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Residual-minimization least-squares method for inverse heat conduction. (English) Zbl 0858.65098
Summary: A numerical method is systematically developed for resolving an inverse heat conduction problem in the presence of noisy discrete data. This paper illustrates the effect of imposing constraints on the unknown function of interest. A Volterra integral equation of the first kind is derived and used as the starting point for residual-minimization, least squares methodology. Symbolic manipulation is exploited for purposes of augmenting the computational methodology.
Preliminary indications suggest that the imposition of physical constraints on the system drastically reduces the level of mathematical sophistication needed for accurately approximating the unknown function of interest. These constraints are actually available in many design studies or from models which are derived by physical processes.

MSC:
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
65R20 Numerical methods for integral equations
45D05 Volterra integral equations
35R30 Inverse problems for PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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[1] Ozisik, M.N., Heat conduction, (1993), Wiley New York · Zbl 0625.76091
[2] Hetrick, D.L., Dynamics of nuclear reactors, (1971), University of Chicago
[3] Dunn, W.L.; Maiorino, J.R., On the numerical characteristics of an inverse solution for three-term radiative transfer, J. quant. spectrosc. radiat. transfer, 24, 203-209, (1980)
[4] Dunn, W.L., Inverse Monte Carlo solutions for radiative transfer in inhomogeneous media, J. quant. spectrosc. radiat. transfer, 29, 19-26, (1983)
[5] Kamiuto, K.; Seki, J., Study of the P1 approximation in an inverse scattering problem, J. quant. spectrosc. radiat. transfer, 37, 455-459, (1987)
[6] Ho, C.H.; Ozisik, M.N., Inverse radiation problems in inhomogeneous media, J. quant. spectrosc. radiat. transfer, 40, 553-560, (1988)
[7] McCormick, N.J., Inverse radiative transfer problems: A review, Nucl. sci. engng., 112, 185-198, (1992)
[8] Moutsoglou, A., Solution of an elliptic inverse convection problem using a whole domain regularization technique, J. thermophysic and heat transfer, 4, 341-349, (1990)
[9] Beck, J.V.; Blackwell, B.; St. Clair, C.A., Inverse heat conduction, (1985), Wiley New York · Zbl 0633.73120
[10] Zabaras, N., Inverse finite element techniques for the analysis of solidification processes, Intern. J. numer. methods engng., 29, 1569-1587, (1990)
[11] Zabaras, N.; Ruan, Y.; Richmond, O., Design of two-dimensional Stefan processes with desired freezing front motions, Numer. heat transfer, 21, 307-325, (1992), Part B
[12] Huang, C.H.; Ozisik, M.N.; Sawaf, B., Conjugate gradient method for determining unknown contact conductance during metal casting, Int. J. heat mass transf., 35, 1779-1786, (1992)
[13] Wing, G.M., A primer on integral equations of the first kind, (1991), SIAM Philadelphia, PA
[14] Kress, R., Linear integral equations, (1989), Springer-Verlag Berlin
[15] Linz, P., The solution of Volterra equations of the first kind in the presence of large uncertainties, (), 123-130
[16] Golberg, M.A., A survey of numerical methods for integral equations, (), 1-58 · Zbl 0434.65100
[17] Baker, C.T.H., The numerical treatment of integral equations, (1978), Clarendon Press Oxford · Zbl 0217.53103
[18] Delves, L.M.; Mohamed, J.L., Computational methods for integral equations, (1988), Cambridge University Press Cambridge · Zbl 0662.65111
[19] Radziuk, J., The numerical solution from measurement data of linear integral equations of the first kind, Intern. J. numer. meth. engng., 11, 729-740, (1977) · Zbl 0363.65089
[20] Hendry, W.L., A Volterra integral equation of the first kind, J. math. anal. appl., 54, 266-278, (1976) · Zbl 0346.45004
[21] Pasquetti, R.; Le Niliot, C., Boundary element approach for inverse heat conduction problems: application to a bidimensional transient numerical experiment, Num. heat transfer, 20, 169-189, (1991), Part B
[22] Atkinson, K.E., Numerical analysis, (1989), Wiley New York · Zbl 0456.65064
[23] Frankel, J., A Galerkin solution to a regularized Cauchy singular integro-differential equation, Quart. appl. math., 53, 245-258, (1995) · Zbl 0823.65145
[24] Rivlin, T.J., The Chebyshev polynomials, (1974), Wiley New York · Zbl 0291.33012
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