 1096-1125.

Published online: 2021-06

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  • Abstract

We consider Anderson acceleration (AA) applied to two nonlinear solvers for the stationary Gross-Pitaevskii equation: a Picard type nonlinear iterative solver and a normalized gradient flow method.  We formulate the solvers as fixed point problems and show that they both fit into the recently developed AA analysis framework. This allows us to prove that both methods' linear convergence rates are improved by a factor (less than one) from the gain of the AA optimization problem at each step.  Numerical tests for finding ground state solutions in 1D and 2D show that AA significantly improves convergence behavior in both solvers, and additionally some comparisons between the solvers are drawn. A local convergence analysis for both methods are also provided.


  • Keywords

Gross-Pitaevskii, Anderson acceleration, convergence analysis.

  • AMS Subject Headings

65J15, 65B05, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-13-1096, author = {Dominique Forbes , and Leo G. Rebholz , and Fei Xue , }, title = {Anderson Acceleration of Nonlinear Solvers for the Stationary Gross-Pitaevskii Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {13}, number = {5}, pages = {1096--1125}, abstract = {

We consider Anderson acceleration (AA) applied to two nonlinear solvers for the stationary Gross-Pitaevskii equation: a Picard type nonlinear iterative solver and a normalized gradient flow method.  We formulate the solvers as fixed point problems and show that they both fit into the recently developed AA analysis framework. This allows us to prove that both methods' linear convergence rates are improved by a factor (less than one) from the gain of the AA optimization problem at each step.  Numerical tests for finding ground state solutions in 1D and 2D show that AA significantly improves convergence behavior in both solvers, and additionally some comparisons between the solvers are drawn. A local convergence analysis for both methods are also provided.


}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0270}, url = {http://global-sci.org/intro/article_detail/aamm/19255.html} }
TY - JOUR T1 - Anderson Acceleration of Nonlinear Solvers for the Stationary Gross-Pitaevskii Equation AU - Dominique Forbes , AU - Leo G. Rebholz , AU - Fei Xue , JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1096 EP - 1125 PY - 2021 DA - 2021/06 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2020-0270 UR - https://global-sci.org/intro/article_detail/aamm/19255.html KW - Gross-Pitaevskii, Anderson acceleration, convergence analysis. AB -

We consider Anderson acceleration (AA) applied to two nonlinear solvers for the stationary Gross-Pitaevskii equation: a Picard type nonlinear iterative solver and a normalized gradient flow method.  We formulate the solvers as fixed point problems and show that they both fit into the recently developed AA analysis framework. This allows us to prove that both methods' linear convergence rates are improved by a factor (less than one) from the gain of the AA optimization problem at each step.  Numerical tests for finding ground state solutions in 1D and 2D show that AA significantly improves convergence behavior in both solvers, and additionally some comparisons between the solvers are drawn. A local convergence analysis for both methods are also provided.


Dominique Forbes, Leo G. Rebholz & Fei Xue. (1970). Anderson Acceleration of Nonlinear Solvers for the Stationary Gross-Pitaevskii Equation. Advances in Applied Mathematics and Mechanics. 13 (5). 1096-1125. doi:10.4208/aamm.OA-2020-0270
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