求解第一类算子方程的多重网格算法
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摘要
本文将提出一类适合第一类算子方程的正则化的多重网格算法，它结合一种新的正则参数选取准则，应用Tikhonov正则化来求解粗网格方程保证了求解的稳定性；而在将解向细网格延拓时使用了一种新的光顺处理策略。文中还对实施算法的若干技术问题进行了讨论。数值实验结果表明本文提出的算法是稳定高效的，为第一类算子方程的求解提供了一个新的有效工具。
This thesis is to present and analyze one class of Regularized Multi-grid Algorithms(RMGA) for solving operator equations of the first kind . The RMGA employs Tikhonov's Regularization to solve the corase grids equations with a new choicing parameter's posteriori-method for improving the numerical stability, and adopts a new smoothing strategy to correct the solutions on the fine grids for preserving the high efficiency of MGM.Meanwhile, some key technical problems in the process of implementation of RMGA are disscussed. The algorithm analysis and numerical tests have shown that the algorithm proposed in this paper is stable and highly effective, which provides a new useful tool for solving the operator equations of the first kind.
This thesis is to present and analyze one class of Regularized Multi-grid Algorithms(RMGA) for solving operator equations of the first kind . The RMGA employs Tikhonov's Regularization to solve the corase grids equations with a new choicing parameter's posteriori-method for improving the numerical stability, and adopts a new smoothing strategy to correct the solutions on the fine grids for preserving the high efficiency of MGM.Meanwhile, some key technical problems in the process of implementation of RMGA are disscussed. The algorithm analysis and numerical tests have shown that the algorithm proposed in this paper is stable and highly effective, which provides a new useful tool for solving the operator equations of the first kind.
引文
[1] 肖庭延，于慎根，王彦飞．反问题的数值解法．科学出版社，2003
[2] 苏超伟．偏微分方程逆问题的数值解法及应用．西北工业大学出版社，1995
[3] He guoqiang,Liu linxian.A kind of implicit iterative methods for ill-posed operator equations.Journal of computational Mathematics ,1999,17:275-284
[4] Per Christian Hansen.Numerical tools for analysis and solution of Fredholm intergral equations of the first kind,Inverse Problems,1997,8:849-872
[5] Tikhonov A N,Gonchasky A V(eds). Ⅲ-Posed Problems in the Natural Sciences,Moscow:MIR. 1987
[6] Tikhonov A N, A.V.Goncharsky, Stepanov V V and Yagola A G, Numerical Methods for the Solution of Ⅲ-Posed Problems,Dordrecht:Kluwer,1995.
[7] 李功胜，马逸尘．改进Tikhonov正则化及其正则解的最优渐进估计阶．系统科学与数学，2002，22：158-167
[8] Tikhonov A N.On Solving Incorrectly Posed Problem and Method of Regularization. Dokl.Acad.Nauk USSP,1963,151(3):501-504.
[9] 刘超群，刘志宁，郑小清．多重网格法及其在计算机流体力学中的应用．清华大学出版社，1995
[10] W．哈克布思．多重网格方法．科学出版社，1988
[11] 曹志浩．多格子方法．复旦大学出版社，1988
[12] Wolfgang Hackbusch.Multi-Grid Methods and Applications,Springer-Verlag Berlin.Heidelberg, 1
[13] Groesch C W.The theory of Tikhonov regularization for Fredholm equations of the first kind.Pitman Advanced Publishing Program.1984
[14] Barbara Kaltenbacher.On the regularizing properties of a full multigrid method for ill-posed problems .Inverse Problems 2001,17:767-788
[15] Andreas Rieder.A wavelet multilevel method for ill-posed problems stavilized by Tikhonov.Inverse Problems 2001,17:767-788
[16] Vogel C R.Non-Convergence of the L-Curve Regularization Parameter Selection Method.Inverse Problems ,1996,12 :pp.535-547
[17] Kunish K,Zou J.Iterative choices of regularizarion parameters in linear inverse problems. 1998,14:1247-1264.
[18] 李功胜，马逸尘．应用正则化子建立求解不适定问题正则化方法的探讨．数学进展，29(2000)：532-541
[19] 凌捷，曾文曲，卢建珠．近似数据不适定问题正则参数的后验选择．广东工业大学学报，1999，16：5-9
[20] 戈卢布G H，范洛恩C F著，袁亚湘等译．矩阵计算．科学出版社，2002
[21] 胡家赣．线性代数方程组的迭代解法．科学出版社，1999
[2] 苏超伟．偏微分方程逆问题的数值解法及应用．西北工业大学出版社，1995
[3] He guoqiang,Liu linxian.A kind of implicit iterative methods for ill-posed operator equations.Journal of computational Mathematics ,1999,17:275-284
[4] Per Christian Hansen.Numerical tools for analysis and solution of Fredholm intergral equations of the first kind,Inverse Problems,1997,8:849-872
[5] Tikhonov A N,Gonchasky A V(eds). Ⅲ-Posed Problems in the Natural Sciences,Moscow:MIR. 1987
[6] Tikhonov A N, A.V.Goncharsky, Stepanov V V and Yagola A G, Numerical Methods for the Solution of Ⅲ-Posed Problems,Dordrecht:Kluwer,1995.
[7] 李功胜，马逸尘．改进Tikhonov正则化及其正则解的最优渐进估计阶．系统科学与数学，2002，22：158-167
[8] Tikhonov A N.On Solving Incorrectly Posed Problem and Method of Regularization. Dokl.Acad.Nauk USSP,1963,151(3):501-504.
[9] 刘超群，刘志宁，郑小清．多重网格法及其在计算机流体力学中的应用．清华大学出版社，1995
[10] W．哈克布思．多重网格方法．科学出版社，1988
[11] 曹志浩．多格子方法．复旦大学出版社，1988
[12] Wolfgang Hackbusch.Multi-Grid Methods and Applications,Springer-Verlag Berlin.Heidelberg, 1
[13] Groesch C W.The theory of Tikhonov regularization for Fredholm equations of the first kind.Pitman Advanced Publishing Program.1984
[14] Barbara Kaltenbacher.On the regularizing properties of a full multigrid method for ill-posed problems .Inverse Problems 2001,17:767-788
[15] Andreas Rieder.A wavelet multilevel method for ill-posed problems stavilized by Tikhonov.Inverse Problems 2001,17:767-788
[16] Vogel C R.Non-Convergence of the L-Curve Regularization Parameter Selection Method.Inverse Problems ,1996,12 :pp.535-547
[17] Kunish K,Zou J.Iterative choices of regularizarion parameters in linear inverse problems. 1998,14:1247-1264.
[18] 李功胜，马逸尘．应用正则化子建立求解不适定问题正则化方法的探讨．数学进展，29(2000)：532-541
[19] 凌捷，曾文曲，卢建珠．近似数据不适定问题正则参数的后验选择．广东工业大学学报，1999，16：5-9
[20] 戈卢布G H，范洛恩C F著，袁亚湘等译．矩阵计算．科学出版社，2002
[21] 胡家赣．线性代数方程组的迭代解法．科学出版社，1999