# Block Preconditioning Matrices for the Newton Method to Compute the Dominant λ-Modes Associated with the Neutron Diffusion Equation

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## Abstract

**:**

## 1. Introduction

## 2. The Modified Generalized Block Newton Method

## 3. Preconditioning

## 4. Numerical Results

`++`based on the data structures provided by the library Deal.ii [3] and PETSc [24]. The computer used for the computations was an Intel

^{®}Core™$\mathrm{i}7$-4790 @3.60 GHz with 32 Gb of RAM running on Ubuntu GNU/Linux 16.04 LTS.

#### 4.1. NEACRP Reactor

#### 4.2. Ringhals Reactor

## 5. Conclusions

_{N}equations.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Convergence history of the Modified Generalized Block Newton Method (MGBNM) for the NEACRP reactor.

Mat. | ${\mathit{D}}_{1}$ (cm) | ${\mathit{D}}_{2}$ (cm) | ${\mathbf{\Sigma}}_{\mathit{a}1}$ (cm^{−1}) | ${\mathbf{\Sigma}}_{\mathit{a}2}$ (cm^{−1}) | ${\mathbf{\Sigma}}_{12}$ (cm^{−1}) | $\mathit{\nu}{\mathbf{\Sigma}}_{\mathit{f}1}$ (cm^{−1}) | $\mathit{\nu}{\mathbf{\Sigma}}_{\mathit{f}2}$ (cm^{−1}) |
---|---|---|---|---|---|---|---|

1 | 5.9264 | 8.2289$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.5979$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.7085$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.7988$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.0000 | 0.0000 |

2 | 1.1276 | 1.7053$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.1878$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.9770$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.3161$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.0000 | 0.0000 |

3 | 1.1276 | 1.7053$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.1878$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.9770$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.0081$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.0000 | 0.0000 |

4 | 1.4624 | 3.9052$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 8.4767$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 6.2569$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.9686$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 5.0150$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 8.7712$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

5 | 1.4637 | 3.9485$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 8.8225$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 6.9978$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.9436$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 5.6085$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.0424$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

6 | 1.4650 | 3.9851$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 9.1484$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 7.6850$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.9196$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 6.1819$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.1954$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

7 | 1.4641 | 4.0579$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 9.0869$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 7.7687$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.8526$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 5.5830$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.0289$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

8 | 1.4642 | 4.0946$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 9.1738$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 8.0302$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.8223$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 5.5741$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.0232$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

9 | 1.4642 | 4.1314$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 9.2596$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 8.2924$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.7920$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 5.5650$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.0169$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

10 | 1.4653 | 4.0919$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 9.4097$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 8.4462$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.8288$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 6.1564$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.1807$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

11 | 1.4655 | 4.1277$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 9.4956$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 8.7030$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.7986$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 6.1474$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.1744$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

12 | 5.5576 | 8.7013$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.7375$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.9644$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.4796$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.0000 | 0.0000 |

13 | 5.6027 | 8.6371$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.4169$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.9313$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.5209$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.0000 | 0.0000 |

14 | 1.4389 | 4.0085$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.0954$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 8.8157$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.6493$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 4.9122$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 8.4889$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

15 | 1.4413 | 4.0665$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.1578$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.0250$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.6054$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 6.0593$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.1626$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

Eigenvalue | Value |
---|---|

1 | $1.002$ |

2 | $0.98862$ |

3 | $0.985406$ |

4 | $0.985406$ |

No. It. | Tol | Mean Its. | Setup Time (s) | Total Time (s) |
---|---|---|---|---|

MGBNM | ($\parallel \mathit{b}-\mathit{Ax}\parallel $) | GMRES | ||

1 | $1\times {10}^{-2}$ | 4.5 | 12.0 | 18.0 |

2 | $1\times {10}^{-3}$ | 9.75 | 12.0 | 20.4 |

3 | $1\times {10}^{-5}$ | 20.75 | 12.0 | 25.2 |

4 | $1\times {10}^{-8}$ | 37.5 | 12.0 | 33.2 |

Total | 72.5 | 48.0 | 96.8 |

No. It. | Tol | Mean Its | Setup Time (s) | Total Time (s) |
---|---|---|---|---|

