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A fast Local Level Set Method for Inverse Gravimetry

Communications in Computational Physics 2011年第9期10卷 1044-1070页

作者： Victor Isakov Shingyu Leung Jianliang Qian Department of Mathematics and Statistics Wichita State UniversityWichitaKansasUSA Department of Mathematics Hong Kong University of Science and TechnologyClear Water BayHong Kong SAR Department of Mathematics Michigan State UniversityEast LansingMI 48824USA

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 x3-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 domainΩ.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.

关键词： Level set methods inverse gravimetry fast algorithms numerical continuation

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RADIATION BOUNDARY CONDITIONS FOR MAXWELL'S EQUATIONS:A REVIEW OF ACCURATE TIME-DOMAIN FORMULATIONS

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Journal of Computational Mathematics 2007年第3期25卷 305-336页

作者： Thomas Hagstrom Stephen Lau Department of Mathematics and Statistics University of New MexicoAlbuquerqueNM 87131USA Division of Applied Mathematics Brown University182 George StreetProvidenceRI 02912USA

We review time-domain formulations of radiation boundary conditions for Maxwell＇s equations, focusing on methods which can deliver arbitrary accuracy at acceptable compu-tational cost. Examples include fast evaluations of nonlocal conditions on symmetric and general boundaries, methods based on identifying and evaluating equivalent sources, and local approximations such as the perfectly matched layer and sequences of local boundary conditions. Complexity estimates are derived to assess work and storage requirements as a function of wavelength and simulation time.

关键词： Maxwell＇s equations Radiation boundary conditions fast algorithms Kirchoffformulas Perfectly matched layers.

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