EDF performs numerical simulations in different domains of physics. Several of its softwares use the library MUMPS in order to perform the costly step of solving sparse linear systems in a way that is generic, robust and efficient. The goal of this work is to develop new techniques for improving the performance gains of an existing functionality of MUMPS, the Block Low-Rank (BLR) compression. By combining several formats of floating-point numbers (mixed precision), it is possible to reduce the time and memory complexities, without compromising the accuracy of the result. Based on an error analysis, we design new variants of the LU factorization of dense matrices. We then adapt this work to the case of a sparse matrix factorization with MUMPS. A first implementation uses our mixed-precision BLR compression as a storage format, thus reducing the memory consumption of MUMPS. A second implementation allows combining these memory gains with time gains during the resolution of triangular systems. Finally, we study new techniques for improving the data locality of the BLR triangular solve with multiple right-hand sides, and obtain time reductions within MUMPS.