Quantization improvements for LU Decomposition to Solve Linear Equations

Quantization is an effective method for compressing Deep Neural Networks. Now, it is considered to accelerate the traditional HPC applications. In this article, we present a quantization method (q8gemm) using integer 8-bits(INT8) for matrix multiplication, which is implemented by block techniques to overcome the overflow problem. We choose different scales for one matrix’s each row and those for the other matrix’s each column to keep the numbers in range. With the intel advanced matrix extension cblas_gemm_s8u8s32, our q8gemm can accelerate panel-matrix multiplication in LU factorization. Furthermore, we propose a new mixed-precision solver (q8gesv ¹ ) for the linear equations. Using q8gemm in INT8, q8gesv modifies LU factorization in FP32 and then utilizes GMRES iterative refinement in FP64 to bring the solution back to 64 bits accuracy. Numerical tests show that on two Intel CPU platforms with avx2 and avx512, q8gemm achieves a speedup of 1.313-1.621 over cblas_sgemm of MKL. The execution time of cblas_gemm_s8u8s32 makes up 86.78%-90.85% of the q8gemm. By means of q8gemm, our q8gesv can achieve a speedup of 1.176-1.493 over the optimized Higham’s mixed-precision algorithm utilized in the HPL-AI benchmark and a speedup of 1.1-1.459 over the LAPACKE_dsgesv function from MKL LAPACK. Meanwhile, q8gesv runs up to 2.859X faster on Intel(R) Core(TM) i5-10600KF and to 2.224X faster on Intel(R) Xeon(R) Gold 6130 than the LAPACKE_dgesv function from MKL LAPACK.