均值对奇异特征值分解算法的影响机制

数学物理学报 ›› 2019, Vol. 39 ›› Issue (5): 1205-1212.

均值对奇异特征值分解算法的影响机制

曹艳华^1,*(),李楠¹(),陈梦成²()

¹ 华东交通大学理学院南昌 330013
² 华东交通大学建筑与土木工程学院南昌 330013

收稿日期:2018-06-21 出版日期:2019-10-26 发布日期:2019-11-08
通讯作者: 曹艳华 E-mail:yanhuacao@yeah.net;734699069@qq.com;919367042@qq.com
作者简介:李楠, E-mail:734699069@qq.com|陈梦成, E-mail: 919367042@qq.com
基金资助:
国家自然科学基金(11461026);国家自然科学基金(11661036);国家自然科学基金(51378206)

Influence Mechanism of the Mean Values on POD Method

Yanhua Cao^1,*(),Nan Li¹(),Mengcheng Chen²()

¹ School of Sciences, East China Jiaotong University, Nanchang 330013
² Institute of Architecture and Civil Engineering, East China Jiaotong University, Nanchang 330013

Received:2018-06-21 Online:2019-10-26 Published:2019-11-08
Contact: Yanhua Cao E-mail:yanhuacao@yeah.net;734699069@qq.com;919367042@qq.com
Supported by:
the NSFC(11461026);the NSFC(11661036);the NSFC(51378206)

摘要/Abstract

摘要：

该文研究了均值在奇异特征分解算法（POD）中的影响机制.首先，给出了POD方法中时间系数的相关基本性质.其次，基于矩阵分析理论，引入全信号及去除均值部分信号的相关矩阵，且从理论上推导了均值对POD方法的影响机制，这在以往研究中较少见到.最后，通过数值算例对均值场在POD过程中的影响机理进行验证.

关键词: 奇异特征分解算法, 均值, 全信号

Abstract:

The proper orthogonal decomposition (POD) has been known as an efficient reduction technique in simulating physical process which governed by partial differential equations. In this paper, the basic properties of time coefficients in POD method are studied. Based on the theories and matrix analysis, by introducing the cross-correlation matrix (CROM) of the whole signals and the covariance matrix of the fluctuating parts of signals (in which the mean values have been cancelled, COVM), we derive the influence mechanism of mean values on the POD method in theory. There are fewer relevant studies in previous works. Finally, numerical examples are provided for verification and demonstration the influence mechanism in POD process.

Key words: POD method, Mean values, Whole signals

中图分类号:

O324

曹艳华,李楠,陈梦成. 均值对奇异特征值分解算法的影响机制[J]. 数学物理学报, 2019, 39(5): 1205-1212.

Yanhua Cao,Nan Li,Mengcheng Chen. Influence Mechanism of the Mean Values on POD Method[J]. Acta mathematica scientia,Series A, 2019, 39(5): 1205-1212.

图/表 6

图 1

表 1

图 2

表 2

表 3

表 4

参考文献 17

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模态数	$u^w$	$u_1^w$	$u_2^w$	$u^f$	$u_1^f$	$u_2^f$
1	410(96.2)	229(97.2)	189(99.1)	13.4(82.4)	8.8(89)	4.7(74.9)
2	13.4(3.14)	6.2(2.61)	1.6(0.83)	2.8(17)	1.0(10.4)	1.6(24.78)
3	2.8(0.65)	0.45(0.19)	0.02(0.01)	0.08(0.5)	0.06(0.57)	0.02(0.31)
4	0.068(0.02)	0.038(0.02)	7e-4(3.8e-4)	1.1e-2(0.07)	2e-4(0.02)	7e-4(0.01)

模态数	$u^w$	$u_1^w$	$u_2^w$	$u^f$	$u_1^f$	$u_2^f$
1	0.77	0.49	0.04	0.09	0.01	0.04
2	9.7e-2	1.7e-2	6.6e-3	1.3e-2	7.1e-3	2.3e-3
3	1.2e-2	1.3e-2	2.6e-3	1.3e-2	4.7e-3	1.3e-3
4	8.9e-3	4.7e-3	5.5e-4	6.5e-3	2.6e-3	5.5e-4

模态数	$u^w$	$u_1^w$	$u_2^w$	$u^f$	$u_1^f$	$u_2^f$
1	21.1	14.0	0.42	0.95	0.36	0.41
2	0.97	0.49	0.10	0.14	0.37	0.10
3	0.19	0.16	0.02	0.11	0.08	0.02
4	2.3e-2	8.0e-2	1.1e-2	2.1e-2	3.8e-2	1.1e-2

模态数	$u^w$	$u_1^w$	$u_2^w$	$u^f$	$u_1^f$	$u_2^f$
1	2.7	1.3	0.09	0.33	0.03	0.09
2	3.2e-1	3.9e-2	1.6e-2	4.8e-2	3.1e-2	1.6e-2
3	3.0e-2	2.5e-2	6.0e-3	2.7e-2	7.7e-3	6.0e-3
4	1.4e-2	7.7e-3	1.4e-3	4.8e-3	3.0e-3	1.4e-3

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