信赖域算法在求解第一类Fredholm积分方程中的应用
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摘要
信赖域算法是一种新发展起来的求解不适定问题的方法,而第一类Fredholm积分方程的求解问题,是一类特殊的反问题。反问题求解的特征即不适定性为了得到稳定的数值解,必须采用正则化方法解决该类问题。本文主要研究的是信赖域算法在求解第一类Fredholm积分方程中的应用。具有弱奇异核第一类Fredholm积分方程以及一般二维、三维第一类Fredholm积分方程的求解一直是反问题研究领域的一个重要课题,用信赖域算法求解此类问题是本文的研究重点。
首先,给出Fredholm积分方程的基本模型和应用,阐述了该问题求解的困难所在；其次,系统的描述信赖域算法,并给出收敛性分析；然后,对具有弱奇异核的一维第一类Fredholm积分方程和非奇异核的二维、三维第一类Fredholm积分方程进行积分离散分析和信赖域方法求解,并与迭代Tikhonov正则化方法、TSVD正则化方法的计算结果进行比对。
数值模拟和对实验结果的分析,验证了文中给出的信赖域算法求解具有弱奇异核的一维第一类Fredholm积分方程是可行的。求解非奇异核的二维、三维第一类Fredholm积分方程依赖于核的变化及其真解的光滑度,此外,数据的扰动和网格的剖分也有关系。
A trust region algorithm is a new developed algorithm for solving ill-posed problems, and solving the first kind Fredholm integral equation is a special kind of inverse problem. Ill-posedness is the characteristics of this problem. In order to obtain a stable numerical solution, regularization method must be used.The main idears of this study are using trust region algorithms for solving first kind Fredholm integral equations. For solving with weakly singular Fredholm integral equation kernel and general two-dimensional, three-dimensional first kind Fredholm integral equation has always been an important topic in research fields, and used trust region algorithm for solving such problems is the focus of this research.
First, we gave the basic model of Fredholm integral equations and applications and described the difficulties of the problem-solving; Secondly, the system description of trust region algorithm, and gives convergence analysis;With trust region algorithms solving and discreting for weakly singular kernel of Fredholm integral equation and general two-dimensional, three-dimensional first kind Fredholm integral equation and with the calculation results of Tikhonov regularization method and TSVD Regularization Method compared.
The results of numerical simulation and experimental analysis verify that paper gives the trust region algorithm with a one-dimensional weakly singular kernel Fredholm integral equation of the first category is feasible; solving the kernel of non-singular two-dimensional and three-dimensional first kind Fredholm integral equation depends on the kernel changes and smoothness of true solution. In addition, there are also a relationship with disturbance of data grid and subdivision.
首先,给出Fredholm积分方程的基本模型和应用,阐述了该问题求解的困难所在；其次,系统的描述信赖域算法,并给出收敛性分析；然后,对具有弱奇异核的一维第一类Fredholm积分方程和非奇异核的二维、三维第一类Fredholm积分方程进行积分离散分析和信赖域方法求解,并与迭代Tikhonov正则化方法、TSVD正则化方法的计算结果进行比对。
数值模拟和对实验结果的分析,验证了文中给出的信赖域算法求解具有弱奇异核的一维第一类Fredholm积分方程是可行的。求解非奇异核的二维、三维第一类Fredholm积分方程依赖于核的变化及其真解的光滑度,此外,数据的扰动和网格的剖分也有关系。
A trust region algorithm is a new developed algorithm for solving ill-posed problems, and solving the first kind Fredholm integral equation is a special kind of inverse problem. Ill-posedness is the characteristics of this problem. In order to obtain a stable numerical solution, regularization method must be used.The main idears of this study are using trust region algorithms for solving first kind Fredholm integral equations. For solving with weakly singular Fredholm integral equation kernel and general two-dimensional, three-dimensional first kind Fredholm integral equation has always been an important topic in research fields, and used trust region algorithm for solving such problems is the focus of this research.
