Entropy- and tikhonov-based regularization techniques applied to the backwards heat equation
The goal of this paper is to analyze the performance of different regularization techniques for an inverse heat conduction problem (IHCP): the estimation of the initial condition. The inverse problem is formulated as a nonlinear constrained optimization problem, and a regularization term is added to...
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| Published in | Computers & mathematics with applications (1987) Vol. 40; no. 8; pp. 1071 - 1084 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.10.2000
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0898-1221 1873-7668 |
| DOI | 10.1016/S0898-1221(00)85017-8 |
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| Abstract | The goal of this paper is to analyze the performance of different regularization techniques for an
inverse heat conduction problem (IHCP): the estimation of the initial condition. The inverse problem is formulated as a nonlinear constrained optimization problem, and a regularization term is added to the objective function with the help of a regularization parameter. Three classes of regularization methods have been considered: Tikhonov regularization, maximum entropy principle, and truncated singular value decomposition. Concerning the entropíc methodology, two new techniques are introduced and good results were obtained using synthetic data corrupted with noise. The Morozov's discrepancy principle is used to find out the regularization parameter. |
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| AbstractList | The goal of this paper is to analyze the performance of different regularization techniques for an
inverse heat conduction problem (IHCP): the estimation of the initial condition. The inverse problem is formulated as a nonlinear constrained optimization problem, and a regularization term is added to the objective function with the help of a regularization parameter. Three classes of regularization methods have been considered: Tikhonov regularization, maximum entropy principle, and truncated singular value decomposition. Concerning the entropíc methodology, two new techniques are introduced and good results were obtained using synthetic data corrupted with noise. The Morozov's discrepancy principle is used to find out the regularization parameter. |
| Author | de Campos Velho, H.F. Muniz, W.B. Ramos, F.M. |
| Author_xml | – sequence: 1 givenname: W.B. surname: Muniz fullname: Muniz, W.B. email: wagner@udel.edu organization: Department of Mathematical Sciences University of Delaware, Newark, DE 19716, U.S.A – sequence: 2 givenname: F.M. surname: Ramos fullname: Ramos, F.M. email: fernando@lac.inpe.br – sequence: 3 givenname: H.F. surname: de Campos Velho fullname: de Campos Velho, H.F. email: fernando@lac.inpe.br |
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| Cites_doi | 10.1103/PhysRev.106.620 10.1590/S0102-261X1997000200003 10.1016/0017-9310(94)00119-G |
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| Keywords | Backwards heat equation Tikhonov regularization Maximum entropy principle |
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| References | Morozov (BIB1) 1966; 7 Özisik (BIB2) 1980 Muniz, de Campos Velho, Ramos (BIB4) 1999; 101 (to appear). Shannon, Weaver (BIB7) 1949 Carslaw, Jaeger (BIB3) 1959 (BIB8) 1985 Ramos, de Campos Velho (BIB9) 1996; 2 de Campos Velho, Ramos (BIB10) 1997; 15 F.M. Ramos, H.F. Campos Velho, J.C. Carvalho, N.J. and Ferreira, Novel approaches on entropic regularization Ramos, Giovannini (BIB14) 1995; 38 Tikhonov, Arsenin (BIB5) 1977 Jaynes (BIB6) 1957; 106 Bertero, Boccacci (BIB12) 1998 (BIB13) 1993; 13 (10.1016/S0898-1221(00)85017-8_BIB8) 1985 Shannon (10.1016/S0898-1221(00)85017-8_BIB7) 1949 10.1016/S0898-1221(00)85017-8_BIB11 Tikhonov (10.1016/S0898-1221(00)85017-8_BIB5) 1977 Ramos (10.1016/S0898-1221(00)85017-8_BIB9) 1996; 2 Özisik (10.1016/S0898-1221(00)85017-8_BIB2) 1980 Morozov (10.1016/S0898-1221(00)85017-8_BIB1) 1966; 7 Jaynes (10.1016/S0898-1221(00)85017-8_BIB6) 1957; 106 Muniz (10.1016/S0898-1221(00)85017-8_BIB4) 1999; 101 Ramos (10.1016/S0898-1221(00)85017-8_BIB14) 1995; 38 Bertero (10.1016/S0898-1221(00)85017-8_BIB12) 1998 (10.1016/S0898-1221(00)85017-8_BIB13) 1993; 13 Carslaw (10.1016/S0898-1221(00)85017-8_BIB3) 1959 de Campos Velho (10.1016/S0898-1221(00)85017-8_BIB10) 1997; 15 |
