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Journal of Computational Mathematics

A reconstruction method for the inverse gravimetric problem
Anthony Gerber-Roth1; Alexandre Munnier1; Karim Ramdani1
1 Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France
The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 197-225.

We propose a reconstruction method to solve the inverse gravimetric problem with constant mass density. The method is based on the computation of the harmonic moments of the unknown domain. Convergence results are proved and numerical experiments are provided to illustrate the method and show its efficiency.

Published online:
DOI: 10.5802/smai-jcm.99
Classification: 31A25, 86A22, 35R30
Keywords: inverse gravimetric problem, shape from moments problem, Prony’s system, quadrature domains, partial balayage of measure
Anthony Gerber-Roth 1; Alexandre Munnier 1; Karim Ramdani 1

1 Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {A reconstruction method for the inverse gravimetric problem},
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Anthony Gerber-Roth; Alexandre Munnier; Karim Ramdani. A reconstruction method for the inverse gravimetric problem. The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 197-225. doi : 10.5802/smai-jcm.99. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.99/

[1] Yacin Ameur; Martin Helmer; Felix Tellander On the uniqueness problem for quadrature domains, Comput. Methods Funct. Theory, Volume 21 (2021) no. 3, pp. 473-504 | DOI | MR | Zbl

[2] Kendall Atkinson; Weimin Han Theoretical numerical analysis. A functional analysis framework, Texts in Applied Mathematics, 39, Springer, 2005, xviii+576 pages | DOI | MR

[3] Dmitry Batenkov; Yosef Yomdin On the accuracy of solving confluent Prony systems, SIAM J. Appl. Math., Volume 73 (2013) no. 1, pp. 134-154 | DOI | MR | Zbl

[4] Steven R. Bell Density of quadrature domains in one and several complex variables, Complex Var. Elliptic Equ., Volume 54 (2009) no. 3-4, pp. 165-171 | DOI | MR | Zbl

[5] Alfredo Canelas; Antoine Laurain; Antonio A. Novotny A new reconstruction method for the inverse potential problem, J. Comput. Phys., Volume 268 (2014), pp. 417-431 | DOI | MR | Zbl

[6] Victor G. Cherednichenko Inverse logarithmic potential problem, Inverse and Ill-Posed Problems Series, Walter de Gruyter, 1996 | DOI | MR

[7] Abdellatif El Badia; Tuong Ha-Duong An inverse source problem in potential analysis, Inverse Probl., Volume 16 (2000) no. 3, pp. 651-663 | DOI | MR | Zbl

[8] Avner Friedman Variational principles and free-boundary problems (Pure and Applied Mathematics), 1982, p. ix+710 | Zbl

[9] Stephen J. Gardiner; Tomas Sjödin Convexity and the exterior inverse problem of potential theory, Proc. Am. Math. Soc., Volume 136 (2008) no. 5, pp. 1699-1703 | DOI | MR | Zbl

[10] Stephen J. Gardiner; Tomas Sjödin Partial balayage and the exterior inverse problem of potential theory, Potential theory and stochastics in Albac. Aurel Cornea memorial volume. Conference proceedings, Albac, Romania, September 4–8, 2007 (Theta Series in Advanced Mathematics), Volume 11, The Theta Foundation, 2008, pp. 111-123 | Zbl

[11] Gene H. Golub; Peyman Milanfar; James Varah A stable numerical method for inverting shape from moments, SIAM J. Sci. Comput., Volume 21 (1999) no. 4, pp. 1222-1243 | DOI | MR | Zbl

[12] Björn Gustafsson On quadrature domains and an inverse problem in potential theory, J. Anal. Math., Volume 55 (1990), pp. 172-216 | DOI | MR | Zbl

[13] Björn Gustafsson; Chiyu He; Peyman Milanfar; Mihai Putinar Reconstructing planar domains from their moments, Inverse Probl., Volume 16 (2000) no. 4, pp. 1053-1070 | DOI | MR | Zbl

[14] Björn Gustafsson; Makoto Sakai Properties of some balayage operators, with applications to quadrature domains and moving boundary problems, Nonlinear Anal., Theory Methods Appl., Volume 22 (1994) no. 10, pp. 1221-1245 | DOI | MR | Zbl

[15] Björn Gustafsson; Henrik Shahgholian Existence and geometric properties of solutions of a free boundary problem in potential theory, J. Reine Angew. Math., Volume 473 (1996), pp. 137-179 | MR | Zbl

[16] Björn Gustafsson; Harold S. Shapiro What is a quadrature domain?, Quadrature domains and their applications (Operator Theory: Advances and Applications), Volume 156, Birkhäuser, 2005, pp. 1-25 | DOI | MR | Zbl

[17] Freidrich Hecht New development in FreeFem++, J. Numer. Math., Volume 20 (2012) no. 3-4, pp. 251-265 https://freefem.org/ | MR | Zbl

[18] Frank Hettlich; William Rundell Iterative methods for the reconstruction of an inverse potential problem, Inverse Probl., Volume 12 (1996) no. 3, pp. 251-266 | DOI | MR | Zbl

