Doctor of Philosophy (PhD)
Linear algebra underlies a large proportion of computational problems. With the continuous increase of scale on modern hardware, performance of small sized linear algebra has become increasingly important. To overcome the shortcomings of conventional approaches, we employ a new approach using a microkernel framework provided by ATLAS to improve the performance of a few linear algebra routines for all problem sizes. Our initial research consists of improving the performance of parallel LU factorization in ATLAS for which we were able to achieve up to 2.07x and 2.66x speedup for small problems, up to 91% and 87% of theoretical peak performance for asymptotic problems on a 12-core Intel Xeon and a 32-core AMD Opteron machine, respectively, outperforming all the state-of-the-art libraries at the time. Such performance was achieved via an exhaustive search of all the tuning parameters, which could take days. This motivated us to try to develop a computational model for our LU factorization that could predict those parameters by combining some basic empirical timings and a theoretical model based on the amount of required computations. While our model provided good prediction for mid-to-asymptotic sized problems, there were some unknown factors for small problems that could possibly be answered by extending the ATLAS tuning framework. While this extension is underway, we decided to pursue the model research using simpler serial BLAS-based approach. We investigated and implemented two Level-3 BLAS routines: TRSM and TRMM that are widely used primarily by LAPACK operations like the aforementioned LU factorization. With the microkernel-based approach, we were able to improve the performance of both routines by up to 15% and 73% for square and fat problems, respectively, over prior ATLAS implementations on modern hardware. Finally, with a collaborative research with ARM Inc., we improved the performance of the most important Level-3 BLAS operation GEMM in ATLAS by up to 53% via implementing microkernels for two 64-bit ARM architectures. This automatically improves other BLAS and LAPACK routines that rely on GEMM for high performance.
Document Availability at the Time of Submission
Release the entire work immediately for access worldwide.
Hasan, Md Rakib, "Maintaining High Performance Across All Problem Sizes and Parallel Scales Using Microkernel-based Linear Algebra" (2017). LSU Doctoral Dissertations. 4486.