一种改进的病态问题奇异值修正法

曾小牛, 刘代志, 李夕海, 苏娟, 陈鼎新, 齐玮

曾小牛, 刘代志, 李夕海, 苏娟, 陈鼎新, 齐玮. 一种改进的病态问题奇异值修正法[J]. 武汉大学学报 ( 信息科学版), 2015, 40(10): 1349-1353. DOI: 10.13203/j.whugis20130709
引用本文: 曾小牛, 刘代志, 李夕海, 苏娟, 陈鼎新, 齐玮. 一种改进的病态问题奇异值修正法[J]. 武汉大学学报 ( 信息科学版), 2015, 40(10): 1349-1353. DOI: 10.13203/j.whugis20130709
ZENG Xiaoniu, LIU Daizhi, LI Xihai, SU Juan, CHEN Dingxin, QI Wei. An Improved Singular Value Modification Method for Ill-posed Problems[J]. Geomatics and Information Science of Wuhan University, 2015, 40(10): 1349-1353. DOI: 10.13203/j.whugis20130709
Citation: ZENG Xiaoniu, LIU Daizhi, LI Xihai, SU Juan, CHEN Dingxin, QI Wei. An Improved Singular Value Modification Method for Ill-posed Problems[J]. Geomatics and Information Science of Wuhan University, 2015, 40(10): 1349-1353. DOI: 10.13203/j.whugis20130709

一种改进的病态问题奇异值修正法

基金项目: 国家自然科学基金资助项目(41171351,61302195)。
详细信息
    作者简介:

