Identification of discontinuous coefficients in elliptic problems using total variation regularization

SIAM Journal on Scientific Computing

Volumn 25, Issue 3, 2003, Pages 881-904

  Chan, Tony F ^a   Tai, Xue Cheng ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF BERGEN   (Norway)

Author keywords

Elliptic; Inverse problems; Noise removal; Parameter estimation; Regularization; Total variational norm

Indexed keywords

GEOMETRY; INVERSE PROBLEMS; MATHEMATICAL INSTRUMENTS; MATHEMATICAL MODELS; NUMERICAL ANALYSIS; OSCILLATIONS; PARAMETER ESTIMATION; VISCOSITY OF LIQUIDS;

ELLIPTIC; REGULARIZATION; TOTAL VARIATION NORMS;

PROBLEM SOLVING;

EID: 2942612639     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827599326020     Document Type: Article

References (51)

1. R. ACAR, Identification of the coefficient in elliptic equations, SIAM J. Control Optim., 31 (1993), pp. 1221-1244.
2. L. ALVAREZ, P.-L. LIONS, AND J.-M. MOREL, Image selective smoothing and edge detection by nonlinear diffusion. II, SIAM J. Numer. Anal., 29 (1992), pp. 845-866.
3. R. S. ANDERSSEN AND M. HEGLAND, For numerical differentiation, dimensionality can be a blessing!, Math. Comp., 68 (1999), pp. 1121-1141.
4. R. E. BANK, J. F. BÜRGLER, W. FICHTNER, AND R. K. SMITH, Some upwinding techniques for finite element approximations of convection-diffusion equations, Numer. Math., 58 (1990), pp. 185-202.
5. H. BEN AMEUR, G. CHAVENT, AND J. JAFFRÉ, Refinement and coarsening indicators for adap tive parametrization: Application to the estimation of hydraulic transmissivities, Inverse Problems, 18 (2002), pp. 775-794.
6. G. BRUCHNER, S. HANDROCK-MEYER, AND H. LANGMACH, An inverse problem for 2d ground-water modelling, Inverse Problems, 14 (1998), pp. 835-851.
7. M. BRÜHL AND M. HANKE, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems, 16 (2000), pp. 1029-1042.
8. K. BRUSDAL AND T. MANNSETH, Basis norm rescaling for nonlinear parameter estimation, SIAM J. Sci. Comput., 21 (2000), pp. 2114-2125.
9. K. BRYAN, Numerical recovery of certain discontinuous electrical conductivities, Inverse Prob lems, 7 (1991), pp. 827-840.
10. K. BRYAN, V. LIEPA, AND M. VOGELIUS, Reconstruction of multiple cracks from experimental electrostatic boundary measurements, in Inverse Problems and Optimal Design in Industry, H. W. Engl and J. McLaughlin, eds., B. G. Teubner, Stuttgart, (1994), pp. 147-167.
11. T. F. CHAN, G. H. GOLUB, AND P. MULET, A nonlinear primal-dual method for total variation-based image restoration, in ICAOS '96 (Paris, 1996), Lecture Notes in Control and Inform. Sci. 219, Springer, London, (1996), pp. 241-252.
12. T. F. CHAN, G. H. GOLUB, AND P. MULET, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), pp. 1964-1977.
13. T. F. CHAN AND X.-C. TAI, Augmented Lagrangian and total variation methods for recovering discontinuous coefficients from elliptic equations, in Computational Science for the 21st Century, M. Bristeau, G. Etgen, W. Fitzgibbon, J. L. Lions, J. Periaux, and M. F. Wheeler, eds., John Wiley, New York, 1997, pp. 597-607.
14. T. F. CHAN AND X.-C. TAI, Level Set and Total Variation Regularization for Elliptic Inverse Problems with Discontinuous Coefficients, CAM report 03-15, Mathematics Department, University of California-Los Angeles, Los Angeles, CA, 2003.
15. Z. CHEN AND J. ZOU, An augmented Lagrangian method for identifying discontinuous param eters in elliptic systems, SIAM J. Control Optim., 37 (1999), pp. 892-910.
16. M. CHENEY, D. ISAACSON, AND J. C. NEWELL, Electrical impedance tomography, SIAM Rev., 41 (1999), pp. 85-101.
17. S.-S. CHOW AND R. S. ANDERSSEN, Determination of the transmissivity zonation using a linear functional strategy, Inverse Problems, 7 (1991), pp. 841-851.
18. D. COLTON, J. COYLE, AND P. MONK, Recent developments in inverse acoustic scattering theory, SIAM Rev., 42 (2000), pp. 369-414.
19. D. COLTON, H. W. ENGL, A. K. LOUIS, J. R. MCLAUGHLIN, AND W. RUNDELL, EDS., Surveys on Solution Methods for Inverse Problems, Springer-Verlag, Vienna, 2000.
20. D. COLTON, A. KIRSCH, AND P. MONK, The linear sampling method in inverse scattering theory, in Surveys on Solution Methods for Inverse Problems, D. Colton, H. W. Engl, A. K. Louis, J. R. McLaughlin, and W. Rundell, eds., Springer, Vienna, 2000, pp. 107-118.
21. D. COLTON AND P. MONK, A linear sampling method for the detection of leukemia using microwaves II, SIAM J. Appl. Math., 60 (1999), pp. 241-255.
22. D. C. DOBSON AND F. SANTOSA, An image enhancement technique for electrical impedance tomography, Inverse Problems, 10 (1994), pp. 317-334.
