Heat Source Estimation with the Conjugate Gradient Method in Inverse Linear Diffusive Problems

Su, Jian; Neto, Antônio José da Silva

doi:10.1590/s0100-73862001000300005

Cited by 10 publications

(11 citation statements)

References 17 publications

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“…The application of gradient methods to solve such problems is also discussed in [5,6]. Some problems adjoint to optimal control problems are formulated in [7-9] for multicomponent distributed systems with different control and observation.…”

Section: 24mentioning

confidence: 99%

“…Alifanov et al used this possibility in [2-4] to develop gradient methods for solving inverse heat-conduction problems. Computational algorithms are based on Green functions.The application of gradient methods to solve such problems is also discussed in [5,6]. Some problems adjoint to optimal control problems are formulated in [7-9] for multicomponent distributed systems with different control and observation.…”

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confidence: 99%

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Solution of inverse boundary-value problems for multicomponent parabolic distributed systems

Сергиенко

Deineka

2007

Cybern Syst Anal

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536.24Computational algorithms implementing gradient methods based on solution of direct and adjoint problems in weak formulations are proposed for a number of complex-valued inverse problems of parameter renewal in multicomponent parabolic distributed systems. This approach makes it unnecessary to construct Lagrangian functionals explicitly and to use Green's functions.Keywords: multicomponent parabolic systems, inverse problems, direct and adjoint problems in weak formulations, gradient methods.Lions [1] analyzed optimal control problems for linear distributed systems with quadratic cost functionals and showed that it is possible to use solutions of direct and adjoint problems to obtain Frechet differentials. Alifanov et al. used this possibility in [2-4] to develop gradient methods for solving inverse heat-conduction problems. Computational algorithms are based on Green functions.The application of gradient methods to solve such problems is also discussed in [5,6]. Some problems adjoint to optimal control problems are formulated in [7-9] for multicomponent distributed systems with different control and observation.We propose here algorithms that implement the gradient methods from [2] based on solutions of direct and adjoint problems in weak formulations [7-9] without explicit use of Lagrangian functionals and Green's functions to solve inverse complex data renewal problems for multicomponent parabolic systems.

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Section: 24mentioning

confidence: 99%

mentioning

confidence: 99%

Solution of inverse boundary-value problems for multicomponent parabolic distributed systems

Сергиенко

Deineka

2007

Cybern Syst Anal

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“…The method of parametric identification of heat fluxes that involves combined use of the method of dynamic observers (recurrent filtration) and spline approximation of the quantities sought has been considered in [33,34,41,42,80]. A review and cases of use of methods of iterational regularization [6,13,14,32] for evaluation of internal sources in linear heatconduction problems can be found in [104].…”

Section: Structural Properties Of Dynamic Processes and Classificatiomentioning

confidence: 99%

Structural Properties of Dynamic Systems and Inverse Problems of Mathematical Physics

Borukhov

Тимошпольский

Zayats

et al. 2005

J Eng Phys Thermophys

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A direct relationship between the theory of inverse problems of mathematical physics and the theory of structural properties of dynamic systems is established based on which the inverse problems of mathematical physics and heat conduction are classified and some of the works on them are reviewed. Introduction. The foundations of the theory of inverse problems of mathematical physics were laid in the 1950s-1960s in the works of A. N. Tikhonov, M. M. Lavrent'ev, I. M. Gel'fand, B. M. Levitan, M. G. Krein, V. A. Marchenko, L. D. Faddeev, and many other mathematicians. The special properties of inverse problems are that, unlike primal problems, they do not possess the property of correctness in the sense of Adamar. In this connection, A. N. Tikhonov and his followers have developed the theory of regularization of ill-posed problems and have proposed stable methods of their solution [1-12].In thermophysics, inverse problems occur as problems of either diagnostics of the thermophysical parameters and internal and (or) boundary sources of the processes of transfer or control and synthesis of the above parameters and sources. We emphasize that the "investigation methodology based on solution of inverse problems is one new line in studying heat-and mass-exchange processes and in processing and optimizing thermal regimes of technical objects and technological processes [13]." The problems and methods of solution of the inverse problems of heat exchange have been presented in detail in [14] (this monograph is now classical).In the present work, we review the basic classes of inverse problems of mathematical physics and, in particular, inverse problems of heat conduction; the inverse problems are organized in accordance with the classification (proposed in [15]) of inverse problems of mathematical physics. The basis for the classification used is the scheme of cause-and-effect relations of dynamic systems. The notion of a dynamic system is fundamental for primal problems of mathematical physics. The theory of structural properties and characteristics of systems, such as controllability, observability, reversibility, realizability, and others, has also been developed within the framework of dynamic systems [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. It turned out that these characteristics are directly related to the formulation of a number of classical inverse problems of mathematical physics and inverse problems of heat conduction [15]. Thus, the classification of inverse problems of mathematical physics that is presented in this review links the theory of inverse problems to the theory of dynamic systems in the space of states, which contributes to the interdisciplinary exchange of results and, in particular, to the use of the methods of the theory of dynamic systems in the theory of inverse heat-conduction problems.It should be noted that investigations of the inverse problems of mathematical physics are the focus of numerous works, including monographs (see, e.g., [1,2,8,13,14,). Therefore, in this review, ...

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“…The inverse problem of determining an unknown heat source has been investigated in many papers [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In [12], temperature measurements on the whole domain are required to get the solution of the inverse heat source problem, which is usually impractical in engineering.…”

Section: Introductionmentioning

confidence: 99%

“…A frequent used technique to study inverse problems is the nonlinear least-squares method which minimizes the difference between the experimental measurements and the calculated responses of the system. In solving this problem, several approaches have been adopted such as conjugate gradient method [10,14], Davidson-Fletcher-Powell method, the MonteCarlo technique, covariance analysis and dynamic programming, which all require the gradient computation. It is well known that the gradient based methods converge fast but the gradient computation is sometimes complicated and the convergence depends strongly on the choice of initial guess of the unknown function.…”

Section: Introductionmentioning

confidence: 99%

Estimation of unknown heat source function in inverse heat conduction problems using quantum-behaved particle swarm optimization

Tian

Sun

et al. 2011

International Journal of Heat and Mass Transfer

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Heat Source Estimation with the Conjugate Gradient Method in Inverse Linear Diffusive Problems

Cited by 10 publications

References 17 publications

Solution of inverse boundary-value problems for multicomponent parabolic distributed systems

Solution of inverse boundary-value problems for multicomponent parabolic distributed systems

Structural Properties of Dynamic Systems and Inverse Problems of Mathematical Physics

Estimation of unknown heat source function in inverse heat conduction problems using quantum-behaved particle swarm optimization

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