Reconstruction of high order derivatives by new mollification methods
In this paper, the problem of reconstructing numerical derivatives from noisy data is considered. A new framework of mollification methods based on the L generalized solution regularization methods is proposed. A specific algorithm for the first three derivatives is presented in the paper, in which...
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| Published in | Applied mathematics and mechanics Vol. 29; no. 6; pp. 769 - 778 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Heidelberg
Shanghai University Press
01.06.2008
College of Sciences, Guangdong Ocean University, Zhanjiang 524088, Guangdong Province, P. R. China%College of Sciences, Shanghai University, Shanghai 200444, P. R. China College of Sciences, Shanghai University, Shanghai 200444, P. R. China |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0253-4827 1573-2754 |
| DOI | 10.1007/s10483-008-0608-y |
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| Abstract | In this paper, the problem of reconstructing numerical derivatives from noisy data is considered. A new framework of mollification methods based on the L generalized solution regularization methods is proposed. A specific algorithm for the first three derivatives is presented in the paper, in which a modification of TSVD, termed cTSVD is chosen as the regularization technique. Numerical examples given in the paper verify the theoretical results and show efficiency of the new method. |
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| AbstractList | In this paper, the problem of reconstructing numerical derivatives from noisy data is considered. A new framework of mollification methods based on the L generalized solution regularization methods is proposed. A specific algorithm for the first three derivatives is presented in the paper, in which a modification of TSVD, termed cTSVD is chosen as the regularization technique. Numerical examples given in the paper verify the theoretical results and show efficiency of the new method. O241; In this paper,the problem of reconstructing numerical derivatives from noisy data is considered.A new framework of mollification methods based on the L generalized solution regularization methods is proposed.A specific algorithm for the first three derivatives is presented in the paper,in which a modification of TSVD,termed cTSVD is chosen as the regularization technique.Numerical examples given in the paper verify the theoreticai results and show efficiency of the new method. In this paper, the problem of reconstructing numerical derivatives from noisy data is considered. A new framework of mollification methods based on the L generalized solution regularization methods is proposed. A specific algorithm for the first three derivatives is presented in the paper, in which a modification of TSVD, termed cTSVD is chosen as the regularization technique. Numerical examples given in the paper verify the theoretical results and show efficiency of the new method. |
| Author | 赵振宇 贺国强 |
| AuthorAffiliation | College of Sciences, Shanghai University, Shanghai 200444, P. R. China College of Sciences, Guangdong Ocean University, Zhanjiang 524088, Guangdong Province, P. R. China |
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| Copyright | Shanghai University and Springer-Verlag GmbH 2008 Copyright © Wanfang Data Co. Ltd. All Rights Reserved. |
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| Keywords | numerical differentiation O241 cTSVD method 65D25 47A52 ill-posed problem mollification method generalized solution L generalized solution |
