×

Solution of inverse diffusion problems by operator-splitting methods. (English) Zbl 1014.65095

Summary: The inverse (nonlinear) diffusion problem or backward heat conduction problem is investigated. It is assumed that the direct solution can be satisfactorily modelled, for example by the finite difference method. The nature of the problem and typical approaches to its solution are briefly reviewed.
An operator-splitting method is introduced as a means of solving the inverse diffusion problem. An error analysis of the method is given, particularly for the application of the method to the simple diffusion equation. The method is applied to a range of test problems to illustrate the points of the analysis and to demonstrate the properties and performance of the method.

MSC:

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65Y20 Complexity and performance of numerical algorithms
35K05 Heat equation
35R30 Inverse problems for PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beck, J. V., Inverse Heat Conduction (1985), John Wiley and Sons: John Wiley and Sons New York
[2] Engl, H. W.; Hanke, M.; Neubauer, A., Regularization and Inverse Problems (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0711.34018
[3] Golub, G. H.; Van Loan, C. F., Matrix Computations (1989), The John Hopkins University Press: The John Hopkins University Press Baltimore, MD · Zbl 0733.65016
[4] Hadamard, J., Lectures on Cauchy’s Problem in Linear Partial Differential Equations (1923), Yale University Press: Yale University Press New Haven, CT · JFM 49.0725.04
[5] Kress, R., Linear Integral Equations (1989), Springer-Verlag: Springer-Verlag Berlin
[6] S.M. Kirkup, M. Wadsworth, D.G. Armour, Inverse solution of atomic mixing equations, Report MCS-96-14, Department of Computing and Mathematical Sciences, University of Salford, 1996; S.M. Kirkup, M. Wadsworth, D.G. Armour, Inverse solution of atomic mixing equations, Report MCS-96-14, Department of Computing and Mathematical Sciences, University of Salford, 1996 · Zbl 0909.65112
[7] Kirkup, S. M.; Wadsworth, M.; Armour, D. G.; Badheka, R.; Van Den Berg, J. A., Computational solution of the atomic mixing equations, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 11, 189-205 (1998) · Zbl 0909.65112
[8] Kirkup, S. M.; Wadsworth, M., Computational solution of the atomic mixing equations: special methods and algorithm of impetus II, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 11, 207-219 (1998) · Zbl 0909.65113
[9] Miller, G. F., Fredholm equations of the first kind, (Delves, L. M.; Walsh, J., Numerical Solution of Integral Equations (1974), Clarendon Press: Clarendon Press Oxford)
[10] Mitchell, A. R.; Griffiths, D. F., The Finite Difference Method in Partial Differential Equations (1980), John Wiley and Sons: John Wiley and Sons New York · Zbl 0417.65048
[11] W.H. Press, S.A. Teukolsky, W.T. Vettering, B.P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, second ed., 1992; W.H. Press, S.A. Teukolsky, W.T. Vettering, B.P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, second ed., 1992 · Zbl 0778.65002
[12] Richtmyer, R. D.; Morton, K. W., Difference Methods for Initial Value Problems (1967), John Wiley and Sons: John Wiley and Sons New York · Zbl 0155.47502
[13] Stetter, H. J., The defect correction principle and discretization methods, Numerische Mathematik, 29, 425-443 (1978) · Zbl 0362.65052
[14] Wadsworth, M.; Armour, D. G.; Badheka, R.; Collins, R., A model for atomic mixing, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 3, 157-169 (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.