Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory
Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory
复制标题
DOI：
10.3934/ipi.2020076
复制
发表时间：
2019-12
期刊：
影响因子：
--
通讯作者：
Dan Zhu;R. Renaut;Hongwei Li;Tianyou Liu
中科院分区：
文献类型：
--
作者：
Dan Zhu;R. Renaut;Hongwei Li;Tianyou Liu
提出了一种用于位场数据分离的快速非凸低秩矩阵分解方法。大尺寸的轨迹矩阵，这也是一个块汉克尔矩阵的奇异值分解，得到使用快速随机奇异值分解算法，其中快速块汉克尔矩阵向量乘法实现与最小的内存存储。这种快速块Hankel矩阵随机奇异值分解算法被集成到\texttt{Altproj}算法中，这是一种标准的非凸方法，用于解决鲁棒主成分分析优化问题。改进算法避免了轨迹矩阵的构造。因此，可以计算大尺寸的重力和磁力数据矩阵。此外，它是更有效的比传统的低秩矩阵分解方法，这是基于使用一个不精确的增广拉格朗日乘子算法。所提出的算法也是强大的，因此，算法相关的参数很容易确定。针对不同规模的重磁合成数据矩阵的分离问题，对改进算法和传统算法进行了对比。结果表明，改进后的算法不仅计算效率更高，而且精度更高。此外，解决更大的问题也是可能的。例如，对于所采用的计算环境，大小大于$205 \times 205$的矩阵使用传统方法会产生“内存不足”异常，但大小为$2001\times 2001$的矩阵可以使用新算法在$1062.29$s中计算。最后，将改进的方法应用于安徽省铜陵地区的真实的重磁数据分离。根据分离的异常推断可能显示矿化的区域。
A fast non-convex low-rank matrix decomposition method for potential field data separation is proposed. 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 \texttt{Altproj} algorithm, which is a standard non-convex method for solving the robust principal component analysis optimization problem. The improved algorithm avoids the construction of the trajectory matrix. Hence, gravity and magnetic data matrices of large size can be computed. Moreover, it is more efficient than the traditional low-rank matrix decomposition method, which is based on the use of an inexact augmented Lagrange multiplier algorithm. The presented algorithm is also robust and, hence, algorithm-dependent parameters are easily determined. The improved and traditional algorithms are contrasted for the separation of synthetic gravity and magnetic data matrices of different sizes. The presented results demonstrate that the improved algorithm is not only computationally 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 $205 \times 205$ generate "out of memory" exceptions with the traditional method, but a matrix of size $2001\times 2001$ can be calculated in $1062.29$s with the new algorithm. Finally, the improved method 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.