# VRP-GMRES(m) Iteration Algorithm for Fast Multipole Boundary Element Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fundamental Theory for the GMRES(m) Algorithm

#### 2.1. Galerkin Principle

#### 2.2. Arnoldi Process

- For the given $m>0$ and ${x}_{0}\in {R}^{n}$, compute ${v}_{1}={r}_{0}/\phantom{{r}_{0}\u2225{r}_{0}\u2225}\phantom{\rule{0.0pt}{0ex}}\u2225{r}_{0}\u2225$, ${r}_{0}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}b-A{x}_{0}$.
- When $k=1,2,\cdots ,\phantom{\rule{3.33333pt}{0ex}}m$, compute ${\overline{v}}_{k+1}=A{v}_{k}-{\displaystyle \sum _{i=1}^{k}}{h}_{i,k}{v}_{i}$, ${h}_{i,k}=\left(A{v}_{k},{v}_{i}\right)$ and ${h}_{k+1,k}=\u2225{\overline{v}}_{k+1}\u2225$, respectively. It is obvious that ${\overline{v}}_{k+1}\perp {v}_{i}\phantom{\rule{0.277778em}{0ex}}$$\left(i=1,2,\cdots ,k\right)$.
- If ${h}_{k+1,k}=0$, then ${v}_{k+1}=0$ and stop calculation. Otherwise, let ${v}_{k+1}={\overline{v}}_{k+1}/\phantom{{\overline{v}}_{k+1}{h}_{k+1,k}}\phantom{\rule{0.0pt}{0ex}}{h}_{k+1,k}$.

**Theorem**

**1.**

#### 2.3. GMRES(m) Algorithm

**Theorem**

**2.**

- Give a fixed integer $m\phantom{\rule{0.166667em}{0ex}}<<n$ and an initial value ${x}^{\left(0\right)}\in {R}^{n}$, compute ${r}^{\left(0\right)}=b-A{x}^{\left(0\right)}$, and let ${r}_{0}={r}^{\left(0\right)}$.
- Obtain ${\left\{{v}_{i}\right\}}_{i=1}^{m}$ and ${\overline{H}}_{m}$ through the Arnoldi process.
- Solve the least squares problem$$\underset{y\in {R}^{m}}{\mathrm{min}}\u2225\beta {e}_{1}-{\overline{H}}_{m}y\u2225$$
- Form an iterative process ${x}^{(k+1)}={x}^{\left(k\right)}+{z}_{m}\phantom{\rule{0.166667em}{0ex}}$ or ${r}^{(k+1)}={r}^{\left(k\right)}-A{z}_{m}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}k=0,1,2,\cdots $.
- If $\u2225{r}^{(k+1)}\u2225<\epsilon $, then ${x}^{*}\approx {x}^{\left(k+1\right)}$ and stop calculation. Otherwise, let ${r}_{0}={r}^{(k+1)}$, and go to 2.

## 3. VRP-GMRES(m) Iterative Algorithm and Convergence Analysis

#### 3.1. VRP-GMRES(m) Algorithm

- Give an integer $m\phantom{\rule{0.166667em}{0ex}}<<n$ and an initial value ${x}^{\left(0\right)}\in {R}^{n}$, compute ${r}^{\left(0\right)}=b-A{x}^{\left(0\right)}$, and let ${r}_{0}={r}^{\left(0\right)}$.
- Carry out the Arnoldi process and obtain ${\left\{{v}_{i}\right\}}_{i=1}^{m}$ and ${\overline{H}}_{m}$.
- Solve the least squares problem$$\underset{y\in {R}^{m}}{\mathrm{min}}\u2225\beta {e}_{1}-{\overline{H}}_{m}y\u2225$$
- Form an iterative process by ${x}^{(k+1)}={x}^{\left(k\right)}+\phantom{\rule{0.166667em}{0ex}}{z}_{m},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}k=0,1,2,\cdots $ Compute ${r}^{(k+1)}=b-A{x}^{(k+1)}$.
- If $\u2225\phantom{\rule{0.166667em}{0ex}}{r}^{(k+1)}\u2225<\epsilon $. Then, ${x}^{*}\approx {x}^{\left(k+1\right)}$ and stop. If $\u2225\phantom{\rule{0.166667em}{0ex}}{r}^{(k+1)}\u2225\ge \epsilon $, then let ${r}_{0}={r}^{(k+1)}$, $m=m+1$, and go to 2.

