一种新的病态问题奇异值修正方法

杨秋伟; 陈华; 周聪; 李翠红

doi:10.13442/j.gnss.1008-9268.2020.06.003

一种新的病态问题奇异值修正方法

doi: 10.13442/j.gnss.1008-9268.2020.06.003

杨秋伟^1,2,3, ,,
陈华^1,2,3,
周聪^1,2,3,
李翠红¹

1. 绍兴文理学院土木工程系浙江绍兴 312000;
2. 宁波工程学院建筑与交通工程学院, 浙江宁波 315211;
3. 浙江省土木工程工业化建造工程技术研究中心(宁波工程学院), 浙江宁波 315211

基金项目:

国家自然科学基金（11202138）；宁波市"科技创新2025"重大专项科技项目（2019B10076）

详细信息

作者简介:
杨秋伟(1979-),博士,教授,主要从事结构模型修正、误差处理方面的研究,在国内外杂志发表论文80余篇,SCI、EI期刊30多篇.

通讯作者:
杨秋伟 E-mail:yangqiuwei79@126.com

中图分类号: P207
计量
- 文章访问数: 275
- HTML全文浏览量: 40
- PDF下载量: 9
- 被引次数: 0
出版历程
- 收稿日期: 2020-08-04
- 网络出版日期: 2021-04-09

Singular value correction method for ill conditioned least squares problem in survey adjustment

YANG Qiuwei^{1,2,3
, ,},
CHEN Hua^1,2,3,
ZHOU Cong^1,2,3,
LI Cuihong¹

1. Department of Civil Engineering, Shaoxing University, Shaoxing 312000, China;
2. School of Civil and Transportation Engineering, Ningbo University of Technology, Ningbo 315211, China;
3. Engineering Research Center of Industrial Construction in Civil Engineering of Zhejiang, Ningbo University of Technology, Ningbo 315211, China

摘要

摘要: 针对测量平差中的病态最小二乘问题，提出了统一的奇异值修正公式，以此为基础提出一种新的奇异值修正法.所提方法克服了现有方法需要确定奇异值截断阈值或者修正阈值的缺陷，基本没有增加额外的计算量，计算简单快捷精度高. 另外，所提方法普适性强，对方程组系数矩阵的维数和是否满秩没有特殊的要求，可以适用于多种类型平差方程组的求解. 以两个病态方程为例对所提方法进行了数值验证，并将计算结果与最小二乘解和奇异值截断解进行了比较，结果表明，所提方法可以获得精度更高的计算结果.
- 病态最小二乘问题 /
- 奇异值分解 /
- 奇异值修正公式
Abstract: To solve the ill conditioned least squares problem in survey adjustment, a new singular value correction method is proposed in this paper based on a unified singular value correction formula. The proposed method overcomes the shortcomings of the existing methods, which need to determine the threshold value of singular value truncation or modification. The proposed method is simple and fast in calculation with high accuracy, and does not increase the amount of the computation cost. In addition, the proposed method has strong universality and no special requirements for the dimension and rank of the coefficient matrix of the system of equations. It can be applied to the solution of any type of linear system of equations. Two ill conditioned equations are taken as examples to verify the proposed method. The results are compared with the least square solution and the singular value truncation solution. It has been shown that the proposed method is simple and easy to use, and can obtain more accurate results than the singular value truncation method.
- ill-conditioned least square problem /
- singular value decomposition /
- modified formula

HTML全文

参考文献(18)

[1]	于冬冬,王乐洋.病态总体最小二乘问题的谱修正迭代法[J].大地测量与地球动力学,2015,35(4):702-706.
[2]	杨振,张耀明,周爱华,等.基本解法求解反问题的正则化方法[J].山东理工大学学报(自然科学版),2015(6):20-24.
[3]	郭惠勇,罗乐,盛懋,等.基于岭估计和L曲线的损伤识别技术研究[J].振动与冲击,2015,34(4):200-204.
[4]	葛旭明,伍吉仓.病态总体最小二乘问题的广义正则化[J].测绘学报,2012,41(3):372-377.
[5]	王乐洋,于冬冬.病态总体最小二乘问题的虚拟观测解法[J].测绘学报,2014,43(6):575-581.
[6]	HANSEN P C.Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank[M].SIAM journal on scientific and statistical computing,1990,11(3):503-518.DOI: 10.1137/0911028.
[7]	李睿,于德介,曾威.模式密度方法在结构异常检测中的应用研究[J].振动工程学报,2008,21(4):343-348.
[8]	熊红霞,刘沐宇,刘可文.小波变换与SVD方法在结构损伤监测中的应用[J].公路,2009(3):96-101.
[9]	付玉立,李全海.病态问题中奇异值修正法初探[J].海洋测绘,2011,31(2):42-44,48.
[10]	顾勇为,归庆明.TSVD解算中选择截断参数的新方法[J].测绘科学技术学报,2010,27(3):176-179.
[11]	林东方,朱建军,宋迎春,等.正则化的奇异值分解参数构造法[J].2016,45(8):883-889.
[12]	吴太旗,邓凯亮,黄谟涛,等.一种改进的不适定问题奇异值分解法[J].武汉大学学报(信息科学版),2011,36(8):900-903.
[13]	卢波.病态问题的奇异值分解算法与比较[J].测绘信息与工程,2011,36(4):19-22.
[14]	杨文采.地球物理反演和地震层析成像[M].北京:地质出版社,1989.
[15]	王振杰.大地测量中不适定问题的正则化解法研究[D].武汉:中国科学院测量与地球物理研究所,2003.
[16]	王振杰,欧吉坤.一种新的病态问题奇异值修正方案及其在大地测量中的应用[J].自然科学进展,2004,14(6):672-676.
[17]	LU T D,NING J S.Total least squares adjustment theory and its applications[M].Beijing:China science and technology press,2011:89-102.
[18]	DENG X S,YIN L B,PENG S C,et al.An iterative algorithm for solving ill-conditioned linear least squaress problems[J].Geodesy and geodynamics,2015,6(6):453-459.DOI: 10.1016/j.geog.2015.06.004.