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Column Cholesky Factorization: A=R T R

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for j = 1 : n

for k = 1 : j-1

% cmod(j,k)

for i = j : n

A(i,j) = A(i,j) – A(i,k)*A(j,k);

end;

end;

% cdiv(j)

A(j,j) = sqrt(A(j,j));

for i = j+1 : n

A(i,j) = A(i,j) / A(j,j);

end;

end;

j

R

RT

A

RT

- Column j of A becomes column j of RT

- Full:
- 2-dimensional array of real or complex numbers
- (nrows*ncols) memory

- Sparse:
- compressed column storage
- about (1.5*nzs + .5*ncols) memory

D

for j = 1 : n

for k < j with A(j,k) nonzero

% sparse cmod(j,k)

A(j:n, j) = A(j:n, j) – A(j:n, k)*A(j, k);

end;

% sparse cdiv(j)

A(j,j) = sqrt(A(j,j));

A(j+1:n, j) = A(j+1:n, j) / A(j,j);

end;

j

R

RT

A

RT

- Column j of A becomes column j of RT

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Fill:new nonzeros in factor

Symmetric Gaussian elimination:

for j = 1 to n add edges between j’s higher-numbered neighbors

G+(A)[chordal]

G(A)

- Preorder
- Independent of numerics

- Symbolic Factorization
- Elimination tree
- Nonzero counts
- Supernodes
- Nonzero structure of R

- Numeric Factorization
- Static data structure
- Supernodes use BLAS3 to reduce memory traffic

- Triangular Solves

x

A

RT

R

- Compute factors of A by Gaussian elimination, but ignore fill
- Preconditioner B = RTR A, not formed explicitly
- Compute B-1z by triangular solves (in time nnz(A))
- Total storage is O(nnz(A)), static data structure
- Either symmetric (IC) or nonsymmetric (ILU)

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2

- Allow one or more “levels of fill”
- unpredictable storage requirements

- Allow fill whose magnitude exceeds a “drop tolerance”
- may get better approximate factors than levels of fill
- unpredictable storage requirements
- choice of tolerance is ad hoc

- Partial pivoting (for nonsymmetric A)
- “Modified ILU” (MIC): Add dropped fill to diagonal of U or R
- A and RTR have same row sums
- good in some PDE contexts

- Choice of parameters
- good: smooth transition from iterative to direct methods
- bad: very ad hoc, problem-dependent
- tradeoff: time per iteration (more fill => more time)vs # of iterations (more fill => fewer iters)

- Effectiveness
- condition number usually improves (only) by constant factor (except MIC for some problems from PDEs)
- still, often good when tuned for a particular class of problems

- Parallelism
- Triangular solves are not very parallel
- Reordering for parallel triangular solve by graph coloring

- Goal is to reduce fill, making Cholesky factorization cheaper.
- Minimum degree:
- Eliminate row/col with fewest nzs, add fill, repeat
- Theory: can be suboptimal even on 2D model problem
- Practice: often wins for medium-sized problems

- Nested dissection:
- Find a separator, number it last, proceed recursively
- Theory: approx optimal separators => approx optimal fill and flop count
- Practice: often wins for very large problems

- Banded orderings (Reverse Cuthill-McKee, Sloan, . . .):
- Try to keep all nonzeros close to the diagonal
- Theory, practice: often wins for “long, thin” problems

- Best orderings for direct methods are ND/MD hybrids.

- Nonsymmetric approximate minimum degree:
- p = colamd(A);
- column permutation: lu(A(:,p)) often sparser than lu(A)
- also for QR factorization

- Symmetric approximate minimum degree:
- p = symamd(A);
- symmetric permutation: chol(A(p,p)) often sparser than chol(A)

- Reverse Cuthill-McKee
- p = symrcm(A);
- A(p,p) often has smaller bandwidth than A
- similar to Sparspak RCM

D

- Symmetric matrix permutations don’t change the convergence of unpreconditioned CG
- Symmetric matrix permutations affect the quality of an incomplete factorization – poorly understood, controversial
- Often banded (RCM) is best for IC(0) / ILU(0)
- Often min degree & nested dissection are bad for no-fill incomplete factorization but good if some fill is allowed
- Some experiments with orderings that use matrix values
- e.g. “minimum discarded fill” order
- sometimes effective but expensive to compute

- Dynamic permutation: ILU with row or column pivoting
- E.g. ILUTP (Saad), Matlab “luinc”
- More robust but more expensive than ILUT

- Static permutation: Try to increase diagonal dominance
- E.g. bipartite weighted matching (Duff & Koster)
- Often effective; no theory for quality of preconditioner

- Field is not well understood, to say the least

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PA

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- Represent A as a weighted, undirected bipartite graph (one node for each row and one node for each column)
- Find matching (set of independent edges) with maximum product of weights
- Permute rows to place matching on diagonal
- Matching algorithm also gives a row and column scaling to make all diag elts =1 and all off-diag elts <=1

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A

A

B-1

- Compute B-1 A explicitly
- Minimize || A B-1 – I ||F (in parallel, by columns)
- Variants: factored form of B-1, more fill, . .
- Good: very parallel, seldom breaks down
- Bad: effectiveness varies widely

n1/2

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Time to solve model problem (Poisson’s equation) on regular mesh

n1/2

n1/3

Time and space to solve any problem on any well-shaped finite element mesh