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| References | GorenfloR.VessellaS.Abel integral equations, analysis and application, lecture notes in mathematics[M]1991BerlinSpringer-Verlag LockerJ.PrenterP. M.Regularization with differential operators I, general theory[J]J Math Anal and Appl19807425045290447.6502310.1016/0022-247X(80)90145-6572669 WeiT.LiM.High order numerical derivatives for one-dimensional scattered noisy data[J]Appl Math Comput20061752174417591096.6502710.1016/j.amc.2005.09.0182225621 HeG. Q.A TSVD form for ill-posed equations leading to optimal convergence rates ICM 2002, Abstracts of Short Communication and Poster Sessions[C]2002BeijingHigher Education Press328 KhanI. R.OhbaR.New finite difference formulas for numerical differentiation[J]J Compu Appl Math20001261/22692760971.6501410.1016/S0377-0427(99)00358-11806120 MurioD. A.MejiaC. E.ZhanS.Discrete mollification and automatic numerical differentiation[J]Computers Math Applic199835511610.1016/S0898-1221(98)00001-71612285 EldenL.BerntssonF.ReginskaT.Wavelet and Fourier method for solving the sideways heat equation[J]SIAM J Scient Comp2000216218722050959.6510710.1137/S10648275973313941762037 MurioD. A.The mollification method and the numerical solution of ill-posed problems[M]1993New YorkA Wiley-Interscience Publication, John Wiley & Sons Inc HankeM.ScherzerO.Inverse problems light: numerical differentiation[J]Amer Math Monthly200110865125211002.6502910.2307/26957051840657 HeinzW.HankeM.NeubauerA.Regularization of inverse problems[M]1996DordrechtKluwer Academic Publishers H’aoD. N.A mollification method for ill-posed problems[J]Numer Math199468446950610.1007/s0021100500731301742 WangY. B.HonY. C.ChengJ.Reconstruction of high order derivatives from input data[J]J Inverse Ill-Posed Probl2006142052181115.6502210.1515/1569394067775710852242305 ManselliP.MillerK.Calculation of the surface temperature and heat flux on one side of a wall from measurements on the opposite side[J]Ann Mat Pura Appl198012341611830434.3504810.1007/BF01796543581928 AdamsR. A.Sobolev spaces[M]Pure and Applied Mathematics1975New York/LondonAcademic Press DeansS. R.Radon transform and its applications[M]1983New YorkA Wiley-Interscience Publication, John Wiley & Sons Inc MurioD. A.Numerical method for inverse transient heat conduction problems[J]Revista de la Union Mathematic Argentina198130125360528.65048634023 P. Manselli (608_CR10) 1980; 123 I. R. Khan (608_CR7) 2000; 126 W. Heinz (608_CR6) 1996 Y. B. Wang (608_CR8) 2006; 14 S. R. Deans (608_CR2) 1983 T. Wei (608_CR9) 2006; 175 D. A. Murio (608_CR11) 1981; 30 L. Elden (608_CR13) 2000; 21 D. A. Murio (608_CR4) 1993 D. A. Murio (608_CR5) 1998; 35 M. Hanke (608_CR3) 2001; 108 G. Q. He (608_CR14) 2002 J. Locker (608_CR15) 1980; 74 D. N. H’ao (608_CR12) 1994; 68 R. A. Adams (608_CR16) 1975 R. Gorenflo (608_CR1) 1991 |
| References_xml | – reference: MurioD. A.The mollification method and the numerical solution of ill-posed problems[M]1993New YorkA Wiley-Interscience Publication, John Wiley & Sons Inc – reference: KhanI. R.OhbaR.New finite difference formulas for numerical differentiation[J]J Compu Appl Math20001261/22692760971.6501410.1016/S0377-0427(99)00358-11806120 – reference: AdamsR. A.Sobolev spaces[M]Pure and Applied Mathematics1975New York/LondonAcademic Press – reference: HankeM.ScherzerO.Inverse problems light: numerical differentiation[J]Amer Math Monthly200110865125211002.6502910.2307/26957051840657 – reference: WeiT.LiM.High order numerical derivatives for one-dimensional scattered noisy data[J]Appl Math Comput20061752174417591096.6502710.1016/j.amc.2005.09.0182225621 – reference: HeG. Q.A TSVD form for ill-posed equations leading to optimal convergence rates ICM 2002, Abstracts of Short Communication and Poster Sessions[C]2002BeijingHigher Education Press328 – reference: HeinzW.HankeM.NeubauerA.Regularization of inverse problems[M]1996DordrechtKluwer Academic Publishers – reference: H’aoD. N.A mollification method for ill-posed problems[J]Numer Math199468446950610.1007/s0021100500731301742 – reference: