Graph grammar
and algorithmic transformations
Lecture topics:
Exemplary 1D Finite Difference problem
Frontal and Multi-frontal Direct Solvers
EXEMPLARY SIMPLE 1D NUMERICAL PROBLEM
Find temperature distribution
Finite difference disretization
such that
TRIDIAGONAL MATRIX FOR EXEMPLARY 1D PROBLEM
FRONTAL SOLVER
Frontal matrix
Frontal matrix focuses on forward elimination of the first row
FRONTAL SOLVER
Frontal matrix
2nd row = 2nd row – 1/h2 * 1st row
FRONTAL SOLVER
Frontal matrix
2nd row = 2nd row – 1/h2 * 1st row
FRONTAL SOLVER
Frontal matrix
The first row has been eliminated
Frontral matrix focuses on the elimination of the second row
FRONTAL SOLVER
Frontal matrix
3rd row = 3nd row – [1/h2] / [-2/h2] * 2nd row
FRONTAL SOLVER
Frontal matrix
3rd row = 3nd row – [1/h2] / [-2/h2] * 2nd row
MULTI-FRONTAL SOLVER ALGORITHM
Construct multiple
frontal matrices in such a way
so they sum up to the full matrix
Variables must be split
into parts
MULTI-FRONTAL SOLVER ALGORITHM
First all frontal matrices are constructed
MULTI-FRONTAL SOLVER ALGORITHM
+
First and second frontal matrices are sum up into new 3x3 frontal matrix
Its first and second rows are fully assembled
MULTI-FRONTAL SOLVER ALGORITHM
+
First column is eliminated by using first row
2nd row = 2nd row – [1/h2] * 1st row
MULTI-FRONTAL SOLVER ALGORITHM
+
2nd row = 2nd row – [1/h2] * 1st row
MULTI-FRONTAL SOLVER ALGORITHM
+
+
Third and fourth frontal matrices are sum up into new 3x3 frontal matrix
Now only the second row is fully assembled
MULTI-FRONTAL SOLVER ALGORITHM
+
Change of the ordering
+
MULTI-FRONTAL SOLVER ALGORITHM
+
+
Eliminate entries below the diagonal
2nd row = 2nd row – [1/h2] / [-2/h2] * 1st row
3rd row = 3rd row – [1/h2] / [-2/h2] * 1st row
MULTI-FRONTAL SOLVER ALGORITHM
Why can I substract fully assembled 1st row
from not fully assemled rows 2nd and 3rd ?
+
+
Eliminate entries below the diagonal
2nd row = 2nd row – [1/h2] / [-2/h2] * 1st row
3rd row = 3rd row – [1/h2] / [-2/h2] * 1st row
MULTI-FRONTAL SOLVER ALGORITHM
This is because substraction and addition are interchangable
(now I am substructing from the 2nd not fully assembled row,
and later I will add the remaining part)
Moreover the 1st row which is utilized for the substractions
in other columns contains only zeros
+
+
Eliminate entries below the diagonal
2nd row = 2nd row – [1/h2] / [-2/h2] * 1st row
3rd row = 3rd row – [1/h2] / [-2/h2] * 1st row
MULTI-FRONTAL SOLVER ALGORITHM
+
+
Eliminate entries below the diagonal
2nd row = 2nd row – [1/h2] / [-2/h2] * 1st row
3rd row = 3rd row – [1/h2] / [-2/h2] * 1st row
MULTI-FRONTAL SOLVER ALGORITHM
+
+
+
Fifth and sixth frontal matrices are sum up into new 3x3 frontal matrix
Now the second and third rows are fully assembled
…
PARALLEL MULTI-FRONTAL SOLVER ALGORITHM
Procesor 1
Procesor 2
Procesor 3
Procesor 4
All frontal matrices are generated at the same time
Procesor 5
Procesor 6
PARALLEL MULTI-FRONTAL SOLVER ALGORITHM
+
Procesor 1
Procesor 2
+
Procesor 3
+
Procesor 4
Procesor 5
Summing up and elimination are executed at the same time
over different pairs of frontal matrices
Procesor 6
PARALLEL MULTI-FRONTAL SOLVER ALGORITHM
+
Procesor 1
Procesor 2
+
Procesor 3
+
Procesor 4
Procesor 5
Summing up and elimination are executed at the same time
over different pairs of frontal matrices
Procesor 6
PARALLEL MULTI-FRONTAL SOLVER ALGORITHM
root
+
node
+
+
Procesor 1
Procesor 2
+
Procesor 3
+
Procesor 4
Procesor 5
Procesor 6
The agorithm is recursively repeated until we reach the root of the tree
The algorithm results in upper trianguler matrix stored in distributed manner
Computational complexity = height of the tree = logN (where N = #unknowns-1)
SUMMARY (BASIC IDEA OF PARALLEL SOLVER)
SUMMARY (BASIC IDEA OF PARALLEL SOLVER)
PROCESSOR 0
 1
 1
 2
h
 u  0
  01    
 2 u
0
h  1   
0
1
2nd = 2nd – 1/h2 * 1st
0  u  0
1
1   01    

