A Fast Local Level Set Method for Inverse Gravimetry
Victor Isakov 1, Shingyu Leung 2, Jianliang Qian 3*1 Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas, USA.
2 Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR.
3 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.
Received 10 July 2010; Accepted (in revised version) 2 December 2010
Available online 24 June 2011
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 $x_3$-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 $\Omega$. 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.AMS subject classifications: 52B10, 65D18, 68U05, 68U07
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Key words: Level set methods, inverse gravimetry, fast algorithms, numerical continuation.
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