#### 3.2. Convergence Analysis

**Definition**

**1.**

**Theorem**

**3.**

**Proof of Theorem**

**3.**

**Corollary**

**1.**

**Theorem**

**4.**

**Proof of Theorem**

**4.**

**Theorem**

**5.**

**Proof of Theorem**

**5.**

## 4. Numerical Experiments

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Absolute errors after three iterations. (

**a**) generalized minimal residual method (GMRES)(m) algorithm; (

**b**) Variable Restart Parameter(VRP)-GMRES(m) algorithm.

**Figure 5.**Comparison of the computational efficiency and accuracy under different meshes for the two algorithms ($m=10$). (

**a**) comparison of computational efficiency; and (

**b**) comparison of computational accuracy.

n | m | GMRES(m) | VRP-GMRES(m) |
---|---|---|---|

10 | 8 | 11 | 2 |

20 | 19 | 3 | 2 |

30 | 29 | 3 | 3 |

40 | 39 | 15 | 12 |

50 | 54 | 79 | 14 |

60 | 85 | 164 | 14 |

n | m | GMRES(m) [s] | VRP-GMRES(m) [s] |
---|---|---|---|

10 | 8 | 0.029415 | 0.018254 |

20 | 19 | 0.028982 | 0.013442 |

30 | 29 | 0.092144 | 0.086172 |

40 | 39 | 1.381825 | 0.847108 |

50 | 54 | 23.539635 | 4.683026 |

60 | 85 | 157.340531 | 14.662790 |

n | m | GMRES(m) | VRP-GMRES(m) |
---|---|---|---|

10 | 8 | 2.1069 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 2.9235 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

20 | 19 | 1.1751 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ | 1.1984 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ |

30 | 29 | 2.0093 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 5.4751 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

40 | 39 | 2.4647 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 2.1327e $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ |

50 | 54 | 3.3092 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.0783 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ |

60 | 85 | 2.0593 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 6.5913 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ |

n | m | GMRES(m) | VRP-GMRES(m) |
---|---|---|---|

10 | 8 | stagnated | Convergent |

20 | 19 | stagnated | Convergent |

30 | 29 | stagnated | Convergent |

40 | 39 | stagnated | Convergent |

50 | 54 | stagnated | Convergent |

60 | 85 | stagnated | Convergent |

n | GMRES(m) [s] | VRP-GMRES(m) [s] |
---|---|---|

40 | 1.580847 | 0.829866 |

50 | 5.702555 | 2.445235 |

60 | 16.965306 | 2.445235 |

70 | 43.204335 | 14.072318 |

80 | 100.077638 | 27.731674 |

90 | 189.671624 | 51.908418 |

Body A | Body B | Body C | Sum | |
---|---|---|---|---|

Node number | 152 | 218 | 98 | 468 |

Element number | 150 | 216 | 96 | 462 |

Contact nodes | 36 | 49 | 25 | 110 |

Contact elements | 25 | 36 | 16 | 77 |

Degree number | 648 | 801 | 369 | 1818 |

m | 9 | 11 | 13 | 15 | 17 | 19 |
---|---|---|---|---|---|---|

GMRES(m) | 192 | 173 | 93 | 67 | 55 | 37 |

VRP-GMRES(m) | 26 | 26 | 26 | 26 | 26 | 26 |

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**MDPI and ACS Style**

Yu, C.; Ren, C.; Bai, X.
VRP-GMRES(m) Iteration Algorithm for Fast Multipole Boundary Element Method. *Math. Comput. Appl.* **2016**, *21*, 49.
https://doi.org/10.3390/mca21040049

**AMA Style**

Yu C, Ren C, Bai X.
VRP-GMRES(m) Iteration Algorithm for Fast Multipole Boundary Element Method. *Mathematical and Computational Applications*. 2016; 21(4):49.
https://doi.org/10.3390/mca21040049

**Chicago/Turabian Style**

Yu, Chunxiao, Cuihuan Ren, and Xueting Bai.
2016. "VRP-GMRES(m) Iteration Algorithm for Fast Multipole Boundary Element Method" *Mathematical and Computational Applications* 21, no. 4: 49.
https://doi.org/10.3390/mca21040049