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A Fast Local Level Set Method for Inverse Gravimetry

Published online by Cambridge University Press:  20 August 2015

Victor Isakov ,
Shingyu Leung  and
Jianliang Qian
Victor Isakov*
Affiliation:
Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas, USA
Shingyu Leung*
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR
Jianliang Qian*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
*
Email:victor.isakov@wichita.edu
Email:masyleung@ust.hk
Corresponding author.Email:qian@math.msu.edu
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Abstract

We propose a fast local level set method for the inverse problem of gravimetry. The theoretical foundation for our approach is based on the following uniqueness result: if an open set D is star-shaped or x3-convex with respect to its center of gravity, then its exterior potential uniquely determines the open set D. To achieve this purpose constructively, the first challenge is how to parametrize this open set D as its boundary may have a variety of possible shapes. To describe those different shapes we propose to use a level-set function to parametrize the unknown boundary of this open set. The second challenge is how to deal with the issue of partial data as gravimetric measurements are only made on a part of a given reference domain Ω. To overcome this difficulty we propose a linear numerical continuation approach based on the single layer representation to find potentials on the boundary of some artificial domain containing the unknown set D. The third challenge is how to speed up the level set inversion process. Based on some features of the underlying inverse gravimetry problem such as the potential density being constant inside the unknown domain, we propose a novel numerical approach which is able to take advantage of these features so that the computational speed is accelerated by an order of magnitude. We carry out numerical experiments for both two- and three-dimensional cases to demonstrate the effectiveness of the new algorithm.

Keywords

52B1065D1868U0568U07Level set methodsinverse gravimetryfast algorithmsnumerical continuation
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 4 , October 2011 , pp. 1044 - 1070
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Bertete-Aguirre, H., Cherkaev, E., and Oristaglio, M.. Non-smooth gravity problem with total variation penalization functional. Geophys. J. Int., 149:499–507, 2002.CrossRefGoogle Scholar
[2]Burger, M.. A level set method for inverse problems. Inverse Problems, 17:1327–1356, 2001.Google Scholar
[3]Burger, M. and Osher, S.. A survey on level set methods for inverse problems and optimal design. European J. Appl. Math., 16:263–301, 2005.Google Scholar
[4]Cecil, T., Osher, S. J., and Qian, J.. Simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension. J. Comp. Phys., 213:458–473, 2006.Google Scholar
[5]Cecil, T., Qian, J., and Osher, S. J.. Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions. J. Comp. Phys., 196:327–347, 2004.Google Scholar
[6]De Lillo, T., Isakov, V., Valdivia, N., and Wang, L.. The detection of surface vibrations from interior acoustical pressure. Inverse Problems, 19:507–524, 2003.Google Scholar
[7]Dorn, O. and Lesselier, D.. Level set methods for inverse scattering. Inverse Problems, 22:R67–R131, 2006.Google Scholar
[8]Isakov, V.. Inverse Source Problems. American Mathematical Society, Providence, Rhode Island, 1990.Google Scholar
[9]Jiang, G. S. and Peng, D.. Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput., 21:2126–2143, 2000.Google Scholar
[10]Leung, S., Qian, J., and Osher, S. J.. A level set method for three dimensional paraxial geometrical optics with multiple sources. Comm. Math. Sci., 2:657–686, 2004.Google Scholar
[11]Litman, A., Lesselier, D., and Santosa, F.. Reconstruction of a 2-D binary obstacle by controlled evolution of a level-set. Inverse Problems, 14:685–706, 1998.Google Scholar
[12]Osher, S. J. and Fedkiw, R.. Level Set Methods and Dynamic Implicit Surfaces. Springer, 2003.Google Scholar
[13]Osher, S. J. and Sethian, J. A.. Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys., 79:12–49, 1988.Google Scholar
[14]Peng, D., Merriman, B., Osher, S., Zhao, H. K., and Kang, M.. A pde-based fast local level set method. J. Comput. Phys., 155:410–438, 1999.Google Scholar
[15]Qian, J. and Leung, S.. A level set method for paraxial multivalued traveltimes. J. Comp. Phys., 197:711–736, 2004.Google Scholar
[16]Santosa, F.. A level-set approach for inverse problems involving obstacles. Control, Opti-mizat. Calculus Variat., 1:17–33, 1996.Google Scholar
[17]Shu, C. W. and Osher, S. J.. Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys., 77:439–471, 1988.Google Scholar
[18]Doel, K. van den, Ascher, U., and Leitao, A.. Multiple level sets for piecewise constant surface reconstruction in highly ill-posed problems. J. Sci. Comput., 43:44–66, 2010.Google Scholar
[19]Zhao, H.-K., Chan, T., Merriman, B., and Osher, S. J.. A variational level set approach for multiphase motion. J. Comput. Phys., 127:179–195, 1996.Google Scholar