Identification of space-dependent permeability in nonlinear diffusion equation from interior measurements using wavelet multiscale method

References

• Shah PC, Gavalas GR, Seinfeld JH. Error analysis in history matching, the optimum level of parameterization. Soc. Pet. Eng. J. 1978;6:219–228.
   Google Scholar
• Reynolds AC, He N, Chu L, Oliver D. Reparameterization techniques for generating reservoir descriptions conditioned to variograms and well-test pressure data, SPE 30588. USA: Dallas. Proceedings of the SPE Annual Technical Conference and Exhibition; 1995.
   Google Scholar
• Cuypers M, Dubrule O, Lamy P, Bissell R. Optimal choice of inversion parameters for history matching with the pilot point method. Proceedings of the 6th European Conference on the Mathematics of Oil Recovery. UK: Peebles; 1998.
   Google Scholar
• Grimstad AA, Mannseth T, Nordtvedt JE, Nævdal G. Reservoir characterization through scale splitting. Proceedings of the 7th European Conference on the Mathematics of Oil Recovery, Baveno, Italy; 2000.
   Google Scholar
• Chen Z, Zou J. An augmented Lagrangian method for identifying discontinuous parameters in elliptic systems. SIAM J. Control Optim. 1999;37:892–910.
   Web of Science ®Google Scholar
• Knowles I. Parameter identification for elliptic problems. J. Comput. Appl. Math. 2001;131: 175–194.
   Google Scholar
• Chan TF, Tai XC. Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients. J. Comput. Phys. 2003;193:40–66.
   Web of Science ®Google Scholar
• Gockenbach MS, Khan AA. An abstract framework for elliptic inverse problems: part 1. An output least-squares approach. Math. Mech. Solids. 2007;12:259–276.
   Web of Science ®Google Scholar
• Gockenbach MS, Khan AA. An abstract framework for elliptic inverse problems: part 2. An augmented Lagrangian approach. Math. Mech. Solids. 2009;14:517–539.
   Web of Science ®Google Scholar
• Hayek M, Ackerer P, Sonnendrücker E. A new refinement indicator for adaptive parameterization: application to the estimation of the diffusion coefficient in an elliptic problem. J. Comput. Appl. Math. 2009;224:307–319.
   Web of Science ®Google Scholar
• Kärkkäinen T. Error estimates for distributed parameter identification in parabolic problems with output least squares and Crank–Nicolson method. Appl. Math. 1997;42:259–277.
   Google Scholar
• Keung YL, Zou J. Numerical identifications of parameters in parabolic equations. Inverse Probl. 1998;14:83–100.
   Web of Science ®Google Scholar
• Engl HW, Zou J. A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction. Inverse Probl. 2000;16:1907–1923.
   Web of Science ®Google Scholar
• Kunisch K, Peichl G. Estimation of a temporally and spatially varying diffusion coefficient in a parabolic system by an augmented Lagrangian technique. Numer. Math. 1991;59:473–509.
   Web of Science ®Google Scholar
• Gou B, Zou J. An augmented Lagrangian method for parameter identification in parabolic systems. J. Math. Anal. Appl. 2001;263:49–68.
   Web of Science ®Google Scholar
• Nilssen TK, Tai XC. Parameter estimation with the augmented Lagrangian method for a parabolic equation. J. Optim. Theory Appl. 2005;124:435–453.
   Web of Science ®Google Scholar
• Nilssen TK, Karlsen KH, Mannseth T, Tai XC. Identification of diffusion parameters in a nonlinear convection-diffusion equation using the augmented Lagrangian method. Comput. Geosci. 2009;13:317–329.
   Web of Science ®Google Scholar
• Karalashvili M, Gross S, Marquardt W, Mhamdi A, Reusken A. Identification of transport coefficient models in convection-diffusion equations. SIAM J. Sci. Comput. 2011;33:303–327.
   Web of Science ®Google Scholar
• Malengier B, Keer R. Parameter estimation in convection dominated nonlinear convection-diffusion problems by the relaxation method and the adjoint equation. J. Comput. Appl. Math. 2008;215:477–483.
   Web of Science ®Google Scholar
• Mocenni C, Madeo D, Sparacino E. Linear least squares parameter estimation of nonlinear reaction diffusion equations. Math. Comput. Model. 2011;81:2244–2257.
   Google Scholar
• Ye CR, Sun ZZ. A linearized compact difference scheme for an one-dimensional parabolic inverse problem. Appl. Math. Model. 2009;33:1521–1528.
   Web of Science ®Google Scholar
• Mejia CE, Acosta CD, Saleme KI. Numerical identification of a nonlinear diffusion coefficient by discrete mollification. Comput. Math. Appl. 2011;62:2187–2199.
   Web of Science ®Google Scholar
• Nilssen TK, Mannseth T, Tai XC. Permeability estimation with the augmented Lagrangian method for a nonlinear diffusion equation. Comput. Geosci. 2003;7:27–47.