First, we gave the basic model of Fredholm integral equations and applications and described the difficulties of the problem-solving; Secondly, the system description of trust region algorithm, and gives convergence analysis;With trust region algorithms solving and discreting for weakly singular kernel of Fredholm integral equation and general two-dimensional, three-dimensional first kind Fredholm integral equation and with the calculation results of Tikhonov regularization method and TSVD Regularization Method compared.
The results of numerical simulation and experimental analysis verify that paper gives the trust region algorithm with a one-dimensional weakly singular kernel Fredholm integral equation of the first category is feasible; solving the kernel of non-singular two-dimensional and three-dimensional first kind Fredholm integral equation depends on the kernel changes and smoothness of true solution. In addition, there are also a relationship with disturbance of data grid and subdivision.
引文
[1]傅初黎,刘亭亭.一个不适定问题的正则化及误差估计[J].兰州大学学报,1999,35(1)：18-24.
[2]吉洪诺夫AH,阿尔先宁B只著.不适定问题的解法[M].王秉忱译.北京：地质出版社,1979.
[3]TalentiG,VessellaS. A note on an ill-posed problem for the heat equation [J]. Austral Math Soc,1982,32: 358-368.
[4]M.J.D.Powell.Convergence properties of a Class minimization algrithms in L Man gasarisaian.R.R Meyer and S.M.Robinson,eds,Nonlinear Programming 2 Acdemac[D].Press New York,1975,1-27.
[5]JJ.More.Recent developments in algorithm for trust region methods,in:A.Bachem[M], Math.Prog:The state of the Art,Springer-VerlagBerlin,1983,258-287.
[6]G.A.Schultz,R.B.Schnabel,R.H.Byrd.A family of trust region based algorithms for Unconstrained minimization with strong global convergence,SIAM [J].Numerical Analysis 1985,20(5),409-426.
[7]Levenberg.A method for the Solution of Certain nonlinear problem in least squares[J],Qart.Appl.Math, 1944,2(1):164-166.
[8]D.M.Marquardt.An algorithm for least-squares estimation of nonlinearity equalities[D] SIAMJ. Appl.Math.1963,431-441.
[9]A.Sartenaer.Automatic determination of an intial trust region in nonlinear programming[J], SIAMJ. Sci.Compult,1997,23(6):1788-1803.
[10]X.S.Zhang,J.L.Zhang and L.Z.Liao an adaptive trust region algorithm for unconstrained optimization [J], Science in China,2002,45(5):620-632.
[11]X.S.Zhang,Z.W.Chen,J.L.Zhang.A self-adaptive trust region method for unconstrained Optimization, OR Transactions,2001,5(1):53-62.
[12]J.L.Zhang, Trust region algorithm for nonlinear optimization,Ph.D Thesis Institute of Applied Math matics hinese Academic of Science,2001,5(3):23-30.
[13]张立平,高自友,赖炎连.半定互补间题的全局收敛性算法[D],系统科学与数学,2001,480-485.
[14]X.J Tong,S.Z.Zhou.A trust region algorithm for nonlinear problems of equalities and inequalities [J]数值计算与计算机应用,2001,53-62.
[15]王宜举,修乃华.非线性规划讲义,(2003)手稿.
[16]刘继军.不适定问题的正则化方法及应用[M].北京：科学出版社,2005.
[17]J M.Peng,Global convergence method for monotone variational inequality problems[J]. Optimization Theory and Applications,1995,299-310.
[18]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.
[19]易存晓,胡永才.一类新的带记忆模型的非单调信赖域算法[D],计算机工程与科学,2008,30.
[20]王长枉,宇振盛.求解非线性系统的有效集信赖域方法[J],数学物理学报,2006,26.
[21]R.H.Byrd,R.S.Sehnabel,G.A.Sehultz.Approximate solution of the trust region subproblem over two-dimension subspace[J],Math.Progaming,1988,40:247-263.
[22]D.M.Gay Subroutine for unconstrained minimization using a model/trust region approach[J],ACM Trans.Software,1983,9:503-524.
[23]T.Steihaug.The conjugate gradient method and trust regions in large scale optimization[J] SIAMJ. Numer.Anal,1983,20:626-637.
[24]N.I.M.Gould,S.Lueidi,M.Roma,Ph.l.Toint.Solving the trust region subproblem using The Lanczos[J] Method,SIAMJ-Optimization,1999,504-525.
