A computational method to estimate the unknown coefficient in a wave equation using boundary measurements

References

• Gerver, ML, 1970. Inverse problem for the one-dimensional wave equation, Geophys. J. R. Astronom. Soc. 21 (1970), pp. 337–357.
   Google Scholar
• Bamberger, A, Chavent, G, and Lailly, P, 1979. About the stability of the inverse problem in 1-D wave equations – Application to the interpretation of seismic profiles, Appl. Math. Optim. 5 (1979), pp. 1–47.
   Web of Science ®Google Scholar
• Macbain, JA, and Bendar, JB, 1986. Existence and uniqueness properties for one-dimensional magnetotelluric inversion problem, J. Math. Phys. 27 (1986), pp. 645–649.
   Web of Science ®Google Scholar
• Prilepko, AI, and Orlovskii, DG, 1985. Determination of the evolution parameter of an equation and inverse problems of mathematical physics, I and II, J. Differ. Eqns. 21 (1) (1985), pp. 119–129.
   Google Scholar
• Iamnuvilov, OYu, and Yamamoto, M, 2001. Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Probl. 17 (2001), pp. 717–728.
   Web of Science ®Google Scholar
• Iamnuvilov, OYu, and Yamamoto, M, 2003. Determination of a coefficient in an acoustic equation with a single measurement, Inverse Probl. 19 (2003), pp. 157–171.
   Web of Science ®Google Scholar
• Jellali, D, 2006. An inverse problem for the acoustic wave equation with finite sets of boundary data, J. Inverse Ill-posed Probl. 14 (7) (2006), pp. 665–684.
   Google Scholar
• Isakov, V, 2005. Inverse Problems for Partial Differential Equations. New York: Springer; 2005.
   Google Scholar
• Lin, T, and Ewing, R, 1992. Direct numerical method for an inverse problem of hyperbolic equations, Numer. Methods Partial Differ. Eqns. 8 (1992), pp. 551–574.
   Google Scholar
• Xie, GQ, 1986. A new iterative method for solving the coefficient inverse problem of the wave equation, Commun. Pure Appl. Math. 39 (3) (1986), pp. 307–322.
   Web of Science ®Google Scholar
• Chen, YM, and Xie, GQ, 1986. An iterative method for simultaneous determination of bulk and shear moduli and density variation, J. Comput. Phys. 62 (1986), pp. 143–163.
   Web of Science ®Google Scholar
• Ding, H, Zheng, Z, and Xu, S, 1990. A new approach to inverse problems of wave equations, Appl. Math. Mech. 11 (12) (1990), pp. 1113–1117.
   Google Scholar
• Lapidus, L, and Pinder, GF, 1982. Numerical Solution of Partial Differential Equations in Science and Engineering. New York: Wiley; 1982.
   Google Scholar
• Mitchell, AR, and Griffiths, DF, 1980. The Finite Difference Methods in Partial Differential Equations. Chichester: Wiley; 1980.
   Google Scholar
• Gustafsson, B, 2007. High Order Difference Methods for Time Dependent PDE. Berlin, Heidelberg: Springer; 2007.
   Google Scholar
• Lim, H, Kim, S, and Douglas, J, 2007. Numerical methods for viscous and nonviscous wave equations, Appl. Numer. Math. 57 (2007), pp. 194–212.
   Web of Science ®Google Scholar
• Kim, S, and Lim, H, 2007. High-order schemes for acoustic waveform simulation, Appl. Numer. Math. 57 (2007), pp. 402–414.
   Web of Science ®Google Scholar
• Kim, S, 2006. A 4th-order implicit procedure for the simulation of acoustics. Istanbul, Turkey: Proceedings of the 9th WSEAS International Conference on Applied Mathematics; 2006, pp. 599–604.
   Google Scholar
• Johnson, C, 1987. Numerical Solutions of Partial Differential Equations by the Finite Element Method. New York: Cambridge University Press; 1987.
   Google Scholar
• Gladwell, GML, 1986. "Inverse Problems in Vibration". In: Martin Nijhoff Publishers. Dordrecht; 1986.
   Google Scholar
• Cohen, G, and Joly, P, 1990. Fourth order schemes for the heterogeneous acoustic equation, Comput. Methods Appl. Mech. Eng. 80 (1990), pp. 397–407.
   Web of Science ®Google Scholar
• Cohen, G, and Joly, P, 1993. Construction and analysis of fourth-order finite difference schemes fro the acoustic wave equation in nonhomogeneous media, SIAM J. Numer. Anal. 33 (4) (1993), pp. 1266–1302.
   Google Scholar
• Shubin, G, and Bell, J, 1987. A modified equation approach to construction fourth order methods for acoustic wave equations, SIAM J. Sci. Stat. Comput. 8 (1987), pp. 135–151.
   Web of Science ®Google Scholar
• Adam, Y, 1997. Highly accurate compact implicit methods and boundary conditions, J. Comput. Phys. 24 (1997), pp. 10–22.
   Google Scholar
• Lele, SK, 1992. Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103 (1992), pp. 16–42.
   Web of Science ®Google Scholar
• Wilson, RV, Demuren, AO, and Carpenter, M, Higher-order compact schemes for numerical simulation of incompressible flows, ICASE Rep., No. 98-13, 1998.
   Google Scholar
• Hirsch, RS, 1975. Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys. 19 (1975), pp. 90–109.
   Web of Science ®Google Scholar
• Giering, R, and Kaminski, T, 1998. Recipes for adjoint code construction article, ACM Trans. Math. Softw. 24 (1998), pp. 437–474.
   Web of Science ®Google Scholar
• Giering, R, 1999. Tangent linear and Adjoint Model Compiler. (1999), Available at http://www.autodiff.com/tamc.
   Google Scholar
• Courty, F, Araya, M, Vazquez, M, Pascual, V, Dervieux, A, Bellesso, N, Hascoet, L, and Greborio, R-M, 2003. TAPENADE On-line Automatic Differentiation Engine. France: INRIA; 2003, Available at http://www-sop.inria.fr/tropics/index.html.
   Google Scholar
• Nocedal, J, 1980. Updating quasi-Newton matrices with limited storage, Math. Comput. 35 (1980), pp. 773–782.
   Web of Science ®Google Scholar
• Nocedal, J, Department of Electrical & Computer Engineering. Northwestern University., Available at http://users.ece.northwestern.edu/nocedal/lbfgs.html.
   Google Scholar
• Liu, DC, and Nocedal, J, 1989. On the limited memory method for large scale optimization, Math. Program. B 45 (3) (1989), pp. 503–528.
   Web of Science ®Google Scholar
• Gilbert, JC, and Nocedal, J, 1992. Global convergence properties of conjugate gradient methods for optimization, SIAM J. Optim. 2 (1) (1992), pp. 21–42.
   Web of Science ®Google Scholar