Application of the low-rank approximation technique in the Gauss elimination method for sparse linear systems

A fast direct algorithm for 3D discretized linear systems using the Gauss elimination method together with the nested dissection ordering approach and low-rank approximations is proposed. This algorithm is described for symmetric positive definite matrices and can be easily extended to the case of nonsymmetric systems. In order to store the factor $L$ in the $LU$-decomposition of the original matrix, the large-block representation as well as HSS format (Hierarchically Semiseparable Structure) are used. The construction of a low-rank approximation is based on using the adaptive cross approximation (ACA) approach, which is more efficient compared to the $SVD$ and $QR$ methods. In order to enhance the efficiency of the corresponding solver, a number of Intel MKL BLAS and LAPACK subroutines are used. This solver was implemented for shared memory computing systems. The functional testing shows a high quality of low-rank/HSS approximation. The performance testing demonstrates up to 3 times performance gain in comparison with the Intel MKL PARDISO direct solver. The proposed solver allows one to significantly decrease the memory and time consumption while using the Gauss elimination method.