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CS 290H Lecture 13 Column approximate minimum degree; Other approaches to nonsymmetric LUPowerPoint Presentation

CS 290H Lecture 13 Column approximate minimum degree; Other approaches to nonsymmetric LU

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CS 290H Lecture 13Column approximate minimum degree;Other approaches to nonsymmetric LU

- Final project progress report due today
- Homework 3 due this Sunday 21 November
- Read “Computing the block triangular form of a sparse matrix” (reader #6)

x

Column Preordering for SparsityQ

- PAQT= LU:Q preorders columns for sparsity, P is row pivoting
- Column permutation of A Symmetric permutation of ATA (or G(A))

P

2

3

4

5

3

1

2

3

4

5

1

4

5

2

3

4

5

2

1

Column Intersection Graph- G(A) = G(ATA) if no cancellation(otherwise)
- Permuting the rows of A does not change G(A)

A

ATA

G(A)

2

3

4

5

3

+

1

2

3

4

5

G(A)

1

4

5

2

3

4

2

5

1

+

+

+

Filled Column Intersection Graph- G(A) = symbolic Cholesky factor of ATA
- In PA=LU, G(U) G(A) and G(L) G(A)
- Tighter bound on L from symbolic QR
- Bounds are best possible if A is strong Hall

A

chol(ATA)

x

Column Preordering for SparsityQ

- PAQT= LU:Q preorders columns for sparsity, P is row pivoting
- Column permutation of A Symmetric permutation of ATA (or G(A))
- Symmetric ordering: Approximate minimum degree
- But, forming ATA is expensive (sometimes bigger than L+U).

P

2

3

4

5

1

1

1

2

2

2

3

3

3

4

4

5

4

5

5

Column Approximate Minimum Degree [Matlab 6]row

col

- Eliminate “row” nodes of aug(A) first
- Then eliminate “col” nodes by approximate min degree
- 4x speed and 1/3 better ordering than Matlab-5 min degree, 2x speed of AMD on ATA
- Can also use other orderings, e.g. nested dissection on aug(A)

I

A

row

AT

I

col

G(aug(A))

A

aug(A)

1

2

3

4

5

1

2

3

4

5

1

4

2

2

3

3

4

5

1

Column Elimination Tree- Elimination tree of ATA (if no cancellation)
- Depth-first spanning tree of G(A)
- Represents column dependencies in various factorizations

T(A)

A

chol(ATA)

+

j

T[k]

- If A is strong Hall then, for some pivot sequence, every column modifies its parent in T(A).

- If column j modifies column k, then j T[k].

Shared Memory SuperLU-MT

- 1D data layout across processors
- Dynamic assignment of panel tasks to processors
- Task tree follows column elimination tree
- Two sources of parallelism:
- Independent subtrees
- Pipelining dependent panel tasks

- Single processor “BLAS 2.5” SuperLU kernel

- Good speedup for 8-16 processors
- Scalability limited by 1D data layout

SuperLU-MT Performance Highlight (1999)

3-D flow calculation (matrix EX11, order 16614):

for column j = 1 to n do

solve

pivot: swap ujj and an elt of lj

scale:lj = lj / ujj

j

U

L

A

( )

L 0L I

( )

ujlj

L

= aj for uj, lj

Left-looking Column LU Factorization- Column j of A becomes column j of L and U

1

G(A)

7

1

2

3

4

5

6

7

8

9

6

3

8

1

2

2

5

9

3

4

5

6

9

T(A)

7

8

8

9

7

A

6

3

4

1

2

5

Symmetric-pattern multifrontal factorization1

G(A)

7

6

3

8

2

5

9

9

T(A)

8

7

6

3

4

1

2

5

Symmetric-pattern multifrontal factorizationFor each node of T from leaves to root:

- Sum own row/col of A with children’s Update matrices into Frontal matrix
- Eliminate current variable from Frontal matrix, to get Update matrix
- Pass Update matrix to parent

1

3

7

1

G(A)

7

1

3

6

3

8

7

F1 = A1

=> U1

2

5

9

9

T(A)

8

3

7

7

3

7

6

3

4

1

2

5

Symmetric-pattern multifrontal factorizationFor each node of T from leaves to root:

- Sum own row/col of A with children’s Update matrices into Frontal matrix
- Eliminate current variable from Frontal matrix, to get Update matrix
- Pass Update matrix to parent

1

3

7

1

G(A)

7

1

3

6

3

8

7

F1 = A1

=> U1

2

5

9

9

2

3

9

T(A)

3

9

8

3

7

2

3

7

3

3

9

7

6

3

9

4

1

2

5

F2 = A2

=> U2

Symmetric-pattern multifrontal factorizationFor each node of T from leaves to root:

- Sum own row/col of A with children’s Update matrices into Frontal matrix
- Eliminate current variable from Frontal matrix, to get Update matrix
- Pass Update matrix to parent

1

3

7

1

G(A)

7

1

3

6

3

8

7

F1 = A1

=> U1

2

5

9

2

3

9

2

3

9

9

F2 = A2

=> U2

8

3

3

9

7

3

7

8

9

7

3

3

3

7

8

9

7

9

6

7

3

7

8

4

1

2

5

8

9

9

F3 = A3+U1+U2

=> U3

Symmetric-pattern multifrontal factorizationT(A)

1

G+(A)

7

1

2

3

4

5

6

7

8

9

6

3

8

1

2

2

5

9

3

4

5

6

9

T(A)

7

8

8

9

7

L+U

6

3

4

1

2

5

Symmetric-pattern multifrontal factorization9

T(A)

4

1

7

8

7

6

3

8

6

3

2

5

4

1

2

5

9

Symmetric-pattern multifrontal factorization- Really uses supernodes, not nodes
- All arithmetic happens on dense square matrices.
- Needs extra memory for a stack of pending update matrices
- Potential parallelism:
- between independent tree branches
- parallel dense ops on frontal matrix

MUMPS: distributed-memory multifrontal[Amestoy, Duff, L’Excellent, Koster, Tuma]

- Symmetric-pattern multifrontal factorization
- Parallelism both from tree and by sharing dense ops
- Dynamic scheduling of dense op sharing
- Symmetric preordering
- For nonsymmetric matrices:
- optional weighted matching for heavy diagonal
- expand nonzero pattern to be symmetric
- numerical pivoting only within supernodes if possible (doesn’t change pattern)
- failed pivots are passed up the tree in the update matrix

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