"Application of the lowrank approximation technique in the Gauss elimination
method for sparse linear systems" Solovyev S.A. 
A fast direct algorithm for 3D discretized linear systems using the Gauss elimination method together with the nested dissection ordering approach and lowrank 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 LUdecomposition of the original matrix, the largeblock representation as well as HSS format (Hierarchically Semiseparable Structure) are used. The construction of a lowrank 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 lowrank/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. Keywords: threedimensional problems of mathematical physics, algorithms for sparse linear systems, Gauss elimination method, lowrank approximation, HSS matrix representation, iterative refinement.

