Claude Tadonki

1

• Claude Tadonki
• Mines ParisTech CRI Mathématiques et Systèmes
• Laboratoire de lAccélérateur Linéaire/IN2P3/CNRS
• France
• claude.tadonki_at_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 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 29 2010,
Vitória, Espírito Santo, Brazil.
3
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 29 2010,
Vitória, Espírito Santo, Brazil.
4
Large Scale Kronecker Product on Supercomputers

C. TADONKI
The Kronecker (complexity and recurrence equation)

• 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 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 29 2010,
Vitória, Espírito Santo, Brazil.
6
Large Scale Kronecker Product on Supercomputers

C. 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 Supercomputers

C. TADONKI
Performances
We consider N 6 matrices of orders 30, 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 Supercomputers

C. TADONKI
Performance of the hybrid implementation

• 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 Supercomputers

C. 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.