Claude Tadonki Mines ParisTech – CRI – Mathématiques et Systèmes Laboratoire de l’Accélérateur Linéaire/IN2P3/CNRS France 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – , Vitória, Espírito Santo, Brazil.

Large Scale Kronecker Product on Supercomputers C. TADONKI The Kronecker product (définition and applications) 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – , Vitória, Espírito Santo, Brazil.

Large Scale Kronecker Product on Supercomputers C. TADONKI The Kronecker product (properties and problem formulation) 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – , Vitória, Espírito Santo, Brazil.

Large Scale Kronecker Product on Supercomputers C. TADONKI The Kronecker (complexity and recurrence equation) 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – , Vitória, Espírito Santo, Brazil. Forming the matrix first would require a huge amount of memory yield lot of redundant multiplication, which in total would be Using the so-called normal factorization, we could derive an optimal scheme which reduces the number of floatting point multiplication to

Large Scale Kronecker Product on Supercomputers C. TADONKI The Kronecker product and its applications 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – , Vitória, Espírito Santo, Brazil.

Large Scale Kronecker Product on Supercomputers C. TADONKI Performance issues and heuristic for finding a good topology 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – , Vitória, Espírito Santo, Brazil. The total (parallel) execution time depends on the sizes of the matrices the gap between virtual topology and physical topology the way the task is splitted among the processors (decomposition)

Large Scale Kronecker Product on Supercomputers C. TADONKI Performances 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – , Vitória, Espírito Santo, Brazil. We consider N = 6 matrices of orders 30, 36, 32, 18, 24, 16, thus L = We see that our heuristic yields a significant improvment compare to trivial decompositions we start loosing the scalabily when the number of cores increases (com) We the turn to hybrid implementation

Large Scale Kronecker Product on Supercomputers C. TADONKI Performance of the hybrid implementation 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – , Vitória, Espírito Santo, Brazil. We see that the hybrid implementation is better for larger number of cores for smaller number of cores, the SM implemntation exacerbates on cache misses Need to investigate on the compromise and a better memory layout.

END & QUESTIONS 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – , Vitória, Espírito Santo, Brazil. Large Scale Kronecker Product on Supercomputers C. TADONKI