0  2  u1  0
h 

 2 
  20 



u

 2 1  2 
 h 
 h 
solve
u1  10
1 0 u 0   0 
0 1 u   10

 1   
PROCESSOR 1
 1
 1
 2
h
 u  20
  22    
 2 u
0
h  1   
0
1
2nd = 2nd – 1/h2 * 1st
0  u   0 
1
1   22     20 

0  2  u1   2 
h 

 h 
send 
1
h2
u12 
20
h2
send u1  10
1 0 u 2  20
0 1 u   10

 1   
DIRECT SOLVERS
STATE OF THE ART
For dense matrices = Gaussian elimination or LU factorization
e.g. LAPACK (J. Dongharra) or parallel version PLAPACK (R. van de Geijn)
A =
L
U
For sparse matrices (e.g. resulting from Finite Element or Finite Difference Methods)
• frontal solvers
Irons B., 1970: A frontal solution program for finite-element analysis. International
Journal of Numerical Methods in Engineering, 2, 5-32
• multi-frontal solvers
Duff I. S., Reid J. K., 1983: The multifrontal solution of indefinite sparse symmetric
linear systems. ACM Transactions on Mathematical Software, 9, 302-325
Duff I. S., Reid J. K., 1984: The multifrontal solution of unsymmetric sets of linear
systems. SIAM Journal on Scientific and Statistical Computing, 5, 633-641
PARALLEL DIRECT SOLVERS FOR SPARSE SYSTEMS
STATE OF THE ART
For large sparse matrices: (e.g. resulting from challenging FEM computations)
Sub-structing method solvers, based on the domain decomposition paradigm
• parallel frontal solvers
Scott J. A., 2003: Parallel Frontal Solvers for Large Sparse Linear Systems. ACM
Transactions on Mathematical Software, 29, 4, 395-417
Walsh T., Demkowicz L., 1999: A Parallel frontal Solver for hp-Adaptive Finite
Elements. TICAM Report 99-01
(sequential frontal solver over each sub-domain, as well as for the interface)
• parallel multi-frontal solvers
Multi-frontal Massively Parallel sparse direct Solver (MUMPS)
Amestoy P. R., Duff I. S., L'Excellent J.-Y., 2000: Multifrontal parallel distributed
symmetric and unsymmetric solvers. Computer Methods in Applied Mechanics and
Engineering 184, 501-520
Amestoy P. R., Guermouche A., L'Excellent J.-Y., Pralet S., 2006: Hybrid scheduling
for the parallel solution of linear systems. Parallel Computing, 32, 2, 136-156
Other parallel multi-frontal solvers: PARDISO, Harwell – Boeing