Estimation of discontinuous coefficients and boundary parameters for hyperbolic systems
Authors:
Patricia K. Lamm and Katherine A. Murphy
Journal:
Quart. Appl. Math. 46 (1988), 1-22
MSC:
Primary 35L55; Secondary 35R05, 65P05, 86A15
DOI:
https://doi.org/10.1090/qam/934677
MathSciNet review:
934677
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Abstract: We consider the problem of estimating discontinuous coefficients, including locations of discontinuities, that occur in second-order hyperbolic systems typical of those arising in 1-D surface seismic problems. In addition, we treat the problem of identifying unknown parameters that appear in boundary conditions for the system. A spline-based approximation theory is presented, together with related convergence findings and representative numerical examples.
- H. W. Alt, K.-H. Hoffmann, and J. Sprekels, A numerical procedure to solve certain identification problems, Optimal control of partial differential equations (Oberwolfach, 1982) Internat. Schriftenreihe Numer. Math., vol. 68, Birkhäuser, Basel, 1984, pp. 11–43. MR 759925
- 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, DOI https://doi.org/10.1007/BFb0005642
- H. T. Banks, James M. Crowley, and Karl Kunisch, Cubic spline approximation techniques for parameter estimation in distributed systems, IEEE Trans. Automat. Control 28 (1983), no. 7, 773–786. MR 716919, DOI https://doi.org/10.1109/TAC.1983.1103310
- H. T. Banks, P. M. Kareiva, and P. K. Lamm, Modeling insect dispersal and estimating parameters when mark-release techniques may cause initial disturbances, J. Math. Biol. 22 (1985), no. 3, 259–277. MR 813398, DOI https://doi.org/10.1007/BF00276485
- H. T. Banks and K. Kunisch, An approximation theory for nonlinear partial differential equations with applications to identification and control, SIAM J. Control Optim. 20 (1982), no. 6, 815–849. MR 675572, DOI https://doi.org/10.1137/0320059
- H. T. Banks and Patricia Daniel Lamm, Estimation of variable coefficients in parabolic distributed systems, IEEE Trans. Automat. Control 30 (1985), no. 4, 386–398. MR 786716, DOI https://doi.org/10.1109/TAC.1985.1103955
- H. T. Banks and K. A. Murphy, Estimation of coefficients and boundary parameters in hyperbolic systems, SIAM J. Control Optim. 24 (1986), no. 5, 926–950. MR 854063, DOI https://doi.org/10.1137/0324055
- David L. Brown, A note on the numerical solution of the wave equation with piecewise smooth coefficients, Math. Comp. 42 (1984), no. 166, 369–391. MR 736442, DOI https://doi.org/10.1090/S0025-5718-1984-0736442-3
- Kenneth P. Bube and Robert Burridge, The one-dimensional inverse problem of reflection seismology, SIAM Rev. 25 (1983), no. 4, 497–559. MR 788323, DOI https://doi.org/10.1137/1025122
J. A. Burns and E. M. Cliff, An approximation technique for the control and identification of hybrid systems, Dyn. & Control of Large Flexible Spacecraft, 3rd VPI & SU/AIAA Symposium, pp. 269–284, 1981
- Guy Chavent, About the stability of the optimal control solution of inverse problems, Inverse and improperly posed problems in differential equations (Proc. Conf., Math. Numer. Methods, Halle, 1979) Math. Res., vol. 1, Akademie-Verlag, Berlin, 1979, pp. 45–58. MR 536167
F. Colonius and K. Kunisch, Stability for parameter estimation in two point boundary value problems, Inst. für Math. Bericht No. 50–1984, Tech. Univ. Graz, October, 1984
C. Kravaris and J. Seinfeld, Identification of parameters in distributed parameter systems, SIAM J. Control and Optimization, to appear
- K. Kunisch and L. White, The parameter estimation problem for parabolic equations and discontinuous observation operators, SIAM J. Control Optim. 23 (1985), no. 6, 900–927. MR 809541, DOI https://doi.org/10.1137/0323052