[19] Jeffrey Hokanson Numerically Stable and Statistically Efficient and Algorithms for Large and Scale Exponential and Fitting, Ph. D. Thesis, Rice University (2013) (https://hdl.handle.net/1911/77161) | MR

[20] Victor Isakov Inverse source problems, Mathematical Surveys and Monographs, 34, American Mathematical Society, 1990, xiv+193 pages | DOI | Zbl

[21] Victor Isakov; Shingyu Leung; Jianliang Qian A fast local level set method for inverse gravimetry, Commun. Comput. Phys., Volume 10 (2011) no. 4, pp. 1044-1070 | DOI | MR | Zbl

[22] Victor Isakov; Shingyu Leung; Jianliang Qian A three-dimensional inverse gravimetry problem for ice with snow caps, Inverse Probl. Imaging, Volume 7 (2013) no. 2, pp. 523-544 | DOI | MR | Zbl

[23] Kazufumi Ito; Karl Kunisch Semi–Smooth Newton Methods for Variational Inequalities of the First Kind, ESAIM, Math. Model. Numer. Anal., Volume 37 (2003) no. 1, p. 41–62 | DOI | Numdam | MR | Zbl

[24] Rainer Kress Linear integral equations, Applied Mathematical Sciences, 82, Springer, 2014, xvi+412 pages | DOI | MR

[25] Rainer Kress; William Rundell A nonlinear integral equation and an iterative algorithm for an inverse source problem, J. Integral Equations Appl., Volume 27 (2015) no. 2, pp. 179-197 | DOI | MR | Zbl

[26] Wenbin Li; Wangtao Lu; Jianliang Qian A level-set method for imaging salt structures using gravity data, Geophysics, Volume 81 (2016) no. 2, p. G27-G40 | DOI

[27] Wenbin Li; Jianliang Qian Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods, Inverse Probl. Imaging, Volume 15 (2021) no. 3, pp. 387-413 | DOI | MR | Zbl

[28] Wangtao Lu; Shingyu Leung; Jianliang Qian An improved fast local level set method for three-dimensional inverse gravimetry, Inverse Probl. Imaging, Volume 9 (2015) no. 2, pp. 479-509 | DOI | MR | Zbl

[29] William McLean Strongly elliptic systems and boundary integral equations, Cambridge University Press, 2000, xiv+357 pages | MR

[30] P. Milanfar; G. C. Verghese; W. C. Karl; A. S. Willsky Reconstructing polygons from moments with connections to array processing, IEEE Trans. Signal Process., Volume 43 (1995) no. 2, pp. 432-443 | DOI

[31] V. F. Pisarenko The Retrieval of Harmonics from a Covariance Function, Geophys. J. Int., Volume 33 (1973) no. 3, pp. 347-366 | DOI | Zbl

[32] Roland Potthast A survey on sampling and probe methods for inverse problems, Inverse Probl., Volume 22 (2006) no. 2, p. R1-R47 | DOI | MR | Zbl

[33] Nilson C. Roberty; Carlos J. S. Alves On the identification of star-shape sources from boundary measurements using a reciprocity functional, Inverse Probl. Sci. Eng., Volume 17 (2009) no. 2, pp. 187-202 | DOI | MR | Zbl

[34] Makoto Sakai Quadrature domains, Lecture Notes in Mathematics, 934, Springer, 1982 | DOI | Zbl

[35] Tetsuya Sakurai; Junko Asakura; Hiroto Tadano; Tsutomu Ikegami Error analysis for a matrix pencil of Hankel matrices with perturbed complex moments, JSIAM Lett., Volume 1 (2009), pp. 76-79 | DOI | MR | Zbl

[36] Harold S. Shapiro The Schwarz function and its generalization to higher dimensions, John Wiley & Sons, 1992 | Zbl

[37] Olaf Steinbach Numerical approximation methods for elliptic boundary value problems, Springer, 2008 | DOI

[38] Petre Stoica; Randolph L. Moses Spectral Analysis of Signals, Pearson Prentice Hall, 2005

[39] Ben Stray; Andrew Lamb; Aisha Kaushik; Jamie Vovrosh; Anthony Rodgers; Jonathan Winch; Farzad Hayati; Daniel Boddice; Artur Stabrawa; Alexander Niggebaum; Mehdi Langlois; Yu-Hung Lien; Samuel Lellouch; Sanaz Roshanmanesh; Kevin Ridley; Geoffrey de Villiers; Gareth Brown; Trevor Cross; George Tuckwell; Asaad Faramarzi; Nicole Metje; Kai Bongs; Michael Holynski Quantum sensing for gravity cartography, Nature, Volume 602 (2022) no. 7898, pp. 590-594 | DOI

[40] K. Doel; Uri M. Ascher; Antonio Leitão Multiple level sets for piecewise constant surface reconstruction in highly ill-posed problems, J. Sci. Comput., Volume 43 (2010) no. 1, pp. 44-66 | DOI | MR | Zbl

[41] Dimiter Zidarov Inverse gravimetric problem in geoprospecting and geodesy. Transl. from the Bulgarian, Elsevier, 1990, 284 pages | Zbl

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