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Sparse matrix data structure

- Typically eitherCompressed Sparse Row

(CSR)orCompressed Sparse Column (CSC) - Informally ia-ja format
- CSR is better for matrix-vector multiplies CSC

can be better for factorization - CSR
- Array of all values, row by row
- Array of all column indices, row by row
- Array of pointers to start of each row

Direct Solvers

- Well just peek at Cholesky factorization of SPD

matrices ALLT - In particular, pivoting not required!
- Modern solvers break Cholesky into three phases
- Ordering determine order of rows/columns
- Symbolic factorization determine sparsity

structure of L in advance - Numerical factorization compute values in L
- Allows for much greater optimization

Graph model of elimination

- Take the graph whose adjacency matrix matches A
- Choosing node i to eliminate next in

row-reduction - Subtract off multiples of row i from rows of

neighbours - In graph terms unioning edge structure of i with

all its neighbours - A is symmetric -gt connecting up all neighbours of

i into a clique - New edges are called fill(nonzeros in L that

are zero in A) - Choosing a different sequence can result in

different fill

Extreme fill

- The star graph
- If you order centre last, zero fill O(n) time

and memory - If you order centre first, O(n2) fill O(n3)

time and O(n2) memory

Fill-reducing orderings

- Finding minimum fill ordering is NP-hard
- Two main heuristics in use
- Minimum Degree (greedy incremental)choose node

of minimum degree first - Without many additional accelerations, this is

too slow, but now very efficient e.g. AMD - Nested Dissection (divide-and-conquer)partition

graph by a node separator, order separator last,

recurse on components - Optimal partition is also NP-hard, but very

good/fast heuristic exist e.g. Metis - Great for parallelism e.g. ParMetis

A peek at Minimum Degree

- See George Liu, The evolution of the minimum

degree algorithm - A little dated now, but most of key concepts

explained there - Biggest optimization dont store structure

explicitly - Treat eliminated nodes as quotient nodes
- Edge in L path in A via zero or more eliminated

nodes

A peek at Nested Dissection

- Core operation is graph partitioning
- Simplest strategy breadth-first search
- Can locally improve with Kernighan-Lin
- Can make this work fast by going multilevel

Theoretical Limits

- In 2D (planar or near planar graphs), Nested

Dissection is within a constant factor of

optimal - O(n log n) fill (nnumber of nodes - think s2)
- O(n3/2) time for factorization
- Result due to Lipton Tarjan
- In 3D asymptotics for well-shaped 3D meshes is

worse - O(n5/3) fill (nnumber of nodes - think s3)
- O(n2) time for factorization
- Direct solvers are very competitive in 2D, but

dont scale nearly as well in 3D

Symbolic Factorization

- Given ordering, determining L is also just a

graph problem - Various optimizations allow determination of row

or column counts of L in nearly O(nnz(A)) time - Much faster than actual factorization!
- One of the most important observationsgood

orderings usually results in supernodes columns

of L with identical structure - Can treat these columns as a single block column

Numerical Factorization

- Can compute L column by column with left-looking

factorization - In particular, compute a supernode (block column)

at a time - Can use BLAS level 3 for most of the numerics
- Get huge performance boost, near optimal

Software

- See Tim Daviss list atwww.cise.ufl.edu/research/

sparse/codes/ - Ordering AMD and Metis becoming standard
- Cholesky PARDISO, CHOLMOD,
- General PARDISO, UMFPACK,

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