MGBNM | ($\parallel \mathit{b}-\mathit{Ax}\parallel $) | GMRES | ||

1 | $1\times {10}^{-2}$ | 8.25 | 6.6 | 12.9 |

2 | $1\times {10}^{-3}$ | 13.25 | - | 9.5 |

3 | $1\times {10}^{-5}$ | 23.25 | - | 16.6 |

4 | $1\times {10}^{-8}$ | 41.25 | - | 30.0 |

Total | 86.0 | 6.6 | 70.0 |

**Table 5.**Data for the preconditioner ${\widehat{P}}_{J}$ with the Geometric Multigrid (GMG) for the NEACRP reactor.

No. It. | Tol | Mean Its | Setup Time (s) | Total Time (s) |
---|---|---|---|---|

MGBNM | ($\parallel \mathit{b}-\mathit{Ax}\parallel $) | GMRES | ||

1 | $1\times {10}^{-2}$ | 6.00 | 2.5 | 19.8 |

2 | $1\times {10}^{-3}$ | 9.75 | - | 30.8 |

3 | $1\times {10}^{-5}$ | 12.75 | - | 39.4 |

4 | $1\times {10}^{-8}$ | 20.50 | - | 61.3 |

Total | 49.00 | 2.5 | 151.3 |

Prec. | Its GMRES | Time Setup | Total Time | Max. CPU mem. |
---|---|---|---|---|

P^{ILU} | 72.5 | 48.0 s | 96.8 s | 2062 Mb |

${\widehat{P}}_{J}$^{ILU} | 86.0 | 6.6 s | 70.0 s | 1418 Mb |

${\widehat{P}}_{L}$^{ILU} | 98.0 | 4.4 s | 73.2 s | 787 Mb |

${\widehat{P}}_{Q}$^{ILU} | 100.25 | 1.8 s | 74.4 s | 787 Mb |

Matrix | Time Matvec | Time | Time | Total | |
---|---|---|---|---|---|

Memory | Products | Assembly | Newton | Time | |

Sparse Matrix | 787 Mb | 27 s | 7 s | 51 s | 74 s |

Matrix Free | 319 Mb | 10 s | 4 s | 33 s | 52 s |

**Table 8.**Computational times for the MGBNM with ${\widehat{P}}_{Q}$, the generalized Davidson method, and the Krylov–Schur method for the NEACRP reactor. eigs, eigenvalues.

No. Eigs | MGBNM | Generalized Davidson | Krylov–Schur |
---|---|---|---|

1 | 14 s | 28 s | 27 s |

2 | 23 s | 39 s | 37 s |

4 | 53 s | 48 s | 52 s |

Prec. | Its GMRES | Time Setup | Total Time | Max. CPU mem. |
---|---|---|---|---|

P | 71.5 | 155 s | 408 s | 12.5 Gb |

${\widehat{P}}_{J}$ | 81.0 | 39 s | 331 s | 9.3 Gb |

${\widehat{P}}_{L}$ | 85.2 | 36 s | 348 s | 6.2 Gb |

${\widehat{P}}_{Q}$ | 88.2 | 8 s | 308 s | 3.7 Gb |

**Table 10.**Computational times for the MGBNM with ${\widehat{P}}_{Q}$, the generalized Davidson method, and the Krylov–Schur method for the Ringhals reactor.

No. Eigs | MGBNM | Generalized Davidson | Krylov–Schur |
---|---|---|---|

1 | 100 s | 264 s | 324 s |

2 | 207 s | 294 s | 471 s |

4 | 308 s | 317 s | 528 s |

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**MDPI and ACS Style**

Carreño, A.; Bergamaschi, L.; Martinez, A.; Vidal-Ferrándiz, A.; Ginestar, D.; Verdú, G. Block Preconditioning Matrices for the Newton Method to Compute the Dominant *λ*-Modes Associated with the Neutron Diffusion Equation. *Math. Comput. Appl.* **2019**, *24*, 9.
https://doi.org/10.3390/mca24010009

**AMA Style**

Carreño A, Bergamaschi L, Martinez A, Vidal-Ferrándiz A, Ginestar D, Verdú G. Block Preconditioning Matrices for the Newton Method to Compute the Dominant *λ*-Modes Associated with the Neutron Diffusion Equation. *Mathematical and Computational Applications*. 2019; 24(1):9.
https://doi.org/10.3390/mca24010009

**Chicago/Turabian Style**

Carreño, Amanda, Luca Bergamaschi, Angeles Martinez, Antoni Vidal-Ferrándiz, Damian Ginestar, and Gumersindo Verdú. 2019. "Block Preconditioning Matrices for the Newton Method to Compute the Dominant *λ*-Modes Associated with the Neutron Diffusion Equation" *Mathematical and Computational Applications* 24, no. 1: 9.
https://doi.org/10.3390/mca24010009