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May  2015, 9(2): 479-509. doi: 10.3934/ipi.2015.9.479

An improved fast local level set method for three-dimensional inverse gravimetry

Wangtao Lu 1, , Shingyu Leung 2, and  Jianliang Qian 1,

1. 

Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States, United States
2. 

Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

Received  February 2014 Revised  January 2015 Published  March 2015

We propose an improved fast local level set method for the inverse problem of gravimetry by developing two novel algorithms: one is of linear complexity designed for computing the Frechet derivative of the nonlinear domain inverse problem, and the other is designed for carrying out numerical continuation rapidly so as to obtain fictitious full measurement data from partial measurement. Since the Laplacian kernel is symmetric and translationally invariant, we design certain affine transformations to speed up the computational process in evaluating the Frechet derivative; since it decays rapidly away from diagonal, we carry out low-rank matrix approximation to reduce storage requirements. These properties are eventually translated into an algorithm of linear complexity and linear storage requirement for computing the derivative. Since the single layer density function, used in representing the potential, is smooth and periodic on an artificial hypersurface enclosing the target domain, the spectral expansion is allowed to approximate this density function, which eventually leads to rapid algorithms for carrying out the numerical continuation in both 2-D and 3-D cases. 2-D and 3-D numerical examples illustrate that this improved level-set method is capable of computing high-resolution inversions and handling 3-D large-scale inverse gravimetry problems.
Keywords: Level set method, gravimetry, numerical continuation., inverse problem.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Wangtao Lu, Shingyu Leung, Jianliang Qian. An improved fast local level set method for three-dimensional inverse gravimetry. Inverse Problems and Imaging, 2015, 9 (2) : 479-509. doi: 10.3934/ipi.2015.9.479
References:
[1]

H. Bertete-Aguirre, E. Cherkaev and M. Oristaglio, Non-smooth gravity problem with total variation penalization functional, Geophys. J. Int., 149 (2002), 499-507. doi: 10.1046/j.1365-246X.2002.01664.x.

[2]

M. Burger, A level set method for inverse problems, Inverse Problems, 17 (2001), 1327-1355. doi: 10.1088/0266-5611/17/5/307.

[3]

M. Burger and S. Osher, A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16 (2005), 263-301. doi: 10.1017/S0956792505006182.

[4]

T. Cecil, S. J. Osher and J. Qian, Simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension, J. Comput. Phys., 213 (2006), 458-473. doi: 10.1016/j.jcp.2005.08.020.

[5]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edition, Springer, 2013. doi: 10.1007/978-1-4614-4942-3.

[6]

T. DeLillo, V. Isakov, N. Valdivia and L. Wang, The detection of surface vibrations from interior acoustical pressure, Inverse Problems, 19 (2003), 507-524. doi: 10.1088/0266-5611/19/3/302.

[7]

O. Dorn and D. Lesselier, Level set methods for inverse scattering, Inverse Problems, 22 (2006), R67-R131. doi: 10.1088/0266-5611/22/4/R01.

[8]

S. Hou, K. Solna and H.-K. Zhao, Imaging of location and geometry for extended targets using the response matrix, J. Comput. Phys., 199 (2004), 317-338. doi: 10.1016/j.jcp.2004.02.010.

[9]

V. Isakov, Inverse Source Problems, American Mathematical Society, Providence, Rhode Island, 1990. doi: 10.1090/surv/034.

[10]

V. Isakov, S. Leung and J. Qian, A fast local level set method for inverse gravimetry, Comm. in Computational Physics, 10 (2011), 1044-1070. doi: 10.4208/cicp.100710.021210a.

[11]

V. Isakov, S. Leung and J. Qian, A three-dimensional inverse gravimetry problem for ice with snow caps, {Inverse Problems and Imaging}, 7 (2013), 523-544. doi: 10.3934/ipi.2013.7.523.

[12]

A. Litman, D. Lesselier and F. Santosa, Reconstruction of a 2-D binary obstacle by controlled evolution of a level-set, Inverse Problems, 14 (1998), 685-706. doi: 10.1088/0266-5611/14/3/018.

[13]

W. Lu and Y. Y. Lu, Efficient boundary integral equation method for photonic crystal fibers, Journal of Lightwave Technology, 30 (2012), 1610-1616. doi: 10.1109/JLT.2012.2189355.

[14]

S. J. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.

[15]

J. Qian, L.-T. Cheng and S. J. Osher, A level set based Eulerian approach for anisotropic wave propagations, Wave Motion, 37 (2003), 365-379. doi: 10.1016/S0165-2125(02)00101-4.

[16]

J. Qian and S. Leung, A level set method for paraxial multivalued traveltimes, J. Comput. Phys., 197 (2004), 711-736. doi: 10.1016/j.jcp.2003.12.017.

