A fast algorithm for the large-scale joint inversion of gravity and magnetic data is developed. The algorithm uses a non-linear Gramian constraint to impose correlation between the density and susceptibility of the reconstructed models. The global objective function is formulated in the space of the weighted parameters, but the Gramian constraint is implemented in the original space, and the non-linear constraint is imposed using two separate Lagrange parameters, one for each model domain. It is significant that this combined approach, using the two spaces provides more similarity between the reconstructed models. Moreover, it is shown theoretically that the gradient for the use of the unweighted space is not a scalar multiple of that used for the weighted space, and hence cannot be accounted for by adjusting the Lagrange parameters. It is assumed that the measured data are obtained on a uniform grid and that a consistent regular discretization of the volume domain is imposed. Then, the sensitivity matrices exhibit a block-Toeplitz-Toeplitz-block structure for each depth layer of the model domain, and both forward and transpose operations with the matrices can be implemented efficiently using two dimensional fast Fourier transforms. This makes it feasible to solve for large scale problems with respect to both computational costs and memory demands, and to solve the non-linear problem by applying iterative methods that rely only on matrix–vector multiplications. As such, the use of the regularized reweighted conjugate gradient algorithm, in conjunction with the structure of the sensitivity matrices, leads to a fast methodology for large-scale joint inversion of geophysical data sets. Numerical simulations demonstrate that it is possible to apply a non-linear joint inversion algorithm, with Lp-norm stabilisers, for the reconstruction of large model domains on a standard laptop computer. It is demonstrated, that while the p = 1 choice provides sparse reconstructed solutions with sharp boundaries, it is also possible to use p = 2 in order to provide smooth and blurred models. The methodology is used for inverting gravity and magnetic data obtained over an area in northwest of Mesoproterozoic St Francois Terrane, southeast of Missouri, USA.
A fast methodology for large-scale focusing inversion of gravity and magnetic data using the structured model matrix and the 2-D fast Fourier transform
SUMMARY We discuss the focusing inversion of potential field data for the recovery of sparse subsurface structures from surface measurement data on a uniform grid. For the uniform grid, the model sensitivity matrices have a block Toeplitz Toeplitz block structure for each block of columns related to a fixed depth layer of the subsurface. Then, all forward operations with the sensitivity matrix, or its transpose, are performed using the 2-D fast Fourier transform. Simulations are provided to show that the implementation of the focusing inversion algorithm using the fast Fourier transform is efficient, and that the algorithm can be realized on standard desktop computers with sufficient memory for storage of volumes up to size n ≈ 106. The linear systems of equations arising in the focusing inversion algorithm are solved using either Golub–Kahan bidiagonalization or randomized singular value decomposition algorithms. These two algorithms are contrasted for their efficiency when used to solve large-scale problems with respect to the sizes of the projected subspaces adopted for the solutions of the linear systems. The results confirm earlier studies that the randomized algorithms are to be preferred for the inversion of gravity data, and for data sets of size m it is sufficient to use projected spaces of size approximately m/8. For the inversion of magnetic data sets, we show that it is more efficient to use the Golub–Kahan bidiagonalization, and that it is again sufficient to use projected spaces of size approximately m/8. Simulations support the presented conclusions and are verified for the inversion of a magnetic data set obtained over the Wuskwatim Lake region in Manitoba, Canada.
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- Award ID(s):
- 1913136
- NSF-PAR ID:
- 10274774
- Date Published:
- Journal Name:
- Geophysical Journal International
- Volume:
- 223
- Issue:
- 2
- ISSN:
- 0956-540X
- Page Range / eLocation ID:
- 1378 to 1397
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Summary The Golub‐Kahan bidiagonalization is widely used in the singular value decomposition of rectangular matrices and has been generalized to an iterative solver for symmetric indefinite linear systems with a two‐by‐two block structure. In this work, we present a scalability study of this generalized solver as implemented in a recent release of the parallel numerical library PETSc (Portable, Extensible Toolkit for Scientific Computation). We present an improved solver performance for the two‐dimensional (2D) Stokes equations as compared to previous work. Furthermore, we investigate the performance of different parallel inner solvers in the outer Golub‐Kahan iteration for a three‐dimensional Stokes problem. The study includes parallel sparse direct solvers and multigrid methods. When increasing the number of cores for a fixed total problem size, the solver exhibits good speedups of up to 50% at the 1024 core count. For the tests in which the total problem size grows while the workload in each core stays constant, the parallel performance of the solver scales almost linearly with the increase in the core counts. In particular, the computation time increases only by about 15% when the number of cores increases from 80 to 1024 for a 2D test case.
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/gji/ggaa372