[25]Ph.L.Toint.Towards an efficient sparsity exploiting Newton method for minimization in:I Duff(eds) Sparse matrices and their uses[R],Academic press,1981,57-88.
[26]J.J.More,D.C.Soresen.Computing a trust region step[J],SIAMJ. SeiStatis.Comput 1983,4:553-572.
[27]P.D.Tao,L.T.Hoai An,A D.C.Optimization algorithm for solving the trust region subproblem[J]. SIA-MJ. Optimization,1998,8:476-505.
[28]J.Zhang,C.Xu.A class indefinite dogleg path methods for unconstrained Minimizations[J]. Optimi--zation,1999,9:646-667.
[29]Y.Yuan.On the truncated conjugate gradient method[M],Math programming,2000,87:561-573.
[30]B.S.He.Solving large-scale trust region subproblem[J],Journal of computing Mathematic 2001,20:1-4.
[31]William W.Hager.Mininze a quadratic over asphere[J],SIAM J.Optimization 2002,12:188-208.
[32]袁亚湘.信赖域算法的收敛性[J],计算数学,1994,21.
[33]Marielba Rojas,Sandra A.Santos,Danny C.Sorensen.LSTRS:Matlab Software for LargeScale Trust Region Subproblems and Regularization[J],Technical Report.USA.2003.
[34]邱春雨,陶建红,傅初黎.一个一维非标准逆热传导问题的Fourier正则化方法[J].兰州大学学报,2002,Vol.38(1)：1-5.
[35]金其年,候宗义.线性不适定问题的渐近正则化方法[J].数学年刊,1999,Vol.20 A(3)：365-374.
[36]Tikhonov A N. Regularization of incorrectly posed problems[J]. Soviet Math Dokl,1963,4:1624-1627.
[37]Oleg Grodzevich.Regularization Using a Parameterized Trust Region Subproblem[J].Combinatorics and Optimization.2004,6.
[37]M. ROJAS, S.A. SANTOS, D.C. SORENSEN. A new matrix-free algorithm for the Largescale trust-region subproblem[J]. Technical Report TR99-19, Rice University,Houston, TX,1999.
[39]C.B.SHAW, Jr.Improvement of the resolution of an instrument by numerical solution of an integral equation[J].Math.Anal.1972,37:83-112.
[40]Morozov V A. On Regularization of Ill-Posed Problems and selection of Regularization Parameter[J]. Comp.Math. Phys,1996,6(1):170-175.
[41]Wang Yan fei. Iterative Choices of Regularization Parameter with Perturbed Operator and Noisy Data[R]. AMSS,Chinese Academy of Sciences,2001.
[42]KENYON W E. Petrophsics Principles of Applications of NMR Logging[J].The Log Analysts,1997, 38 (2):21-43.
[43]MORRISS C E,DEUTCH P,etal.Operating Guide for the Combinable Magnetic Resonance Tool [J]. The Log Analysts,1996,37 (6):53-60.
[44]WILL IAMM地球物理数据分析——离散反演理论[M].北京：地质出版社,1988：115-116.
[45]吕和祥,于洪洁.动力学方程的积分型直接积分法,应用数学和力学,2001,22：1-156.
[46]T.Lv,Y Huang.Extra Plation method for solving wealdy singular nonlinear Volter rain tegral equations of the second kind[J] Journal of Mathematieal Analysis And Applieations.2006,32(4):225-237.
[47]Jian J B,Liang Y M. Finitely convergent algorithm of generalized gradient projection for Systems of nonlinear inequalities[J]. Neural Parallel Sci Comput,2004,12:207-218.
[48]Jian J B, Zhang X L, Quan R. A new finitely convergent algorithm for systems of nonlinear Applied Mathematics Letters,2007,20:405-411.
[49]Jian J B, Cheng W X, Ke X Y. Finitely convergent "-generalized projection algorithm for Nonlinear systems[J]. J Math Anal Appl,2007,332:1446-1459.
[50]袁亚湘,孙文瑜.最优化理论与方法[M].北京：科学出版社,1997.
