Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Boundary shape identification problems in two-dimensional domains related to thermal testing of materials

Authors: H. T. Banks and Fumio Kojima
Journal: Quart. Appl. Math. 47 (1989), 273-293
MSC: Primary 65M99; Secondary 73U05, 93B30
DOI: https://doi.org/10.1090/qam/998101
MathSciNet review: 998101
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the identification of the geometrical structure of the system boundary for a two-dimensional diffusion system. The domain identification problem treated here is converted into an optimization problem based on a fit-to-data criterion and theoretical convergence results for approximate identification techniques are discussed. Results of numerical experiments to demonstrate the efficacy of the theoretical ideas are reported.

References [Enhancements On Off] (What's this?)

  • [1] O. Axelsson and V. A. Barker, Finite element solution of boundary value problems, Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. Theory and computation. MR 758437
  • [2] H. T. Banks, On a variational approach to some parameter estimation problems, Distributed parameter systems (Vorau, 1984) Lect. Notes Control Inf. Sci., vol. 75, Springer, Berlin, 1985, pp. 1–23. MR 897549, https://doi.org/10.1007/BFb0005642
  • [3] H. T. Banks and K. Ito, A theoretical framework for convergence and continuous dependence of estimates in inverse problems for distributed parameter systems, Appl. Math. Lett. 1 (1988), no. 1, 13–17. MR 947163, https://doi.org/10.1016/0893-9659(88)90166-8
  • [4] H. T. Banks and K. Ito, A unified framework for approximation in inverse problems for distributed parameter systems, Control Theory Adv. Tech. 4 (1988), no. 1, 73–90. MR 941397
  • [5] D. Begis and R. Glowinski, Application de la méthode des éléments finis à l’approximation d’un problème de domaine optimal. Méthodes de résolution des problèmes approchés, Appl. Math. Optim. 2 (1975/76), no. 2, 130–169 (French). MR 0443372, https://doi.org/10.1007/BF01447854
  • [6] Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR 507062
  • [7] Denise Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl. 52 (1975), no. 2, 189–219. MR 0385666, https://doi.org/10.1016/0022-247X(75)90091-8
  • [8] D. M. Heath, C. S. Welch, and W. P. Winfree, Quantitative thermal diffusivity measurements of composites, in Review of Progress in Quantitative Nondestructive Evaluation, D. G. Thompson and D. E. Chimenti (eds.), Plenum Publ., Vol. 5B, 1986, pp. 1125-1132
  • [9] J.-L. Lions, Optimal control of systems governed by partial differential equations., Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971. MR 0271512
  • [10] Olivier Pironneau, Optimal shape design for elliptic systems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR 725856
  • [11] E. Polak, Computational methods in optimization. A unified approach, Mathematics in Science and Engineering, Vol. 77, Academic Press, New York-London, 1971. MR 0282511
  • [12] J. B. Rosen, The gradient projection method for nonlinear programming. I. Linear constraints, J. Soc. Indust. Appl. Math. 8 (1960), 181–217. MR 0112750
  • [13] J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim. 2 (1980), no. 7-8, 649–687 (1981). MR 619172, https://doi.org/10.1080/01630563.1980.10120631
  • [14] Y. Sunahara, Sh. Aihara, and F. Kojima, A Method for Spatial Domain Identification of Distributed Parameter Systems under Noisy Observations, in Proc. 9th IFAC World Congress, Budapest, Hungary, 1984, Pergamon Press, New York, 1984
  • [15] Y. Sunahara and F. Kojima, Boundary Identification for a Two Dimensional Diffusion System under Noisy Observations, in Proc. 4th IFAC Symp. Control of Distributed Parameter Systems, UCLA, California, 1986, Pergamon Press, New York, 1986

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 65M99, 73U05, 93B30

Retrieve articles in all journals with MSC: 65M99, 73U05, 93B30

Additional Information

DOI: https://doi.org/10.1090/qam/998101
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society