Abstract
We develop an abstract framework and convergence theory for Galerkin approximation for inverse problems involving the identification of nonautonomous, in general nonlinear, distributed parameter systems. We provide a set of relatively easily verified conditions which are sufficient to guarantee the existence of optimal solutions and their approximation by a sequence of solutions to a sequence of approximating finite-dimensional identification problems. Our approach is based upon the theory of monotone operators in Banach spaces and is applicable to a reasonably broad class of nonlinear distributed systems. Operator theoretic and variational techniques are used to establish a fundamental convergence result. An example involving evolution systems with dynamics described by nonstationary quasi-linear elliptic operators along with some applications and numerical results are presented and discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
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.
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.
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.
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.
Banks, H. T., and P. D. Lamm, Estimation of variable coefficients in parabolic distributed systems, IEEE Trans. Automat. Control, 30 (1985), 386–398.
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, Birkhaüser-Verlag, Basel, 1989, pp. 1–20.
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.
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.
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.
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.
Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.
Crandall, M. G., and A. Pazy, Nonlinear evolution equations in Banach space, Israel J. Math., 11 (1972), 57–94.
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.
Goldstein, J. A., Approximation of nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 24 (1972), 558–573.
Hille, E., and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications, Volume XXXI, American Mathematical Society, Providence, RI, 1957.
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.
Kluge, R., and H. Langmach, On the determination of some rheologic properties of mechanical media, Abh. Akad. Wiss. DDR, 6 (1978), 141–158.
Langmach, H., On the determination of functional parameters in some parabolic differential equations, Abh. Akad. Wiss. DDR, 6 (1978), 175–184.
Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971.
Oden, J. T., and J. N. Reddy, Mathematical Theory of Finite Elements, Wiley, 1976.
Okubo, A., Diffusion and Ecological Problems: Mathematical Models, Springer-Verlag, New York, 1980.
Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
Raviart, P. A., Sur l'approximation de certaines équations d'évolution linéaires et non linéaires, J. Math. Pures Appl., 46 (1967), 109–183.
Schultz, M. H., Spline Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1973.
Shang, G., and G. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973.
Slattery, J. C., Quasi-linear heat and mass transfer, I. The constitutive equations, Appl. Sci Res., A, 12 (1963), 51–56.
Slattery, J. C., Quasi-linear heat and mass transfer, II. Analysis of experiments, Appl. Sci Res, A., 12 (1963), 57–65.
Swartz, B. K., and R. S. Varga, Error bounds for spline and L-spline interpolation, J. Approx. Theory, 6 (1972), 6–49.
Weiss, G. H., Equations for the age structure of growing populations, Bull. Math. Biophys., 30 (1985), 427–435.
Author information
Authors and Affiliations
Additional information
Communicated by I. Lasiecka
Part of this research was carried out while the first and third authors were visiting scientists at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center. Hampton, VA, which is operated under NASA Contracts NAS1-17070 and NAS1-18107. Also, a portion of this research was carried out with computational resources made available through a grant to the second and third authors from the San Diego Supercomputer Center operated for the National Science Foundation by General Atomics, San Diego, CA. The research of the first author was supported in part under Grants NSF MCS-8504316, NASA NAG-1-517, AFOSR-84-0398, and AFOSR-F49620-86-C-0111. The second author's research was supported in part by the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund. The research of the third author was supported in part under Grants AFOSR-84-0393 and AFOSR-87-0356.
Rights and permissions
About this article
Cite this article
Banks, H.T., Reich, S. & Rosen, I.G. Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems. Appl Math Optim 24, 233–256 (1991). https://doi.org/10.1007/BF01447744
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01447744