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Volume 13, Issue 3
Parallel Smoothed Aggregation Multilevel Schwarz Preconditioned Newton-Krylov Algorithms for Poisson-Boltzmann Problems

Shang-Rong Cai, Jun-Yi Xiao, Yu-Chieh Tseng & Feng-Nan Hwang

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 745-769.

Published online: 2020-03

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  • Abstract

We study a multilevel Schwarz preconditioned Newton-Krylov algorithm to solve the Poisson-Boltzmann equation with applications in multi-particle colloidal simulation. The smoothed aggregation-type coarse mesh space is introduced in collaboration with the one-level Schwarz method as a composite preconditioner for accelerating the convergence of a Krylov subspace method for solving the Jacobian system at each Newton step. The important feature of the proposed solution algorithm is that the geometric mesh information needed for constructing the multilevel preconditioner is the same as the one-level Schwarz method on the fine mesh. Other components, such as the definition of the coarse mesh, all the mesh transfer operators, and the coarse mesh problem, are taken care of by the Trillinos/ML packages of the Sandia National Laboratories in the United States. After algorithmic parameter tuning, we show that the proposed smoothed aggregation multilevel Newton-Krylov-Schwarz (NKS) algorithm numerically outperforms than smoothed aggregation multigrid method and one-level version of the NKS algorithm with satisfactory parallel performances up to a few thousand cores. Besides, we investigate how the electrostatic forces between particles for the separation distance depend on the radius of spherical colloidal particles and valence ratios of cation and anion in a cubic system.

  • AMS Subject Headings

65F08, 49M15, 65M55, 68W10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hwangf@math.ncu.edu.tw (Feng-Nan Hwang)

  • BibTex
  • RIS
  • TXT
@Article{NMTMA-13-745, author = {Cai , Shang-RongXiao , Jun-YiTseng , Yu-Chieh and Hwang , Feng-Nan}, title = {Parallel Smoothed Aggregation Multilevel Schwarz Preconditioned Newton-Krylov Algorithms for Poisson-Boltzmann Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {3}, pages = {745--769}, abstract = {

We study a multilevel Schwarz preconditioned Newton-Krylov algorithm to solve the Poisson-Boltzmann equation with applications in multi-particle colloidal simulation. The smoothed aggregation-type coarse mesh space is introduced in collaboration with the one-level Schwarz method as a composite preconditioner for accelerating the convergence of a Krylov subspace method for solving the Jacobian system at each Newton step. The important feature of the proposed solution algorithm is that the geometric mesh information needed for constructing the multilevel preconditioner is the same as the one-level Schwarz method on the fine mesh. Other components, such as the definition of the coarse mesh, all the mesh transfer operators, and the coarse mesh problem, are taken care of by the Trillinos/ML packages of the Sandia National Laboratories in the United States. After algorithmic parameter tuning, we show that the proposed smoothed aggregation multilevel Newton-Krylov-Schwarz (NKS) algorithm numerically outperforms than smoothed aggregation multigrid method and one-level version of the NKS algorithm with satisfactory parallel performances up to a few thousand cores. Besides, we investigate how the electrostatic forces between particles for the separation distance depend on the radius of spherical colloidal particles and valence ratios of cation and anion in a cubic system.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0174}, url = {http://global-sci.org/intro/article_detail/nmtma/15783.html} }
TY - JOUR T1 - Parallel Smoothed Aggregation Multilevel Schwarz Preconditioned Newton-Krylov Algorithms for Poisson-Boltzmann Problems AU - Cai , Shang-Rong AU - Xiao , Jun-Yi AU - Tseng , Yu-Chieh AU - Hwang , Feng-Nan JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 745 EP - 769 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0174 UR - https://global-sci.org/intro/article_detail/nmtma/15783.html KW - Poisson-Boltzmann equation, domain decomposition, Newton-Krylov-Schwarz algorithm, smoothed aggregation, parallel computing. AB -

We study a multilevel Schwarz preconditioned Newton-Krylov algorithm to solve the Poisson-Boltzmann equation with applications in multi-particle colloidal simulation. The smoothed aggregation-type coarse mesh space is introduced in collaboration with the one-level Schwarz method as a composite preconditioner for accelerating the convergence of a Krylov subspace method for solving the Jacobian system at each Newton step. The important feature of the proposed solution algorithm is that the geometric mesh information needed for constructing the multilevel preconditioner is the same as the one-level Schwarz method on the fine mesh. Other components, such as the definition of the coarse mesh, all the mesh transfer operators, and the coarse mesh problem, are taken care of by the Trillinos/ML packages of the Sandia National Laboratories in the United States. After algorithmic parameter tuning, we show that the proposed smoothed aggregation multilevel Newton-Krylov-Schwarz (NKS) algorithm numerically outperforms than smoothed aggregation multigrid method and one-level version of the NKS algorithm with satisfactory parallel performances up to a few thousand cores. Besides, we investigate how the electrostatic forces between particles for the separation distance depend on the radius of spherical colloidal particles and valence ratios of cation and anion in a cubic system.

Shang-Rong Cai, Jun-Yi Xiao, Yu-Chieh Tseng & Feng-Nan Hwang. (2020). Parallel Smoothed Aggregation Multilevel Schwarz Preconditioned Newton-Krylov Algorithms for Poisson-Boltzmann Problems. Numerical Mathematics: Theory, Methods and Applications. 13 (3). 745-769. doi:10.4208/nmtma.OA-2019-0174
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