Super large-scale magnetic data inversion

In this paper, indexed kernel matrix with wavelet compression method (IKMWC) is presented to solve the huge amount of computer memory for the kernel matrix and CPU time for the multiplication of the dense kernel matrix to vectors which is caused by super large-scale magnetic data inversion problems with more than 104 data and 106 mesh cells. We restrict the mesh model with horizontally regular cells and set the observations located over each cell's center point in a flat surface. A great number of equivalent elements are generated in the kernel matrix of this kind of mesh model. Thus a new three-dimension kernel matrix formed by only storing the unequal elements of the original one is called an indexed kernel matrix (IKM). Since the elements in this indexed kernel matrix are far more less than those in the original one, the computer memory demands are reduced greatly. A second important feature of the algorithm we presented here is the use of wavelet transformation to the indexed kernel matrix. To keep the index relationship between the indexed kernel matrix and the original one, the wavelet transformation is applied only on the depth dimension of the IKM. By thresholding the small wavelet coefficients, a sparse representation of the IKM is formed to further reduce the required computer memory for the IKM to 1/5 similar to 1/10. Using the fast algorithms for sparse matrix-vector multiplication also reduce the CPU time to 1/5 similar to 1/10. Our method is tested on synthetic example which shows that, the IKMWC method has efficient computation performance in solving super large-scale magnetic data inversion problems.