确定热方程未知源问题的超阶正则化方法

数学物理学报 ›› 2020, Vol. 40 ›› Issue (3): 717-724.

确定热方程未知源问题的超阶正则化方法

赵振宇^1,^2,^3,*(),林日光²,李志²,梅端²

¹ 山东理工大学数学与统计学院山东淄博 255049
² 广东海洋大学数学与计算机学院广东湛江 524008
³ 南方海洋科学与工程广东省实验室(湛江) 广东湛江 524000

收稿日期:2019-02-18 出版日期:2020-06-26 发布日期:2020-07-15
通讯作者: 赵振宇 E-mail:wozitianshanglai@163.com
基金资助:
南方海洋科学与工程广东省实验室（湛江）资助项目(ZJW-2019-04);广东海洋大学创新强校工程项目(Q18306)

A Super Order Regularization Method for Determination of an Unknown Source in the Heat Equation

Zhenyu Zhao^1,^2,^3,*(),Riguang Lin²,Zhi Li²,Duan Mei²

¹ School of Mathematics and Statistics, Shandong University of Technology, Shandong Zibo 255049
² Faculty of Mathematics and Computer Science, Guangdong Ocean University, Guangdong Zhanjiang 524088
³ Southern Marine Science and Engineering Guangdong Laboratory(Zhanjiang), Guangdong Zhanjiang 524088

Received:2019-02-18 Online:2020-06-26 Published:2020-07-15
Contact: Zhenyu Zhao E-mail:wozitianshanglai@163.com
Supported by:
the Fund of Southern Marine Science and Engineering Guangdong Laboratory (Zhanjiang)(ZJW-2019-04);the Project of Enhancing School with Innovation of Guangdong Ocean University(Q18306)

摘要/Abstract

摘要：

该文研究了热传导方程中未知源的确定问题.针对问题的不适定性，提出了一种结合超阶惩罚项的Tikhonov正则化方法.在由偏差原理选取正则化参数情况下，方法能够在不同光滑条件下获得最优收敛阶.计算过程不需要事先知道光滑度和精确解的先验界.数值试验表明，该方法是有效和稳定的.

关键词: 不适定问题, 超阶正则化, 热方程, 未知源, 偏差原理

Abstract:

The problem for determining an unknown source in the heat equation is considered in this paper. We present a Tikhonov regularization method with a super order penalty term to deal with illposedness of the problem. The regularization parameter is chosen by a discrepancy principle and the order optimal error bounds can be obtained for various smooth conditions. The smoothness parameter and the a priori bound of exact solution are not needed in the numerical process. Numerical tests show that the proposed method is effective and stable.

Key words: Ill-posed problem, Super order regularization, Heat equation, Unknown source, Discrepancy principle

中图分类号:

O124

赵振宇,林日光,李志,梅端. 确定热方程未知源问题的超阶正则化方法[J]. 数学物理学报, 2020, 40(3): 717-724.

Zhenyu Zhao,Riguang Lin,Zhi Li,Duan Mei. A Super Order Regularization Method for Determination of an Unknown Source in the Heat Equation[J]. Acta mathematica scientia,Series A, 2020, 40(3): 717-724.

