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 | References | Similar Articles | Additional Information

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.

**[1]**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) MR**759925****[2]**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 MR**897549****[3]**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) MR**716919****[4]**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) MR**813398****[5]**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) MR**675572****[6]**H. T. Banks and Patricia Daniel Lamm,*Estimation of variable coefficients in parabolic distributed systems*, IEEE Trans. Auto. Control.**30**, 386-398 (1985) MR**786716****[7]**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) MR**854063****[8]**D. L. Brown,*A note on the numerical solution of the wave equation with piecewise smooth coefficients*, Math. Comp.**42**, 369-391 (1984) MR**736442****[9]**K. P. Bube and R. Burridge,*The one-dimensional inverse problem of reflection seismology*, SIAM Review**25**, 497-559 (1983) MR**788323****[10]**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**[11]**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 MR**536167****[12]**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**[13]**C. Kravaris and J. Seinfeld,*Identification of parameters in distributed parameter systems*, SIAM J. Control and Optimization, to appear**[14]**K. Kunisch and L. W. White,*Identification under approximation for an elliptic boundary value problem*, April, 1985; SIAM J. Control and Optimization, submitted MR**809541****[15]**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) MR**872448****[16]**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 MR**0528467****[17]**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences**44**, Springer-Verlag, N. Y., 1983 MR**710486****[18]**P. M. Prenter,*Splines and variational methods*, John Wiley & Sons, N.Y., 1975 MR**0483270****[19]**M. H. Schultz,*Spline analysis*, Prentice-Hall, Englewood Cliffs, N.J., 1973 MR**0362832****[20]**I. Stakgold,*Green's Functions and Boundary Value Problems*, John Wiley & Sons, 1979 MR**537127**

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DOI:
https://doi.org/10.1090/qam/934677

Article copyright:
© Copyright 1988
American Mathematical Society