| References_xml | – volume: 7 start-page: 414 year: 1966 end-page: 417 ident: BIB1 article-title: On the solution of functional equations by the method of regularization publication-title: Soviet Math. Dokl. – volume: 106 start-page: 620 year: 1957 end-page: 630 ident: BIB6 article-title: Information theory and statistical mechanics publication-title: Physical Review – year: 1959 ident: BIB3 publication-title: Conduction of Heat in Solids – year: 1985 ident: BIB8 publication-title: Maximum-Entropy and Bayesian Methods in Inverse Problems, Fundamental Theories of Physics – reference: F.M. Ramos, H.F. Campos Velho, J.C. Carvalho, N.J. and Ferreira, Novel approaches on entropic regularization, – volume: 13 year: 1993 ident: BIB13 publication-title: NAG Fortran Library Mark – year: 1977 ident: BIB5 publication-title: Solutions of Ill-Posed Problems – year: 1949 ident: BIB7 publication-title: The Mathematical Theory of Communication – year: 1998 ident: BIB12 publication-title: Introduction to Inverse Problems in Imaging – volume: 15 start-page: 133 year: 1997 end-page: 143 ident: BIB10 article-title: Numerical inversion of two-dimensional geoelectric conductivity distributions from eletromagnetic ground data publication-title: Brazilian Journal of Geophysics – reference: (to appear). – volume: 2 start-page: 195 year: 1996 end-page: 206 ident: BIB9 article-title: Reconstruction of geoelectric conductivity distributions using a first-order minimum entropy technique publication-title: Proc. of the 2 – volume: 38 start-page: 101 year: 1995 end-page: 111 ident: BIB14 article-title: Résolution d'un problem inverse multidimensional de diffusion de la chaleur par la méthode des eléments analytiques et par le principe de l'entropie maximale publication-title: International Journal of Heat and Mass Transfer – volume: 101 start-page: 153 year: 1999 end-page: 171 ident: BIB4 article-title: A comparison of some inverse methods for estimating the initial condition of the heat equation publication-title: Journal of Computational and Applied Mathematics – year: 1980 ident: BIB2 publication-title: Heat Conduction – volume: 7 start-page: 414 year: 1966 ident: 10.1016/S0898-1221(00)85017-8_BIB1 article-title: On the solution of functional equations by the method of regularization publication-title: Soviet Math. Dokl. – volume: 101 start-page: 153 year: 1999 ident: 10.1016/S0898-1221(00)85017-8_BIB4 article-title: A comparison of some inverse methods for estimating the initial condition of the heat equation publication-title: Journal of Computational and Applied Mathematics – year: 1977 ident: 10.1016/S0898-1221(00)85017-8_BIB5 – volume: 106 start-page: 620 year: 1957 ident: 10.1016/S0898-1221(00)85017-8_BIB6 article-title: Information theory and statistical mechanics publication-title: Physical Review doi: 10.1103/PhysRev.106.620 – volume: 2 start-page: 195 year: 1996 ident: 10.1016/S0898-1221(00)85017-8_BIB9 article-title: Reconstruction of geoelectric conductivity distributions using a first-order minimum entropy technique – year: 1985 ident: 10.1016/S0898-1221(00)85017-8_BIB8 – year: 1980 ident: 10.1016/S0898-1221(00)85017-8_BIB2 – year: 1949 ident: 10.1016/S0898-1221(00)85017-8_BIB7 – volume: 15 start-page: 133 year: 1997 ident: 10.1016/S0898-1221(00)85017-8_BIB10 article-title: Numerical inversion of two-dimensional geoelectric conductivity distributions from eletromagnetic ground data publication-title: Brazilian Journal of Geophysics doi: 10.1590/S0102-261X1997000200003 – ident: 10.1016/S0898-1221(00)85017-8_BIB11 – year: 1959 ident: 10.1016/S0898-1221(00)85017-8_BIB3 – volume: 38 start-page: 101 year: 1995 ident: 10.1016/S0898-1221(00)85017-8_BIB14 article-title: Résolution d'un problem inverse multidimensional de diffusion de la chaleur par la méthode des eléments analytiques et par le principe de l'entropie maximale publication-title: International Journal of Heat and Mass Transfer doi: 10.1016/0017-9310(94)00119-G – year: 1998 ident: 10.1016/S0898-1221(00)85017-8_BIB12 – volume: 13 year: 1993 ident: 10.1016/S0898-1221(00)85017-8_BIB13 publication-title: NAG Fortran Library Mark |
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inverse heat conduction problem (IHCP): the estimation of... |
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| SubjectTerms | Backwards heat equation Maximum entropy principle Tikhonov regularization |
| Title | Entropy- and tikhonov-based regularization techniques applied to the backwards heat equation |
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