Abstract
In this paper we consider the analytic and numerical analysis of a class of inverse problems arising from input-output systems governed by partial differential equations of parabolic type where the inputs and outputs are given as point actuators and sensors. In particular, it is assumed that an unknown input produces an (approximately) known or desired output. The output is given as a finite set of data from which it is desired to (approximately) reconstruct the effecting input. It is well known that this type of scenario typically leads to ill-posed problems about which there is a vast literature.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
V. YA. Arsenin and A.A. Timonov, “On the Construction of Regularizing Operators Close to an Optimal Operator for One-Dimensional and Multi-Dimensional Integral Equations of ConvoLution Type of the First Kind,” Soviet Math. Doklady,32(1985), 566–570.
C.T.H. Baker, The Numerical Treatment of Integral Equations, OxFord Univ. Press, OxFord, 1977.
H.T. Banks and K. Kunish, “The Linear Regulator Problem for Parabolic Systems,” SIAM J. Control and Optimization, 22 (1984), 684–698.
H.T. Banks and K. Kunish, Estimation Techniques for Distributed Parameter Systems, Birkhäuser, 1989.
J. Baumeister, Stable Solution of Inverse Problems, Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1987.
J.V. Beck, B. Blackwell, C.R. St. Clair, Jr., Inverse Heat Conduction:, Ill-posed Problems, Wiley, 1985.
J.V. Beck, “Nonlinear Estimation Applied to the Nonlinear Heat Conduction Problem,” Int. J. Heat Mass Trans. 13, 1970, 703–716.
C.I. Byrnes, D.S. Gilliam, “Asymptotic Properties of Root Locus for Distributed Parameter Systems,” Proc. of IEEE Conf. on Dec. and Control, Austin, 1988.
C.I. Byrnes, D.S. Gilliam, “Boundary Feedback Stabilization of Distributed Parameter Systems”, Signal Processing,Scattering and Operator Theory and Numerical Methods, Proceedings of the International Symposium MTNS-89, M. Kasashoek, J. van Schuppen and A. Ran, eds., Birkhäuser, Boston, 1990, 421–428.
J.R. CANNON, The One-Dimensional Heat Equation, Encyclopedia of mathematics and its Applications, Vol. 23, Addison—Wesley, 1984.
A. Carasso “Determining Surface Temperatures from Interior Observations,”SIAM J. Appl. Math.42,3, (1982), 558–574.
Carl De Boor, A Practical Guide to Splines, Applied Math. Sciences, 27, Springer-Verlag, 1978.
B. Davies and B. Martin, “Numerical Inversion of the Laplace Transform: A Survey and Comparison of Methods,” J. of Comp. Phys., 33, (1979), 1–32.
A.R. Davies, M. Iqbal, K. Maleknejad and T.C. Redshaw, “A Comparison of Statistical Regularization and Fourier Extrapolation Methods for Numerical DeconvoLution, in Numerical Treatment of Inverse Problems in Differential and Integral Equations,” P.Deuflhard and E.Hairer (eds.), Birkhauser,1983, pp.320–334.
P.J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, London, 1984.
H.W. Engl and C.W. Groetsch, (eds), Inverse and Ill-posed Problems, Academic Press, N.Y., (1987).
H.W. Engl and P. Manselli, “Stability Estimates and Regularization for an Inverse Heat Conduction Problem in Semi-Infinte and Finite Time Intervals,” Numer. Fund. Anal. and Optim., 10 (1989), 517–540.
D.S. Gilliam, B.A. Mair, C.F. Martin, “A ConvoLution Method for Regular Parabolic Equations,” to appear in SIAM J. Control and Optim..
D.S. Gilliam, B.A. Mair, C.F. Martin, “A ConvoLution Method for Inverse Heat Conduction Problems,” Math. Systems Theory, 12, 1988, 49–60.
D.S. Gilliam, B.A. Mair, C.F. Martin, “Observability and Determination of Surface Temperature, Part 1,” Int.J. Control, 48, 6, 1988, 2249–2264.
D.S. Gilliam, B.A. Mair, “Stability of a ConvoLution Method for Inverse Heat Conduction Problems,” submitted.
H. Haario and E. Somersalo, “A Regularization Method for Integral Equations of the First Kind,” Theory and Applications of Inverse Problems, pp. 16–26, (ed. H. Harris), Longman, 1988.
M.M. Lavrent’ev, V. G. Romanov and S.P. Shishiatskii, Ill-Posed Problems of Mathematical Physics and Analysis, Translations of Mathematical Monographs, vol.64, American Math. Soc. 1986.
J. Locker and P.M. Prenter, “Regularization with Differential Operators, I: General Theory,” Journal of Math. Anal. and Appl., 74 (1980), 504–529.
J. Locker and P.M. Prenter, “Regularization with Differential Operators, II: Weak Least Squares Finite Element SoLutions to First Kind Integral Equations,” SIAM J. Numer. Anal. 17 (1980).
B.A. Mair, “On the Recovery of Surface Temperature and Heat FLux via ConvoLutions,” Computation and Control, Birkhäuser, Boston 1989, 197–208.
V.A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer—Verlag, New York, 1984.
D.A. Murio, “Paramter Selection by Discrete Mollification and the Numerical SoLution of the Inverse Heat Conduction Problem,” Jour. of Comp. and Appl. Math 22, 1988, 25–34.
D.A. Murio and C.C. Roth, “An Integral SoLution for the Inverse Heat Conduction Problem after the Method of Weber,” Comut. Math. Applic., 15, 1, 1988, 39–51.
L.E. Payne, Improperly Posed Problems in Partial Differential Equations, CBMS Regional conference Series, 22, SIAM, Philadelphia, 1975.
J. Philip, “DeconvoLution of Gaussian and Other Kernels,” in Numerical Treatment of Inverse Problems in Differential and Integral Equations, P. Deuflhard and E.Hairer (eds.), Birkhauser,1983, pp.335–354.
T.I. Seidman, “Ill-Posed Problems Arising in Boundary Control and Observation for Diffusion Equations,” Inverse and Improperly Posed Problems in Differential Equations, (Proc. Conf. Math. Numer. Methods, Halle,1979) pp. 233–247, Math. Research, 1, Akademie-Verlag, Berlin, 1979.
A.N. Tikhonov, V.Y. Arsenin, SoLutions of Ill-Posed Problems Wiley, 1971.
J.M. Varah, “A Practical Examination of Some Numerical Methods for Linear Discrete Ill-Posed Problems,” SIAM Review, 21, 1, (1979), 100–111.
C.R. Vogel, “Optimal Choice of a Truncation Level for the Truncated SVD SoLution of Linear First Kind Integral Equations When Data Are Noisy,” SIAM Journ. Numer. Anal.,23,1(1986),109–117.
G. Wahba, “Practical Approximate SoLutions to Linear Operator Equations When the Data Are Noisy,” SIAM Journ. Numer. Anal., 14(1977),651–667.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer Science+Business Media New York
About this chapter
Cite this chapter
Gilliam, D.S., Lund, J.R., Mair, B.A., Martin, C.F. (1991). A Regularization Method for Inverse Heat Conduction Problems. In: Bowers, K., Lund, J. (eds) Computation and Control II. Progress in Systems and Control Theory, vol 11. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0427-5_10
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0427-5_10
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-3611-1
Online ISBN: 978-1-4612-0427-5
eBook Packages: Springer Book Archive