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Large Scale Kronecker Product on Supercomputers

This workshop presentation discusses the Kronecker product and its applications in large-scale matrix computations on supercomputers. It covers the properties and problem formulation of the Kronecker product, performance issues, and heuristics for finding an optimal topology. The presentation also explores the performance of hybrid implementations and the impact of memory layout.

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Large Scale Kronecker Product on Supercomputers

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  1. Claude Tadonki Mines ParisTech – CRI – Mathématiques et Systèmes Laboratoire de l’Accélérateur Linéaire/IN2P3/CNRS France claude.tadonki@u-psud.fr 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – 29 2010, Vitória, Espírito Santo, Brazil.

  2. Large Scale Kronecker Product on SupercomputersC. 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 – 29 2010, Vitória, Espírito Santo, Brazil.

  3. Large Scale Kronecker Product on SupercomputersC. 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 – 29 2010, Vitória, Espírito Santo, Brazil.

  4. Large Scale Kronecker Product on SupercomputersC. TADONKI The Kronecker (complexity and recurrenceequation) • 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 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – 29 2010, Vitória, Espírito Santo, Brazil.

  5. Large Scale Kronecker Product on SupercomputersC. 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 – 29 2010, Vitória, Espírito Santo, Brazil.

  6. Large Scale Kronecker Product on SupercomputersC. TADONKI Performance issues and heuristic for finding a good topology • 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) 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – 29 2010, Vitória, Espírito Santo, Brazil.

  7. Large Scale Kronecker Product on SupercomputersC. TADONKI Performances Weconsider N = 6 matrices of orders30, 36, 32, 18, 24, 16, thus L = 159 252 480 • 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 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – 29 2010, Vitória, Espírito Santo, Brazil.

  8. Large Scale Kronecker Product on SupercomputersC. TADONKI Performance of the hybridimplementation • 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. 2nd Workshop on Architecture and Multi-Core Applications 23rd International Symposium on Computer Architecture and High Performance Computing (SBAC PAD 2011) October, 26 – 29 2010, Vitória, Espírito Santo, Brazil.

  9. Large Scale Kronecker Product on SupercomputersC. TADONKI 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 – 29 2010, Vitória, Espírito Santo, Brazil.

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