确定热方程未知源问题的超阶正则化方法

Abstract

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

CLC Number:

O124

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.

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Figures/Tables 1

$\delta_1$	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

References 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

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

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