A Fast Local Level Set Method for Inverse Gravimetry (Communications in Computational Physics, 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 uniquenessresult: 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 thispurpose constructively, the first challenge is how to parametrize this open set D as itsboundary may have a variety of possible shapes. To describe those different shapes wepropose to use a level-set function to parametrize the unknown boundary of this openset. The second challenge is how to deal with the issue of partial data as gravimetricmeasurements are only made on a part of a given reference domain Ω. To overcomethis difficulty, we propose a linear numerical continuation approach based on the singlelayer representation to find potentials on the boundary of some artificial domaincontaining the unknown set D. The third challenge is how to speed up the level set inversionprocess. Based on some features of the underlying inverse gravimetry problemsuch as the potential density being constant inside the unknown domain, we propose anovel numerical approach which is able to take advantage of these features so that thecomputational speed is accelerated by an order of magnitude. We carry out numericalexperiments for both two- and three-dimensional cases to demonstrate the effectivenessof the new algorithm.