GorenfloR.VessellaS.Abel integral equations, analysis and application, lecture notes in mathematics[M]1991BerlinSpringer-Verlag – reference: MurioD. A.Numerical method for inverse transient heat conduction problems[J]Revista de la Union Mathematic Argentina198130125360528.65048634023 – reference: WangY. B.HonY. C.ChengJ.Reconstruction of high order derivatives from input data[J]J Inverse Ill-Posed Probl2006142052181115.6502210.1515/1569394067775710852242305 – reference: ManselliP.MillerK.Calculation of the surface temperature and heat flux on one side of a wall from measurements on the opposite side[J]Ann Mat Pura Appl198012341611830434.3504810.1007/BF01796543581928 – reference: DeansS. R.Radon transform and its applications[M]1983New YorkA Wiley-Interscience Publication, John Wiley & Sons Inc – reference: EldenL.BerntssonF.ReginskaT.Wavelet and Fourier method for solving the sideways heat equation[J]SIAM J Scient Comp2000216218722050959.6510710.1137/S10648275973313941762037 – reference: MurioD. A.MejiaC. E.ZhanS.Discrete mollification and automatic numerical differentiation[J]Computers Math Applic199835511610.1016/S0898-1221(98)00001-71612285 – reference: LockerJ.PrenterP. M.Regularization with differential operators I, general theory[J]J Math Anal and Appl19807425045290447.6502310.1016/0022-247X(80)90145-6572669 – volume: 30 start-page: 25 issue: 1 year: 1981 ident: 608_CR11 publication-title: Revista de la Union Mathematic Argentina – volume: 21 start-page: 2187 issue: 6 year: 2000 ident: 608_CR13 publication-title: SIAM J Scient Comp doi: 10.1137/S1064827597331394 – volume-title: Regularization of inverse problems[M] year: 1996 ident: 608_CR6 – volume: 14 start-page: 205 year: 2006 ident: 608_CR8 publication-title: J Inverse Ill-Posed Probl doi: 10.1515/156939406777571085 – volume: 74 start-page: 504 issue: 2 year: 1980 ident: 608_CR15 publication-title: J Math Anal and Appl doi: 10.1016/0022-247X(80)90145-6 – start-page: 328 volume-title: A TSVD form for ill-posed equations leading to optimal convergence rates ICM 2002, Abstracts of Short Communication and Poster Sessions[C] year: 2002 ident: 608_CR14 – volume-title: Pure and Applied Mathematics year: 1975 ident: 608_CR16 – volume-title: Abel integral equations, analysis and application, lecture notes in mathematics[M] year: 1991 ident: 608_CR1 doi: 10.1007/BFb0084665 – volume-title: The mollification method and the numerical solution of ill-posed problems[M] year: 1993 ident: 608_CR4 doi: 10.1002/9781118033210 – volume: 68 start-page: 469 issue: 4 year: 1994 ident: 608_CR12 publication-title: Numer Math doi: 10.1007/s002110050073 – volume: 35 start-page: 1 issue: 5 year: 1998 ident: 608_CR5 publication-title: Computers Math Applic doi: 10.1016/S0898-1221(98)00001-7 – volume: 123 start-page: 161 issue: 4 year: 1980 ident: 608_CR10 publication-title: Ann Mat Pura Appl doi: 10.1007/BF01796543 – volume-title: Radon transform and its applications[M] year: 1983 ident: 608_CR2 – volume: 108 start-page: 512 issue: 6 year: 2001 ident: 608_CR3 publication-title: Amer Math Monthly doi: 10.1080/00029890.2001.11919778 – volume: 175 start-page: 1744 issue: 2 year: 2006 ident: 608_CR9 publication-title: Appl Math Comput doi: 10.1016/j.amc.2005.09.018 – volume: 126 start-page: 269 issue: 1/2 year: 2000 ident: 608_CR7 publication-title: J Compu Appl Math doi: 10.1016/S0377-0427(99)00358-1 |
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| SubjectTerms | Algorithms Applications of Mathematics Classical Mechanics Derivatives Fluid- and Aerodynamics Mathematical Modeling and Industrial Mathematics Mathematical models Mathematics Mathematics and Statistics Partial Differential Equations Reconstruction Regularization 数值分化 数值分析 求解方法 软化作用 |
| Title | Reconstruction of high order derivatives by new mollification methods |
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