带Neumann边界条件的Helmholtz方程柯西问题的一种新的正则化方法

兰州理工大学学报 ›› 2023, Vol. 49 ›› Issue (4): 151-156.

带Neumann边界条件的Helmholtz方程柯西问题的一种新的正则化方法

余亚辉^*1, 李振平^1,2

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

收稿日期:2022-05-18 出版日期:2023-08-28 发布日期:2023-08-29
通讯作者: 余亚辉(1982-),男,河南舞阳人,副教授. Email: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项目(2019GGJS241)

A new regularization method for the Cauchy problem of the Helmholtz equation with Neumann boundary conditions

YU Ya-hui¹, LI Zhen-ping^1,2

1. Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471023, China;
2. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received:2022-05-18 Online:2023-08-28 Published:2023-08-29

摘要/Abstract

摘要： 考虑矩形域上带Neumann边界条件的Helmholtz方程的柯西问题,该问题是一类严重不适定的偏微分方程反问题,即它的解不连续依赖于输入数据.基于经典的Tikhonov正则化方法利用自设计过滤化子修改核函数的思想,提出一种新的正则化求解方法,给出该问题基于分离变量的近似解,对正则化参数的先验和后验两种选取规则下精确解与近似解进行误差分析,得到满足收敛性和稳定性的Hölder型误差估计.

关键词: Helmholtz方程柯西问题, Neumann边界条件, 不适定问题, 正则化, 后验参数选取

Abstract: A Cauchy problem for the Helmholtz equation with Neumann boundary conditions in a rectangle domain is discussed in this paper. This problem is a serious ill-posed inverse problem of partial differential equations, that is, the solution does not depend continuously on the input data. Based on the classical Tikhonov regularization method using the idea of modifying the kernel function by self-designed filters subsets, a new regularization method is presented for approximating the solution to this problem based on themethod of separating variables. After performing error analysis on the exact and approximate solutions under both a priori and a posteriori selection rules of the regularization parameters, the Hlder-type error estimates satisfying convergence and stability are obtained.

Key words: Cauchy problem for Helmholtz equation, Neumann boundary condition, ill-posed problem, regularization, a posteriori parameter choice rule

中图分类号:

O175

余亚辉, 李振平. 带Neumann边界条件的Helmholtz方程柯西问题的一种新的正则化方法[J]. 兰州理工大学学报, 2023, 49(4): 151-156.

YU Ya-hui, LI Zhen-ping. A new regularization method for the Cauchy problem of the Helmholtz equation with Neumann boundary conditions[J]. Journal of Lanzhou University of Technology, 2023, 49(4): 151-156.

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