TY - JOUR

T1 - Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory

AU - Zhu, Dan

AU - Renaut, Rosemary A.

AU - Li, Hongwei

AU - Liu, Tianyou

N1 - Funding Information:
2020 Mathematics Subject Classification. Primary: 65F22, 65F55; Secondary: 86A20, 86A22. Key words and phrases. Gravity and magnetic data, Low-rank method, Potential field separation, Fast algorithm with minimal memory storage, Tongling, Anhui province, China. The first author is supported by the National Key R & D Program of China 2018YFC1503705; The second author is supported by the NSF grant DMS 1913136; The third author is supported by the National Key R & D Program of China 2018YFC1503705 and Hubei Subsurface Multi-scale Imaging Key Laboratory (China University of Geosciences) SMIL-2018-06; ∗ Corresponding author: Rosemary A. Renaut.
Funding Information:
The first author is supported by the National Key R & D Program of China 2018YFC1503705; The second author is supported by the NSF grant DMS 1913136; The third author is supported by the National Key R & D Program of China 2018YFC1503705 and Hubei Subsurface Multiscale Imaging Key Laboratory (China University of Geosciences) SMIL-2018-06;The authors are appreciative of the reviewers? comments that assisted with improving the paper. Rosemary Renaut acknowledges the support of NSF grant DMS 1913136: ?Approximate Singular Value Expansions and Solutions of Ill-Posed Problems?. Dan Zhu and Hongwei Li acknowledge the support of the National Key R&D Program of China (2018YFC1503705). Hongwei Li acknowledges the support of Hubei Subsurface Multi-scale Imaging Key Laboratory (China University of Geosciences) (SMIL-2018-06). We also acknowledge Anhui Geology and Mineral Exploration Bureau 321 Geological Team for providing the data sets that were used for the real data experiments.
Publisher Copyright:
© 2021, American Institute of Mathematical Sciences. All rights reserved.

PY - 2021

Y1 - 2021

N2 - A fast non-convex low-rank matrix decomposition method for potential field data separation is presented. The singular value decomposition of the large size trajectory matrix, which is also a block Hankel matrix, is obtained using a fast randomized singular value decomposition algorithm in which fast block Hankel matrix-vector multiplications are implemented with minimal memory storage. This fast block Hankel matrix randomized singular value decomposition algorithm is integrated into the Altproj algorithm, which is a standard non-convex method for solving the robust principal component analysis optimization problem. The integration of this improved estimation for the partial singular value decomposition avoids the construction of the trajectory matrix in the robust principal component analysis optimization problem. Hence, gravity and magnetic data matrices of large size can be computed and potential field data separation is achieved with better computational efficiency. The presented algorithm is also robust and, hence, algorithm-dependent parameters are easily determined. The performance of the algorithm, with and without the efficient estimation of the low rank matrix, is contrasted for the separation of synthetic gravity and magnetic data matrices of different sizes. These results demonstrate that the presented algorithm is not only computa-tionally more efficient but it is also more accurate. Moreover, it is possible to solve far larger problems. As an example, for the adopted computational environment, matrices of sizes larger than 205 × 205 generate “out of memory” exceptions without the improvement, whereas a matrix of size 2001 × 2001 can now be calculated in 1062.29s. Finally, the presented algorithm is applied to separate real gravity and magnetic data in the Tongling area, Anhui province, China. Areas which may exhibit mineralizations are inferred based on the separated anomalies.

AB - A fast non-convex low-rank matrix decomposition method for potential field data separation is presented. The singular value decomposition of the large size trajectory matrix, which is also a block Hankel matrix, is obtained using a fast randomized singular value decomposition algorithm in which fast block Hankel matrix-vector multiplications are implemented with minimal memory storage. This fast block Hankel matrix randomized singular value decomposition algorithm is integrated into the Altproj algorithm, which is a standard non-convex method for solving the robust principal component analysis optimization problem. The integration of this improved estimation for the partial singular value decomposition avoids the construction of the trajectory matrix in the robust principal component analysis optimization problem. Hence, gravity and magnetic data matrices of large size can be computed and potential field data separation is achieved with better computational efficiency. The presented algorithm is also robust and, hence, algorithm-dependent parameters are easily determined. The performance of the algorithm, with and without the efficient estimation of the low rank matrix, is contrasted for the separation of synthetic gravity and magnetic data matrices of different sizes. These results demonstrate that the presented algorithm is not only computa-tionally more efficient but it is also more accurate. Moreover, it is possible to solve far larger problems. As an example, for the adopted computational environment, matrices of sizes larger than 205 × 205 generate “out of memory” exceptions without the improvement, whereas a matrix of size 2001 × 2001 can now be calculated in 1062.29s. Finally, the presented algorithm is applied to separate real gravity and magnetic data in the Tongling area, Anhui province, China. Areas which may exhibit mineralizations are inferred based on the separated anomalies.

KW - Anhui province

KW - China

KW - Fast algorithm with minimal memory storage

KW - Gravity and magnetic data

KW - Low-rank method

KW - Potential field separa-tion

KW - Tongling

UR - http://www.scopus.com/inward/record.url?scp=85099471075&partnerID=8YFLogxK

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U2 - 10.3934/ipi.2020076

DO - 10.3934/ipi.2020076

M3 - Article

AN - SCOPUS:85099471075

SN - 1930-8337

VL - 15

SP - 159

EP - 183

JO - Inverse Problems and Imaging

JF - Inverse Problems and Imaging

IS - 1

ER -