[51]Tong X J, Zhou S Z. A trust region algorithm for nonlinear problems of equalities and Inequalities [J].数值计算与计算机应用,2001,1：53-62.
[52]Jian J B, Liang Y M. Finitely convergent algorithm of generalized gradient projection for systems of nonlinear inequalities [J]. Neural Parallel Sci Compute,2004,12:207-218.
[2]吉洪诺夫AH,阿尔先宁B只著.不适定问题的解法[M].王秉忱译.北京：地质出版社,1979.
[3]TalentiG,VessellaS. A note on an ill-posed problem for the heat equation [J]. Austral Math Soc,1982,32: 358-368.
[4]M.J.D.Powell.Convergence properties of a Class minimization algrithms in L Man gasarisaian.R.R Meyer and S.M.Robinson,eds,Nonlinear Programming 2 Acdemac[D].Press New York,1975,1-27.
[5]JJ.More.Recent developments in algorithm for trust region methods,in:A.Bachem[M], Math.Prog:The state of the Art,Springer-VerlagBerlin,1983,258-287.
[6]G.A.Schultz,R.B.Schnabel,R.H.Byrd.A family of trust region based algorithms for Unconstrained minimization with strong global convergence,SIAM [J].Numerical Analysis 1985,20(5),409-426.
[7]Levenberg.A method for the Solution of Certain nonlinear problem in least squares[J],Qart.Appl.Math, 1944,2(1):164-166.
[8]D.M.Marquardt.An algorithm for least-squares estimation of nonlinearity equalities[D] SIAMJ. Appl.Math.1963,431-441.
[9]A.Sartenaer.Automatic determination of an intial trust region in nonlinear programming[J], SIAMJ. Sci.Compult,1997,23(6):1788-1803.
[10]X.S.Zhang,J.L.Zhang and L.Z.Liao an adaptive trust region algorithm for unconstrained optimization [J], Science in China,2002,45(5):620-632.
[11]X.S.Zhang,Z.W.Chen,J.L.Zhang.A self-adaptive trust region method for unconstrained Optimization, OR Transactions,2001,5(1):53-62.
[12]J.L.Zhang, Trust region algorithm for nonlinear optimization,Ph.D Thesis Institute of Applied Math matics hinese Academic of Science,2001,5(3):23-30.
[13]张立平,高自友,赖炎连.半定互补间题的全局收敛性算法[D],系统科学与数学,2001,480-485.
[14]X.J Tong,S.Z.Zhou.A trust region algorithm for nonlinear problems of equalities and inequalities [J]数值计算与计算机应用,2001,53-62.
[15]王宜举,修乃华.非线性规划讲义,(2003)手稿.
[16]刘继军.不适定问题的正则化方法及应用[M].北京：科学出版社,2005.
[17]J M.Peng,Global convergence method for monotone variational inequality problems[J]. Optimization Theory and Applications,1995,299-310.
[18]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.
[19]易存晓,胡永才.一类新的带记忆模型的非单调信赖域算法[D],计算机工程与科学,2008,30.
[20]王长枉,宇振盛.求解非线性系统的有效集信赖域方法[J],数学物理学报,2006,26.
[21]R.H.Byrd,R.S.Sehnabel,G.A.Sehultz.Approximate solution of the trust region subproblem over two-dimension subspace[J],Math.Progaming,1988,40:247-263.
[22]D.M.Gay Subroutine for unconstrained minimization using a model/trust region approach[J],ACM Trans.Software,1983,9:503-524.
[23]T.Steihaug.The conjugate gradient method and trust regions in large scale optimization[J] SIAMJ. Numer.Anal,1983,20:626-637.
[24]N.I.M.Gould,S.Lueidi,M.Roma,Ph.l.Toint.Solving the trust region subproblem using The Lanczos[J] Method,SIAMJ-Optimization,1999,504-525.
[25]Ph.L.Toint.Towards an efficient sparsity exploiting Newton method for minimization in:I Duff(eds) Sparse matrices and their uses[R],Academic press,1981,57-88.
[26]J.J.More,D.C.Soresen.Computing a trust region step[J],SIAMJ. SeiStatis.Comput 1983,4:553-572.