    曾小牛,博士生,研究方向为地球物理数据分析与处理。E-mail:xiaoniuzeng@163.com

  • 中图分类号: P207.2;P223.0

An Improved Singular Value Modification Method for Ill-posed Problems

Funds: The National Natural Science Foundation of China, Nos. 41171351, 61302195.
  • 摘要: 基于Gaussian滤波函数和Tikhonov滤波函数的联系,选择Gaussian滤波函数作为正则化矩阵,提出了一种改进的病态问题奇异值修正法——Tikhonov-Gaussian法。通过球体重力模型数据的向下延拓仿真实验,验证了改进的奇异值修正法优于标准的Tikhonov修正法。
    Abstract: Based on the relationship between the Gaussian and Tikhonov filter functions, we choose the Gaussian filter function as the regularization matrix, and propose an improved singular values modification method for ill-posed problem, we term it the Tikhonov-Gaussian method. Simulation results from the downward continuation of the model gravity data show that the improved singular values modification method has significant quality advantages in comparison to the standard Tikhonov method.
  • [1] Wang Yanfei. Computational Methods for Inverse Problems and Their Applications[M]. Beijing: Higher Education Press, 2007(王彦飞. 反演问题的计算方法及其应用[M]. 北京: 高等教育出版社, 2007)
    [2] Hansen R O.Rank-Deficient and Discrete Ill-posed Problems[M]. Philadelphia: Siam,1998
    [3] Tikhonov A N, Arsenin N Y. Solutions of Ill-posed Problems[M]. New York: Wiley, 1977
    [4] Xu P L. Truncated SVD Methods for Discrete Linear Ill-posed Problems[J]. Geophysical Journal International, 1998, 135(2): 505-514
    [5] Kelley C T.Iterative Methods for Linear and Nonlinear Equations[M]. Philadelphia: Siam,1995
    [6] Kirsch A.An Introduction to the Mathematical Theory of Inverse Problems[M]. New York: Springer, 2011
    [7] Gu Yongwei, Gui Qingming, Bian Shaofeng, et al. Comparison Between Tikhonov Regularization and Truncated SVD in Geophysics[J]. Geomatics and Information Science of Wuhan University, 2005, 30(3): 238-241(顾勇为, 归庆明, 边少锋, 等. 地球物理反问题中两种正则化方法的比较[J]. 武汉大学学报·信息科学版, 2005, 30(3): 238-241)
    [8] Wang Zhenjie, Ou Jikun, Liu Lintao. A Method for Resolving Ill-conditioned Problems—Two-Step Solution[J]. Geomatics and Information Science of Wuhan University, 2005, 30(9): 821-824(王振杰, 欧吉坤, 柳林涛. 一种解算病态问题的方法——两步解法[J]. 武汉大学学报·信息科学版, 2005, 30(9): 821-824)
    [9] Wu Taiqi, Deng Kailiang, Huang Motao, et al. An Improved Singular Values Decomposition Method for Ill-posed Problem[J]. Geomatics and Information Science of Wuhan University, 2011, 36(8): 900-903(吴太旗, 邓凯亮, 黄谟涛, 等. 一种改进的不适定问题奇异值分解法[J]. 武汉大学学报·信息科学版, 2011, 36(8): 900-903)
    [10] Engl H W, Hanke M, Neubauer A. Regularization of Inverse Problems[M]. Netherlands: Kluwer Academic Publishers, 1996
    [11] Wang Y F, Yang C C, Li X W. Regularizing Kernel-based BRDF Model Inversion Method for Ill-posed Land Surface Parameter Retrieval Using Smoothness Constraint[J]. Journal of Geophysical Research, 2008, 113: D13101
    [12] Jiang Peng, Peng Lihui, Xiao Deyun. Tikhonov Regularization Based on Second Order Derivative Matrix for Electrical Capacitance Tomography Image Reconstruction[J]. Journal of Chemical Industry and Engineering, 2008, 59(2): 405-409(江鹏,彭黎辉,萧德云. 采用二阶导数阵作为正则化的电容成像图像重建算法[J]. 化工学报, 2008, 59(2): 405-409)
    [13] Bauer F, Lukas M A. Comparing Parameter Choice Methods for Regularization of Ill-posed Problems[J]. Mathematics and Computers in Simulation, 2011, 81(9): 1 795-1 841
    [14] Morozov V A. Methods for Solving Incorrectly Posed Problems[M]. New York: Springer-Verlag, 1984
    [15] Hansen P C, O'Leary D P. The Use of the L-curve in the Regularization of Discrete Ill-posed Problems[J]. SIAM J Sci Comput, 1993, 14(6): 1 487-1 503
    [16] Golub G H, Heath M, Wahba G. Generalized Cross-validation as a Method for Choosing a Good Ridge Parameter[J]. Technometrics, 1979, 21(2): 215-223
    [17] Reginska T. A Regularization Parameter in Discrete Ill-posed Problems[J]. SIAM J Sci Comput, 1996, 17(3): 740-749
    [18] Guo Chengbao, Xiao Changhan, Liu Daming. Research on the Continuations of Magnetic Field of Magnetic Object Based on Integral Equation Method and Singlar Value Decomposion[J]. Acta Physica Sinica, 2008, 57(7): 4 182-4 188(郭成豹, 肖昌汉, 刘大明. 基于积分方程法和奇异值分解的磁性目标磁场延拓技术研究[J]. 物理学报, 2008, 57(7): 4 182-4 188)
    [19] Zeng Xiaomin, Li Xihai, Su Juan, et al. An Adaptive Iterative Method for Downward Continuation of Potential Field Data from a Horizontal Plane[J]. Geophysics, 2013, 78(4): J43-J52
    [20] Liu Dongjia, Hong Tianqiu, Jia Zhihai, et al. Wave Number Domain Iteration Method for Downward of Potential Fields and Its Convergence[J]. Chinese Journal of Geophysics, 2009, 52(6): 1 599-1 605(刘东甲, 洪天求, 贾志海, 等. 位场向下延拓的波数域迭代法及其收敛性[J]. 地球物理学报, 2009, 52(6): 1 599-1 605)
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出版历程
  • 收稿日期:  2013-11-25
  • 发布日期:  2015-10-04

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