[17]

J. Qian and S. Leung, A local level set method for paraxial multivalued geometric optics, SIAM J. Sci. Comp., 28 (2006), 206-223. doi: 10.1137/030601673.

[18]

F. Santosa, A level-set approach for inverse problems involving obstacles, Control, Optimizat. Calculus Variat., 1 (1996), 17-33.

[19]

K. van den Doel, U. Ascher and A. Leitao, Multiple level sets for piecewise constant surface reconstruction in highly ill-posed problems, J. Sci. Comput., 43 (2010), 44-66. doi: 10.1007/s10915-009-9341-x.

[20]

H.-K. Zhao, T. Chan, B. Merriman and S. J. Osher, A variational level set approach for multiphase motion, J. Comput. Phys., 127 (1996), 179-195. doi: 10.1006/jcph.1996.0167.

show all references

References:
[1]

H. Bertete-Aguirre, E. Cherkaev and M. Oristaglio, Non-smooth gravity problem with total variation penalization functional, Geophys. J. Int., 149 (2002), 499-507. doi: 10.1046/j.1365-246X.2002.01664.x.

[2]

M. Burger, A level set method for inverse problems, Inverse Problems, 17 (2001), 1327-1355. doi: 10.1088/0266-5611/17/5/307.

[3]

M. Burger and S. Osher, A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16 (2005), 263-301. doi: 10.1017/S0956792505006182.

[4]

T. Cecil, S. J. Osher and J. Qian, Simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension, J. Comput. Phys., 213 (2006), 458-473. doi: 10.1016/j.jcp.2005.08.020.

[5]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edition, Springer, 2013. doi: 10.1007/978-1-4614-4942-3.

[6]

T. DeLillo, V. Isakov, N. Valdivia and L. Wang, The detection of surface vibrations from interior acoustical pressure, Inverse Problems, 19 (2003), 507-524. doi: 10.1088/0266-5611/19/3/302.

[7]

O. Dorn and D. Lesselier, Level set methods for inverse scattering, Inverse Problems, 22 (2006), R67-R131. doi: 10.1088/0266-5611/22/4/R01.

[8]

S. Hou, K. Solna and H.-K. Zhao, Imaging of location and geometry for extended targets using the response matrix, J. Comput. Phys., 199 (2004), 317-338. doi: 10.1016/j.jcp.2004.02.010.

[9]

V. Isakov, Inverse Source Problems, American Mathematical Society, Providence, Rhode Island, 1990. doi: 10.1090/surv/034.

[10]

V. Isakov, S. Leung and J. Qian, A fast local level set method for inverse gravimetry, Comm. in Computational Physics, 10 (2011), 1044-1070. doi: 10.4208/cicp.100710.021210a.

[11]

V. Isakov, S. Leung and J. Qian, A three-dimensional inverse gravimetry problem for ice with snow caps, {Inverse Problems and Imaging}, 7 (2013), 523-544. doi: 10.3934/ipi.2013.7.523.

[12]

A. Litman, D. Lesselier and F. Santosa, Reconstruction of a 2-D binary obstacle by controlled evolution of a level-set, Inverse Problems, 14 (1998), 685-706. doi: 10.1088/0266-5611/14/3/018.

[13]

W. Lu and Y. Y. Lu, Efficient boundary integral equation method for photonic crystal fibers, Journal of Lightwave Technology, 30 (2012), 1610-1616. doi: 10.1109/JLT.2012.2189355.

[14]

S. J. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.

[15]

J. Qian, L.-T. Cheng and S. J. Osher, A level set based Eulerian approach for anisotropic wave propagations, Wave Motion, 37 (2003), 365-379. doi: 10.1016/S0165-2125(02)00101-4.

[16]

J. Qian and S. Leung, A level set method for paraxial multivalued traveltimes, J. Comput. Phys., 197 (2004), 711-736. doi: 10.1016/j.jcp.2003.12.017.

[17]

J. Qian and S. Leung, A local level set method for paraxial multivalued geometric optics, SIAM J. Sci. Comp., 28 (2006), 206-223. doi: 10.1137/030601673.

[18]

F. Santosa, A level-set approach for inverse problems involving obstacles, Control, Optimizat. Calculus Variat., 1 (1996), 17-33.

[19]

K. van den Doel, U. Ascher and A. Leitao, Multiple level sets for piecewise constant surface reconstruction in highly ill-posed problems, J. Sci. Comput., 43 (2010), 44-66. doi: 10.1007/s10915-009-9341-x.

[20]

H.-K. Zhao, T. Chan, B. Merriman and S. J. Osher, A variational level set approach for multiphase motion, J. Comput. Phys., 127 (1996), 179-195. doi: 10.1006/jcph.1996.0167.

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