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Proper generalized decomposition solution of the parameterized Helmholtz problem: application to inverse geophysical problems
Author(s) -
Signorini Marianna,
Zlotnik Sergio,
Díez Pedro
Publication year - 2016
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5313
Subject(s) - parameterized complexity , helmholtz free energy , inverse problem , solver , parametric statistics , helmholtz equation , parametrization (atmospheric modeling) , mathematics , inverse , decomposition , mathematical optimization , process (computing) , computer science , algorithm , mathematical analysis , geometry , boundary value problem , physics , ecology , statistics , quantum mechanics , biology , radiative transfer , operating system
Summary The identification of the geological structure from seismic data is formulated as an inverse problem. The properties and the shape of the rock formations in the subsoil are described by material and geometric parameters, which are taken as input data for a predictive model. Here, the model is based on the Helmholtz equation, describing the acoustic response of the system for a given wave length. Thus, the inverse problem consists in identifying the values of these parameters such that the output of the model agrees the best with observations. This optimization algorithm requires multiple queries to the model with different values of the parameters. Reduced order models are especially well suited to significantly reduce the computational overhead of the multiple evaluations of the model. In particular, the proper generalized decomposition produces a solution explicitly stating the parametric dependence, where the parameters play the same role as the physical coordinates. A proper generalized decomposition solver is devised to inexpensively explore the parametric space along the iterative process. This exploration of the parametric space is in fact seen as a post‐process of the generalized solution. The approach adopted demonstrates its viability when tested in two illustrative examples. Copyright © 2016 John Wiley & Sons, Ltd.