- Patricia K. Lamm, Estimation of discontinuous coefficients in parabolic systems: applications to reservoir simulation, SIAM J. Control Optim. 25 (1987), no. 1, 18–37. MR 872448, DOI https://doi.org/10.1137/0325002
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- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
- P. M. Prenter, Splines and variational methods, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1975. Pure and Applied Mathematics. MR 0483270
- Martin H. Schultz, Spline analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Automatic Computation. MR 0362832
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H. W. Alt, K.-H. Hoffman, and J. Sprekels, A numerical procedure to solve certain identification problems, Int. Ser. Numer. Math. 68, 11–43 (1984)
H. T. Banks, On a variational approach to some parameter estimation problems, ICASE Rep. No. 85–32, ICASE, M.S. 132C, NASA Langley Research Center, Hampton, VA 23665, June 1985
H. T. Banks, J. M. Crowley, and K. Kunisch, Cubic spline approximation techniques for parameter estimation in distributed systems, IEEE Trans. Auto. Control 28, 773–786 (1983)
H. T. Banks, P. M. Kareiva, P. K. Lamm, Modeling insect dispersal and estimating parameters when mark-release techniques may cause initial disturbances, J. Math. Biology 22, 259–277 (1985)
H. T. Banks and K. Kunisch, An approximation theory for nonlinear partial differential equations with applications to identification and control, SIAM J. Control and Optimization 20, 815–849 (1982)
H. T. Banks and Patricia Daniel Lamm, Estimation of variable coefficients in parabolic distributed systems, IEEE Trans. Auto. Control. 30, 386–398 (1985)
H. T. Banks and K. A. Murphy, Estimation of coefficients and boundary parameters in hyperbolic systems, LCDS Rep. 84-5, February 1984, Brown University; SIAM J. Control and Optimization 24, 926–950 (1986)
D. L. Brown, A note on the numerical solution of the wave equation with piecewise smooth coefficients, Math. Comp. 42, 369–391 (1984)
K. P. Bube and R. Burridge, The one-dimensional inverse problem of reflection seismology, SIAM Review 25, 497–559 (1983)
J. A. Burns and E. M. Cliff, An approximation technique for the control and identification of hybrid systems, Dyn. & Control of Large Flexible Spacecraft, 3rd VPI & SU/AIAA Symposium, pp. 269–284, 1981
G. Chavent, About the stability of the optimal control solution of inverse problems, Inverse and Improperly Posed Problems in Differential Equations, G. Anger, editor, Akademie-Verlag, Berlin, pp. 45–58, 1979
F. Colonius and K. Kunisch, Stability for parameter estimation in two point boundary value problems, Inst. für Math. Bericht No. 50–1984, Tech. Univ. Graz, October, 1984
C. Kravaris and J. Seinfeld, Identification of parameters in distributed parameter systems, SIAM J. Control and Optimization, to appear
K. Kunisch and L. W. White, Identification under approximation for an elliptic boundary value problem, April, 1985; SIAM J. Control and Optimization, submitted
P. K. Lamm, Estimation of discontinuous coefficients in parabolic systems: applications to reservoir simulation, Jan. 1984; SIAM J. Control and Optimization 25, 18–37 (1987)
J. L. Lions, Some aspects of modeling problems in distributed parameter systems, Proc. IFIP Working Conf. (Rome 1976) Ruberti, ed., Lecture notes in Control and Info. 1, pp. 11–41, Springer-Verlag, Berlin, 1978
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences 44, Springer-Verlag, N. Y., 1983
P. M. Prenter, Splines and variational methods, John Wiley & Sons, N.Y., 1975
M. H. Schultz, Spline analysis, Prentice-Hall, Englewood Cliffs, N.J., 1973
I. Stakgold, Green’s Functions and Boundary Value Problems, John Wiley & Sons, 1979
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Article copyright:
© Copyright 1988
American Mathematical Society