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

Wangtao Lu; Shingyu Leung; Jianliang Qian

doi:10.3934/ipi.2015.9.479

Article Contents

2015, Volume 9, Issue 2: 479-509. Doi: 10.3934/ipi.2015.9.479

This issue Previous Article Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data Next Article A study of the one dimensional total generalised variation regularisation problem

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

1.
Department of Mathematics, Michigan State University, East Lansing, MI 48824
2.
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

Received: January 31, 2014

Revised: December 31, 2014

Published: April 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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