图/表 1

参考文献 22

1	Burykin A A , Denisov A M . Determination of the unknown sources in the heat-conduction equation. Comput Math Model, 1997, 8 (4): 309- 313
2	Cannon J R , DuChateau P . Structural identification of an unknown source term in a heat equation. Inverse Probl, 1998, 14 (3): 535- 551
3	Cannon J R , Perez-Esteva S . Uniqueness and stability of 3d heat sources. Inverse Probl, 1991, 7, 57- 62 doi: 10.1088/0266-5611/7/1/006
4	Chen B Q , Zhao Z Y , Li Z , Meng Z H . Numerical differentiation by a fourier extension method with super-order regularization. Appl Math Comput, 2018, 334, 1- 10 doi: 10.1016/j.cam.2017.11.024
5	Choulli M , Yamamoto M . Conditional stability in determining a heat source. J Inverse Ill-Posed Probl, 2004, 12 (3): 233- 243
6	Dou F F , Fu C L , Yang F L . Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation. J Comput Appl Math, 2009, 230 (2): 728- 737
7	Badia A E , Duong T H . On an inverse source problem for the heat equation. Application to a pollution detection problem. J Inverse Ill-Posed Probl,, 2002, 10 (6): 858- 600
8	Engl H W , Hanke M , Neubauer A . Regularization of Inverse Problems. Netherlands: Springer, 1996
9	Farcas A , Lesnic D . The boundary-element method for the determination of a heat source dependent on one variable. J Engrg Math, 2006, 54 (4): 375- 388 doi: 10.1007/s10665-005-9023-0
10	Li G S . Data compatibility and conditional stability for an inverse source problem in the heat equation. Appl Math Comput, 2006, 173 (1): 566- 581
11	Ling L , Yamamoto M M , Hon Y C , Takeuchi T . Identification of source locations in two-dimensional heat equations. Inverse Probl, 2006, 22 (4): 1289- 1305
12	Nair M T , Pereverzev S V , Tautenhahn U . Regularization in hilbert scales under general smoothing conditions. Inverse Problems, 2005, 21 (21): 1851- 1862
13	Park H M , Chung J S . A sequential method of solving inverse natural convection problems. Inverse Probl, 2002, 18 (3): 529- 546
14	Ryabenḱii V S , Tsynkov S V , Utyuzhnikov S V . Inverse source problem and active shielding for composite domains. Appl Math Lett, 2007, 20 (5): 511- 515 doi: 10.1016/j.aml.2006.05.019
15	Yamamoto M . Conditional stability in determination of force terms of heat equations in a rectangle. Math Comput Modelling, 1993, 18 (1): 79- 88
16	Yan L , Fu C L , Yang F L . The method of fundamental solutions for the inverse heat source problem. Eng Anal Bound Elem, 2008, 32 (3): 216- 222 doi: 10.1016/j.enganabound.2007.08.002
17	Yan L , Yang F L , Fu C L . A meshless method for solving an inverse spacewise-dependent heat source problem. J Comput Physics, 2009, 228 (1): 123- 136
18	Yang F . The truncation method for identifying an unknown source in the poisson equation. Appl Math Comput, 2011, 22, 9334- 9339
19	Yang F , Fu C L . The modified regularization method for identifying the unknown source on poisson equation. Appl Math Modelling, 2012, 2, 756- 763
20	Yi Y , Murio D A . Source term identification in 1-D IHCP. Computers Math Appl, 2004, 47 (12): 1921- 1933 doi: 10.1016/j.camwa.2002.11.025
21	Zhao Z Y , You L . A modified Tikhonov regularization method for identifying an unknown source in the heat equation. Acta Math Sci, 2014, 34A, 186- 192
22	Meng Z H , You L , Zhao Z Y , Xie O . Determination of an unknown source in the heat equation by the method of Tikhonov regularization in hilbert scales. J Appl Math Physics, 2014, 2 (2): 42095

$\delta_1$	M2			M1
	$q=2$	$q=4$	$q=8$
1e-1	9.18e-2	8.68e-2	8.90e-2	9.68e-2
1e-2	1.81e-2	1.54e-2	1.47e-2	1.41e-2
1e-3	2.91e-3	2.24e-3	1.98e-3	1.58e-3
1e-4	5.99e-4	2.81e-4	2.25e-4	1.75e-4

[1]	李亚峰,辛巧,穆春来. 带有指数型非线性项的离散泊松方程和热方程[J]. 数学物理学报, 2019, 39(4): 773-784.
[2]	王志,闫理坦,余显烨. 分数噪声驱动的随机热方程解的局部时[J]. 数学物理学报, 2019, 39(3): 582-595.
[3]	谢瓯, 孟泽红, 赵振宇, 由雷. 求解拉普拉斯方程柯西问题的截断赫尔米特展开方法[J]. 数学物理学报, 2017, 37(3): 457-468.
[4]	杨帆, 傅初黎, 李晓晓, 任玉鹏. 一类非线性反向热传导问题的Fourier正则化方法[J]. 数学物理学报, 2017, 37(1): 62-71.
[5]	杨帆, 傅初黎, 李晓晓. 修正的Helmholtz方程未知源识别的Fourier截断正则化方法[J]. 数学物理学报, 2014, 34(4): 1040-1047.
[6]	赵振宇, 由雷. 求解热源识别问题的修正吉洪诺夫正则化方法[J]. 数学物理学报, 2014, 34(1): 186-192.
[7]	冯立新, 李媛. 确定热流的非标准逆热传导问题的小波正则化方法[J]. 数学物理学报, 2012, 32(4): 709-719.
[8]	杨帆, 傅初黎, 李晓晓. 修正的Helmholtz方程未知源识别反问题[J]. 数学物理学报, 2012, 32(3): 557-565.
[9]	于航. 热方程数据同化问题的非典型方法[J]. 数学物理学报, 2011, 31(4): 857-865.
[10]	韩传峰, 张超 . 用复参数迭代法解不适定自共轭紧算子方程[J]. 数学物理学报, 2011, 31(4): 887-891.
[11]	李洪芳, 傅初黎, 熊向团, 南楠. 一类求解抛物型方程侧边值问题的最优滤波方法[J]. 数学物理学报, 2009, 29(2): 245-252.
[12]	傅初黎, 邱春雨, 赵华. 求解一般抛物方程侧边值问题的Fourier正则化方法[J]. 数学物理学报, 2005, 25(6): 806-810.
[13]	黄小为, 吴传生, 朱华平. 基于奇异值分解建立的一种新的正则化方法[J]. 数学物理学报, 2005, 25(3): 331-336.
[14]	王林峰. 完备流形上带位势的热核估计[J]. 数学物理学报, 2004, 4(6): 699-713.
[15]	邱春雨, 傅初黎. 一个抛物型方程不适定问题的小波正则化方法[J]. 数学物理学报, 2002, 22(3): 361-372.