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Authors: Tondi, R.* 
Cavazzoni, C.* 
Danecek, P.* 
Morelli, A.* 
Title: Parallel ‘large’ dense matrix problems: application to 3D joint inversion of seismological and gravity data
Issue Date: Nov-2012
Series/Report no.: /48 (2012)
DOI: 10.1016/j.cageo.2012.05.026
Keywords: Parallel
Dense matrix
Block-cyclic distribution
Inverse problem
Probability density function
Gravity field
Shear-wave velocity structure
Density structure
Subject Classification04. Solid Earth::04.01. Earth Interior::04.01.02. Geological and geophysical evidences of deep processes 
04. Solid Earth::04.03. Geodesy::04.03.04. Gravity anomalies 
04. Solid Earth::04.06. Seismology::04.06.07. Tomography and anisotropy 
05. General::05.01. Computational geophysics::05.01.03. Inverse methods 
05. General::05.01. Computational geophysics::05.01.05. Algorithms and implementation 
Abstract: To obtain accurate and reliable estimations of the major lithological properties of the rock within a studied volume, geophysics uses the joint information provided by different geophysical datasets (e.g. gravimetric, magnetic, seismic). Representation of the different types of information entering the problem using probability density functions can provide the mathematical framework to formulate their combination. The maximum likelihood estimator of the resulting joint posterior probability density functions leads to the solution of the problem. However, one key problem appears to limit the use of this solver to an extensive range of real applications: information coming from potential fields that implies the presence of dense matrices in the resolving estimator. It is well known that dense matrix systems rapidly challenge both the algorithms and the computing platforms, and are not suited to high-resolution 3D geophysical analysis. In this study, we propose a procedure that allows us to obtain fast and reliable solutions of the joint posterior probability density functions in the presence of large gravity datasets and using sophisticated model parametrization. As it is particularly CPUconsuming, this 3D problem makes use of parallel computing to improve the performance and the accuracy of the simulations. Analysis of the correctness of the results, and the performance on different parallel environments, shows the portability and the efficiency of the code. This code is applied to a real experiment, where we succeed in recovering a 3D shear-wave velocity and density distribution within the upper mantle of the European continent, satisfying both the seismological and gravity data. On a multiprocessor machine, we have been able to handle forward and inverse calculations with a dense matrix of 215.66 Gb in 18 min, 20 s and 20 min, 54 s, respectively.
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