[27]P.D.Tao,L.T.Hoai An,A D.C.Optimization algorithm for solving the trust region subproblem[J]. SIA-MJ. Optimization,1998,8:476-505.
[28]J.Zhang,C.Xu.A class indefinite dogleg path methods for unconstrained Minimizations[J]. Optimi--zation,1999,9:646-667.
[29]Y.Yuan.On the truncated conjugate gradient method[M],Math programming,2000,87:561-573.
[30]B.S.He.Solving large-scale trust region subproblem[J],Journal of computing Mathematic 2001,20:1-4.
[31]William W.Hager.Mininze a quadratic over asphere[J],SIAM J.Optimization 2002,12:188-208.
[32]袁亚湘.信赖域算法的收敛性[J],计算数学,1994,21.
[33]Marielba Rojas,Sandra A.Santos,Danny C.Sorensen.LSTRS:Matlab Software for LargeScale Trust Region Subproblems and Regularization[J],Technical Report.USA.2003.
[34]邱春雨,陶建红,傅初黎.一个一维非标准逆热传导问题的Fourier正则化方法[J].兰州大学学报,2002,Vol.38(1)：1-5.
[35]金其年,候宗义.线性不适定问题的渐近正则化方法[J].数学年刊,1999,Vol.20 A(3)：365-374.
[36]Tikhonov A N. Regularization of incorrectly posed problems[J]. Soviet Math Dokl,1963,4:1624-1627.
[37]Oleg Grodzevich.Regularization Using a Parameterized Trust Region Subproblem[J].Combinatorics and Optimization.2004,6.
[37]M. ROJAS, S.A. SANTOS, D.C. SORENSEN. A new matrix-free algorithm for the Largescale trust-region subproblem[J]. Technical Report TR99-19, Rice University,Houston, TX,1999.
[39]C.B.SHAW, Jr.Improvement of the resolution of an instrument by numerical solution of an integral equation[J].Math.Anal.1972,37:83-112.
[40]Morozov V A. On Regularization of Ill-Posed Problems and selection of Regularization Parameter[J]. Comp.Math. Phys,1996,6(1):170-175.
[41]Wang Yan fei. Iterative Choices of Regularization Parameter with Perturbed Operator and Noisy Data[R]. AMSS,Chinese Academy of Sciences,2001.
[42]KENYON W E. Petrophsics Principles of Applications of NMR Logging[J].The Log Analysts,1997, 38 (2):21-43.
[43]MORRISS C E,DEUTCH P,etal.Operating Guide for the Combinable Magnetic Resonance Tool [J]. The Log Analysts,1996,37 (6):53-60.
[44]WILL IAMM地球物理数据分析——离散反演理论[M].北京：地质出版社,1988：115-116.
[45]吕和祥,于洪洁.动力学方程的积分型直接积分法,应用数学和力学,2001,22：1-156.
[46]T.Lv,Y Huang.Extra Plation method for solving wealdy singular nonlinear Volter rain tegral equations of the second kind[J] Journal of Mathematieal Analysis And Applieations.2006,32(4):225-237.
[47]Jian J B,Liang Y M. Finitely convergent algorithm of generalized gradient projection for Systems of nonlinear inequalities[J]. Neural Parallel Sci Comput,2004,12:207-218.
[48]Jian J B, Zhang X L, Quan R. A new finitely convergent algorithm for systems of nonlinear Applied Mathematics Letters,2007,20:405-411.
[49]Jian J B, Cheng W X, Ke X Y. Finitely convergent "-generalized projection algorithm for Nonlinear systems[J]. J Math Anal Appl,2007,332:1446-1459.
[50]袁亚湘,孙文瑜.最优化理论与方法[M].北京：科学出版社,1997.
[51]Tong X J, Zhou S Z. A trust region algorithm for nonlinear problems of equalities and Inequalities [J].数值计算与计算机应用,2001,1：53-62.
[52]Jian J B, Liang Y M. Finitely convergent algorithm of generalized gradient projection for systems of nonlinear inequalities [J]. Neural Parallel Sci Compute,2004,12:207-218.