   Web of Science ®Google Scholar
• Zhao JJ, Liu T, Liu SS. An adaptive homotopy method for permeability estimation of a nonlinear diffusion equation. Inverse Probl. Sci. Eng. (preprint 2012); in press. Available at http://www.tandfonline.com/doi/full/10.1080/17415977.2012.712524#tabModule.
   Google Scholar
• Ou YH, Hasanov A, Liu ZH. Inverse coefficient problems for nonlinear parabolic differential equations. Acta Math. Sin. (Engl. Ser.). 2008;24:1617–1624.
   Web of Science ®Google Scholar
• Kolb P, Collino F, Lailly P. Pre-stack inversion of a 1-D medium. Proc. IEEE. 1986;74:498–508.
   Web of Science ®Google Scholar
• Hackbusch W. Multi-grid methods and applications. Berlin: Spinger-Verlag; 1985.
   Google Scholar
• Fulton SR, Ciesielski PE, Schubert WH. Multigrid methods for elliptic problems: a review. J. Atmos. Sci. 1986;114:943–959.
   Google Scholar
• Borcea L. A nonlinear multigrid for imaging electrical conductivity and permittivity at low frequency. Inverse Probl. 2001;17:329–359.
   Web of Science ®Google Scholar
• Ye JC, Bouman CA, Webb KJ, Millane RP. Nonlinear multigrid algorithms for Bayesian optical diffusion tomography. IEEE Trans. Image. Process. 2001;10:909–922.
   Web of Science ®Google Scholar
• Oh S, Milstein AB, Bouman CA, Webb KJ. Multigrid algorithms for optimization and inverse problems, Proceedings of the SPIE/IS &T Conference on Computational Imaging; 2003. Santa Clara, CA; 2003.
   Google Scholar
• Oh S, Milstein AB, Bouman CA, Webb KJ. A general framework for nonlinear multigrid inversion. IEEE Trans. Image. Process. 2005;14:125–140.
   PubMed Web of Science ®Google Scholar
• Daubechies I. Ten lectures on wavelets. Philadelphia, PA: SIAM; 1992.
   Google Scholar
• Liu J. A multiresolution method for distributed parameter estimation. SIAM J. Sci. Comput. 1993;14:389–405.
   Web of Science ®Google Scholar
• Fu HS, Han B. A wavelet multiscale method for the inverse problems of a two-dimensional wave equation. Inverse Probl. Sci. Eng. 2004;12:643–656.
   Web of Science ®Google Scholar
• Fu HS, Han B, Gai GQ. A wavelet multiscale-homotopy method for the inverse problem of two-dimensional acoustic wave equation. Appl. Math. Comput. 2007;190:576–582.
   Web of Science ®Google Scholar
• Zhang XM, Liu KA, Liu JQ. The wavelet multiscale method for inversion of porosity in the fluid-saturated porous media. Appl. Math. Comput. 2006;180:419–427.
   Web of Science ®Google Scholar
• He Y, Han B. A wavelet adaptive-homotopy method for inverse problem in the fluid-saturated porous media. Appl. Math. Comput. 2009;208:189–196.
   Web of Science ®Google Scholar
• Zhang XM, Liu JQ, Liu KA. Porosity inversion of 1-D two-phase medium with wavelet multiscale method. Acta Phys. Sin. 2008;57:654–660.
   Web of Science ®Google Scholar
• Zhang XM, Zhou CY, Liu JQ, Liu KA. Multiparameter identification of fluid-saturated porous medium with the wavelet multiscale method. J. Porous Media. 2009;12:255–264.
   Web of Science ®Google Scholar
• Ding L, Han B, Liu JQ. A wavelet multiscale method for inversion of Maxwell equations. Appl. Math. Mech.-Engl. Ed. 2009;30:1035–1044.
   Web of Science ®Google Scholar
• DuChateau P, Thelwell R, Butters G. Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient. Inverse Probl. 2004;20:601–625.
   Web of Science ®Google Scholar
• Hasanov A, DuChateau P, Pektas B. An adjoint problem approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equation. J. Inverse Ill-posed Probl. 2006;14:435–463.
   Google Scholar
• Hasanov A. Convexity argument for monotone potential operators and its application. Nonlinear Anal. TWA. 2000;41:907–919.
   Web of Science ®Google Scholar
• Hasanov A, Tatar S. Semi-analytic inversion method for determination of elastoplastic properties of power hardening materials from limited torsional experiment. Inverse Probl. Sci. Eng. 2010;18:265–278.
   Web of Science ®Google Scholar
• Mallat SG. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 1989;11:674–693.
   Web of Science ®Google Scholar
• Nilssen TK. Parameter estimation with the augmented Lagrangian method and a study of some fourth order problems [Ph.D. dissertation]. University of Bergen; 2001.
   Google Scholar