您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (3): 77-84.doi: 10.6040/j.issn.1671-9352.0.2022.304

• • 上一篇    下一篇

后验参数软化法求解Helmholtz方程柯西问题

李振平1,2,余亚辉1*   

  1. 1.洛阳理工学院数学与物理教学部, 河南 洛阳 471023;2.西北师范大学数学与统计学院, 甘肃 兰州 730070
  • 发布日期:2023-03-02
  • 作者简介:李振平(1982— ),女,博士研究生,研究方向为应用偏微分方程及数值解. E-mail:henanlizhp@163.com*通信作者简介:余亚辉(1982— ),男,硕士,副教授,研究方向为应用数学. E-mail:yuyahui@lit.edu.cn
  • 基金资助:
    河南省高等学校青年骨干教师培养计划资助项目(2019GGJS241)

A mollification method with a posteriori parameter selection for solving the Cauchy problem of the Helmholtz equation

LI Zhen-ping1,2, YU Ya-hui1*   

  1. 1. Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471023, Henan, China;
    2. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2023-03-02

PDF (PC)

154

摘要： 为了恢复解的稳定性,提出一种基于Gauss核的后验参数软化正则化方法,得到精确解与近似解之间的稳定性误差估计,并作数值实验,验证了该方法的有效性。

关键词: Helmholtz方程的柯西问题, 软化方法, 后验参数选取, 误差估计

Abstract: In order to restore the stability of the solution, a mollification regularization method with a posteriori selection of regularization parameter based on Gaussian kernel function is proposed. The stability error estimate between the exact solution and its approximation solution is obtained. Numerical experiments are carried out to verify the effectiveness of the proposed method.

Key words: Cauchy problem for the Helmholtz equation, mollification method, posteriori parameter selection, error estimate

中图分类号: 