23. I. EKELAND AND R. TEMAM, Convex Analysis and Variational Problems, North-Holland, Ams terdam, 1976.
24. R. EWING, ED., The Mathematics of Reservoir Simulation, Frontiers in Appl. Math, 1, SIAM, Philadelphia, 1984.
25. E. GIUSTI, Minimal Surfaces and Functions of Bounded Variations, Birkhäuser, Basel, 1984.
26. K. J. HERSVIK AND M. S. ESPEDAL, Adaptive hierarchical upscaling of flow in heterogeneous reservoirs based on an a posteriori error estimate, Comput. Geosci., 2 (1998), pp. 311-336.
27. K. ITO AND K. KUNISCH, The augmented Lagrangian method for parameter estimation in elliptic systems, SIAM J. Control Optim., 28 (1990), pp. 113-136.
28. C. JOHNSON, U. NÄVERT, AND J. PITKÄRANTA, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg., 45 (1984), pp. 285-312.
29. Y. L. KEUNG AND J. ZOU, Numerical identifications of parameters in parabolic systems, Inverse Problems, 14 (1998), pp. 83-100.
30. A. KIRSCH, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), pp. 1489-1512.
31. A. KIRSCH, New characterizations of solutions in inverse scattering theory, Appl. Anal., 76 (2000), pp. 319-350.
32. K. ITO, K. KUNISCH, AND Z. LI, Level-set function approach to an inverse interface problem, Inverse Problems, 17 (2001), pp. 1225-1242.
33. C. KRAVARIS AND J. H. SEINFELD, Identification of parameters in distributed parameter systems by regularization, SIAM J. Control Optim., 23 (1985), pp. 217-241.
34. K. KUNISCH AND X.-C. TAI, Nonoverlapping domain decomposition methods for inverse prob lems, in Proceedings of the 9th International Conference on Domain Decomposition Meth ods, 1996, Bergen, Norway, P. Bjørstad, M. Espedal, and D. Keyes, eds., ddm.org, pp. 517-524.
35. K. KUNISCH AND X.-C. TAI, Sequential and parallel splitting methods for bilinear control prob lems in Hilbert spaces, SIAM J. Numer. Anal., 34 (1997), pp. 91-118.
36. K. KUNISCH AND L. W. WHITE, Identifiability under approximation for an elliptic boundary value problem, SIAM J. Control Optim., 25 (1987), pp. 279-297.
37. T. LIN AND E. RAMIREZ, A numerical method for parameter identification of a boundary value problem, Appl. Anal., 69 (1998), pp. 349-379.
38. M. T. NAIR, M. HEGLAND, AND R. S. ANDERSSEN, The trade-off between regularity and stability in Tikhonov regularization, Math. Comp., 66 (1997), pp. 193-206.
39. B. F. NIELSEN AND A. TVEITO, An Upscaling Method for One-Phase Flow in Heterogeneous Reservoirs: A Weighted Output Least Squares Approach, Tech. Report 8, Informatic In stitute, University of Oslo, Oslo, Norway, 1996.
40. S. OSHER AND L. I. RUDIN, Feature-oriented image enhancement using shock filters, SIAM J. Numer. Anal., 27 (1990), pp. 919-940.
41. L. I. RUDIN, S. OSHER, AND E. FATEMI, Nonlinear total variation based noise removal algo rithms, Phys. D, 60 (1992), pp. 259-268.
42. F. SANTOSA, A level-set approach for inverse problems involving obstacles, ESAIM Contrôle Optim. Calc. Var., 1 (1995/96), pp. 17-33 (electronic).
43. O. SCHERZER, H. W. ENGL, AND R. S. ANDERSSEN, Parameter identification from boundary measurements in a parabolic equation arising from geophysics, Nonlinear Anal., 20 (1993), pp. 127-156.
44. X.-C. TAI, J. FROYEN, M. E. ESPEDAL, AND T. F. CHAN, Overlapping domain decomposition and multigrid methods for inverse problems, in Domain Decomposition Methods 10, Tenth International Conference on Domain Decomposition Methods, Contemp. Math., 218, AMS, Providence, RI, 1998, pp. 523-529.
45. X.-C. TAI AND T. KARKAINEN, Identification of a nonlinear parameter in a parabolic equation from a linear equation, Comput. Appl. Math., 14 (1995), pp. 157-184.
46. X.-C. TAI AND P. NEITTAANMAKI, Error estimates for numerical identification of distributed parameters, J. Comput. Math. 10, Suppl. (1992), pp. 66-78.
47. E. VAINIKKO, Courant elements for the inverse problem in the filtration coefficient, in Finite Element Methods; Fifty Years of the Courant Element, K. Krizek, N. Neittaanmaki, and R. Stenberg, eds., Lecture Notes in Pure and Appl. Math. 164, Marcel Dekker, New York, Hong Kong, 1994, pp. 451-461.
48. C. R. VOGEL, Sparse matrix computations arising in distributed parameter identification, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 1027-1037.
49. C. R. VOGEL AND M. E. OMAN, Iterative methods for total variation denoising, SIAM J. Sci. Comput., 17 (1996), pp. 227-238.
50. W. W.-G. YEH, Review of parameter identification procedures in groundwater hydrology: The inverse problems, Water Resources Research, 22 (1986), pp. 95-108.
51. W. P. ZIEMER, Weakly Differentiable Functions, Springer-Verlag, Berlin, New York, 1989.