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Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems. (English) Zbl 0739.65098
Over the years the first author, together with a variety of co-authors, has developed computational techniques for treating distributed parameter inverse problems — the estimation of unknown functions appearing in the specification of partial differential equations on the basis of over- specified data. The present paper presents a unified approach to these questions with several examples.
Generalizing an earlier approach for linear problems based on the Trotter-Kato theorem, the present approach is based on an appropriate theorem in the book by V. Barbu [Semigroups of nonlinear contractions in Banach spaces (1974; Zbl 0276.47044)] and applies to a reasonably broad class of nonlinear distributed systems. The approach formulates these problems via minimization over compact admissible parameter sets and shows convergence for the estimates constructed using increasingly accurate data and Galerkin approximations for the system.
Reviewer: T.Seidman

MSC:
65Z05 Applications to the sciences
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
35R30 Inverse problems for PDEs
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[1] Babuska, I., and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, ed.), Academic Press, New York, 1972, pp. 3-359.
[2] Banks, H. T., Computational techniques for inverse problems in size-structured stochastic population models, LCDS-CCS Report 87-41, Division of Applied Mathematics, Brown University, Providence, RI (1987), and Proceedings of the IFIP Conference on Optimal Control of Systems Governed by Partial Differential Equations, Santiago de Compostela, Spain, July 6-9, 1987 (A. Bermudez, ed.), Lecture Notes in Control and Information Sciences, Vol. 114, Springer-Verlag, Berlin, 1989, pp. 3-10.
[3] Banks, H. T., L. W. Botsford, F. Kappel, and C. Wang, Modeling and estimation in size-structured population models, LCDS-CCS Report 87-13, Division of Applied Mathematics, Brown University, Providence, RI (1987), and in Mathematical Ecology (T. G. Hallam et al., eds.), World Scientific, Singapore, 1988, pp. 521-541.
[4] Banks, H. T., J. M. Crowley, and I. G. Rosen, Methods for the identification of material parameters in distributed models for flexible structures, Mat. Apl. Comput., 5(2) (1986), 139-168. · Zbl 0631.93017
[5] Banks, H. T., and K. Ito, A unified framework for approximation and inverse problems for distributed parameter systems, Control Theory Adv. Technol., 4 (1988), 73-90.
[6] Banks, H. T., and P. D. Lamm, Estimation of variable coefficients in parabolic distributed systems, IEEE Trans. Automat. Control, 30 (1985), 386-398. · Zbl 0586.93007
[7] Banks, H. T., C. K. Lo, S. Reich, and I. G. Rosen, Numerical studies of identification in nonlinear distributed parameter systems, Proceedings of the Fourth International Conference on the Identification and Control of Distributed Parameter Systems, Vorau, Austria, July 10-16, 1988, International Series of Numerical Mathematics, Vol. 91, Birkha?ser-Verlag, Basel, 1989, pp. 1-20.
[8] Banks, H. T., and K. A. Murphy, Quantitative modeling of growth and dispersal in population models, in Mathematical Topics in Population Biology, Morphogenesis and Neurosciences, Lecture Notes in Biology and Mathematics, Vol. 71, Springer-Verlag, Berlin, 1987, pp. 98-109. · Zbl 0629.92017
[9] Banks, H. T., S. Reich, and I. G. Rosen, An approximation theory for the identification of nonlinear distributed parameter systems, LCDS-CCS Report 88-8, Division of Applied Mathematics, Brown University, Providence, RI (1988) and SIAM J. Control Optim., 28 (1990), 552-569. · Zbl 0722.47058
[10] Banks, H. T., and I. G. Rosen, Fully discrete approximation methods for the estimation of parabolic systems and boundary parameters, Acta Appl. Math., 7 (1986), 1-34. · Zbl 0611.65082
[11] Banks, H. T., and I. G. Rosen, Numerical schemes for the estimation of functional parameters in distributed models for mixing mechanisms in lake and sea sediment cores, Inverse Problems, 3 (1987), 1-23. · Zbl 0644.76047
[12] Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. · Zbl 0328.47035
[13] Crandall, M. G., and A. Pazy, Nonlinear evolution equations in Banach space, Israel J. Math., 11 (1972), 57-94. · Zbl 0249.34049
[14] Csipke, R., The Identification of Time-Varying Parameters in a Distributed Model for Biological Mixing in Deep-Sea Sediment Cores, Master’s Thesis, Department of Mathematics, University of Southern California, Los Angeles, May 1990.
[15] Goldstein, J. A., Approximation of nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 24 (1972), 558-573. · Zbl 0231.47037
[16] Hille, E., and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications, Volume XXXI, American Mathematical Society, Providence, RI, 1957. · Zbl 0078.10004
[17] Kluge, R., and H. Langmach, On some problems of determination of functional parameters in partial differential equations, in Distributed Parameter Systems: Modeling and Identification, Springer Lecture Notes in Control and Information Sciences, Vol. 1, Springer-Verlag, Berlin, 1978, pp. 298-309. · Zbl 0371.49005
[18] Kluge, R., and H. Langmach, On the determination of some rheologic properties of mechanical media, Abh. Akad. Wiss. DDR, 6 (1978), 141-158. · Zbl 0425.73004
[19] Langmach, H., On the determination of functional parameters in some parabolic differential equations, Abh. Akad. Wiss. DDR, 6 (1978), 175-184. · Zbl 0437.35008
[20] Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971. · Zbl 0203.09001
[21] Oden, J. T., and J. N. Reddy, Mathematical Theory of Finite Elements, Wiley, 1976. · Zbl 0336.35001
[22] Okubo, A., Diffusion and Ecological Problems: Mathematical Models, Springer-Verlag, New York, 1980. · Zbl 0422.92025
[23] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. · Zbl 0516.47023
[24] Raviart, P. A., Sur l’approximation de certaines ?quations d’?volution lin?aires et non lin?aires, J. Math. Pures Appl., 46 (1967), 109-183. · Zbl 0198.49901
[25] Schultz, M. H., Spline Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1973. · Zbl 0333.41009
[26] Shang, G., and G. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973. · Zbl 0356.65096
[27] Slattery, J. C., Quasi-linear heat and mass transfer, I. The constitutive equations, Appl. Sci Res., A, 12 (1963), 51-56. · Zbl 0112.41603
[28] Slattery, J. C., Quasi-linear heat and mass transfer, II. Analysis of experiments, Appl. Sci Res, A., 12 (1963), 57-65. · Zbl 0112.41603
[29] Swartz, B. K., and R. S. Varga, Error bounds for spline and L-spline interpolation, J. Approx. Theory, 6 (1972), 6-49. · Zbl 0242.41008
[30] Weiss, G. H., Equations for the age structure of growing populations, Bull. Math. Biophys., 30 (1985), 427-435. · Zbl 0165.23304
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