  • O241.82
[1] REGINSKA T, REGINSKI K. Approximate solution of a Cauchy problem for the Helmholtz equation[J]. Inverse Probl, 2006, 22(3):975-989.
[2] FENG Xiaoli, FU Chuli, CHENG Hao. A regularization method for solving the Cauchy problem for the Helmholtz equation[J]. Appl Math Model, 2011, 35(7):3301-3315.
[3] XIONG Xiangtuan, FU Chuli. Two approximate methods of a Cauchy problem for the Helmholtz equation[J]. Comput Appl Math, 2007, 26(2):285-307.
[4] ZHANG H W, QIN H H, WEI T. A quasi-reversibility regularization method for the Cauchy problem of the Helmholtz equation[J]. Int J Comput Math, 2011, 88(4):839-850.
[5] QIN H H, WEI T. Two regularization methods for the Cauchy problems of the Helmholtz equation[J]. Appl Math Model, 2010, 34(4):947-967.
[6] REGINSKA T. Regularization methods for a mathematical model of laser beams[J]. Eur J Math Comput Appl, 2014, 1(2):39-49.
[7] ZHANG Yuanxiang, FU Chuli, DENG Zhiliang. An a posteriori truncation method for some Cauchy problems associated with Helmholtz-type equations[J]. Inverse Prob Sci Eng, 2013, 21(7):1151-1168.
[8] FU Chuli, FENG Xiaoli, QIAN Zhi. The Fourier regularization for solving the Cauchy problem for the Helmholtz equation[J]. Appl Numer Math, 2009, 59(10):2625-2640
[9] QIAN Ailin, XIONG Xiangtuan, WU Yujiang. On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation[J]. J Comput Appl Math, 2010, 233(8):1969-1979.
[10] XIONG Xiangtuan. A regularization method for a Cauchy problem of the Helmholtz equation[J]. J Comput Appl Math, 2010, 233(8):1723-1732.
[11] HE Shangqin, FENG Xiufang. A mollification regularization method with the Dirichlet kernel for two Cauchy problems of three-dimensional Helmholtz equation[J]. Int J Comput Math, 2020, 97(11):2320-2336.
[12] HE Shangqin, DI Congna, YANG Li. The mollification method based on a modified operator to the ill-posed problem for 3D Helmholtz equation with mixed boundary[J]. Appl Numer Math, 2021, 160:422-435.
[13] LI Zhenping, XU Chao, LAN Man, et al. A mollification method for a Cauchy problem for the Helmholtz equation[J]. Int J Comput Math, 2018, 95(11):2256-2268.
[14] QIAN Zhi, FENG Xiaoli. A fractional Tikhonov method for solving a Cauchy problem of Helmholtz equation[J]. Appl Anal, 2017, 96(10):1656-1668.
[15] DINH N H. A mollification method for ill-posed problems[J]. Numer Math, 1994, 68(4):469-506.
[16] DINH N H, DUC N. Stability results for the heat equation backward in time[J]. J Math Anal Appl, 2009, 353(2):627-641.
[17] MURIO D A. The mollification method and the numerical solution of ill-posed problems[M]. New York: John Wiley and Sons Inc, 1993: 60-130.
[18] YANG Fan, FU Chuli. A mollification regularization method for the inverse spatial-dependent heat source problem[J]. J Comput Appl Math, 2014, 255:555-567.
[19] 邓志亮. 两类不适定问题的正则化方法研究[D]. 兰州: 兰州大学, 2010. DENG Zhiliang. The study of the regularization methods for two classes of ill-posed problems[D]. Lanzhou: Lanzhou University, 2010.
[20] 丁凤霞, 程浩. 椭圆方程柯西问题磨光正则化参数的后验选取[J]. 山东大学学报(理学版), 2018, 53(2):18-24. DING Fengxia, CHENG Hao. A posteriori choice rule for the mollification regularization parameter for the Cauchy problem of an elliptic equation[J]. Journal of Shandong University(Natural Science), 2018, 53(2):18-24.
[21] QIAN Zhi, FENG Xiaoli. Numerical solution of a 2D inverse heat conduction problem[J]. Inverse Probl Sci Eng, 2013, 21(3):467-484.
[1] 李维智,李宛珊,李建良. 等离子体模型ADI-FDTD格式的最大模误差估计及外推[J]. 《山东大学学报(理学版)》, 2021, 56(9): 96-110.
[2] 王凤霞,熊向团. 非齐次热方程侧边值问题的拟边值正则化方法[J]. 《山东大学学报(理学版)》, 2021, 56(6): 74-80.
[3] 戚斌,程浩. Neumann边界条件分数阶扩散波方程的源项辨识[J]. 《山东大学学报(理学版)》, 2021, 56(6): 64-73.
[4] 杨彩杰,孙同军. 抛物型最优控制问题的Crank-Nicolson差分方法[J]. 《山东大学学报(理学版)》, 2020, 55(6): 115-121.
[5] 郑瑞瑞,孙同军. 一类捕食与被捕食模型最优控制问题的有限元方法的先验误差估计[J]. 《山东大学学报(理学版)》, 2020, 55(1): 23-32.
[6] 陈雅文,熊向团. 抛物方程非特征柯西问题的分数次Tikhonov方法[J]. 《山东大学学报(理学版)》, 2019, 54(12): 120-126.
[7] 丁凤霞,程浩. 椭圆方程柯西问题磨光正则化参数的后验选取[J]. 山东大学学报（理学版）, 2018, 53(2): 18-24.
[8] 于金彪,任永强,曹伟东,鲁统超,程爱杰,戴涛. 可压多孔介质流的扩展混合元解法[J]. 山东大学学报（理学版）, 2017, 52(8): 25-34.
[9] 史艳华1,石东洋2*. 伪双曲方程类Wilson非协调元逼近[J]. J4, 2013, 48(4): 77-84.
[10] 吴志勤1,石东伟2. 带弱奇异核非线性积分微分方程的收敛性分析[J]. J4, 2012, 47(8): 60-63.
[11] 李娜,高夫征*,张天德. Sobolev方程的扩展混合体积元方法[J]. J4, 2012, 47(6): 34-39.
[12] 于金彪1,2,席开华3*,戴涛2,鲁统超3,杨耀忠2,任永强3,程爱杰3. 两相多组分流有限元方法的收敛性[J]. J4, 2012, 47(2): 19-25.
[13] 周家全,孙应德,张永胜. Burgers方程的非协调特征有限元方法[J]. J4, 2012, 47(12): 103-108.
[14] 乔保民,梁洪亮. 一类非线性广义神经传播方程Adini元的超收敛分析[J]. J4, 2011, 46(8): 42-46.
[15] 郑瑞瑞,孙同军. 一类线性Sobolev方程的迎风混合元方法[J]. J4, 2011, 46(4): 23-28.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发