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Automatic Performance Tuning of Sparse-Matrix-Vect

or-Multiplication (SpMV) and Iterative Sparse

Solvers

- James Demmel
- www.cs.berkeley.edu/demmel/cs267_Spr09

Outline

- Motivation for Automatic Performance Tuning
- Results for sparse matrix kernels
- Sparse Matrix Vector Multiplication (SpMV)
- Sequential and Multicore results
- OSKI Optimized Sparse Kernel Interface
- Tuning Entire Sparse Solvers

Berkeley Benchmarking and OPtimization (BeBOP)

- Prof. Katherine Yelick
- Current members
- Kaushik Datta, Mark Hoemmen, Marghoob Mohiyuddin,

Shoaib Kamil, Rajesh Nishtala, Vasily Volkov,

Sam Williams, - Previous members
- Hormozd Gahvari, Eun-Jim Im, Ankit Jain, Rich

Vuduc, many undergrads, - Many results here from current, previous students
- bebop.cs.berkeley.edu

Automatic Performance Tuning

- Goal Let machine do hard work of writing fast

code - What does tuning of dense BLAS, FFTs, signal

processing, have in common? - Can do the tuning off-line once per

architecture, algorithm - Can take as much time as necessary (hours, a

week ) - At run-time, algorithm choice may depend only on

few parameters (matrix dimensions, size of FFT,

etc.) - Cant always do tuning off-line
- Algorithm and implementation may strongly depend

on data only known at run-time - Ex Sparse matrix nonzero pattern determines both

best data structure and implementation of

Sparse-matrix-vector-multiplication (SpMV) - Part of search for best algorithm just be done

(very quickly!) at run-time

Source Accelerator Cavity Design Problem (Ko via

Husbands)

Linear Programming Matrix

A Sparse Matrix You Encounter Every Day

SpMV with Compressed Sparse Row (CSR) Storage

Matrix-vector multiply kernel y(i) ? y(i)

A(i,j)x(j) for each row i for kptri to

ptri1 do yi yi valkxindk

Matrix-vector multiply kernel y(i) ? y(i)

A(i,j)x(j) for each row i for kptri to

ptri1 do yi yi valkxindk

Example The Difficulty of Tuning

- n 21200
- nnz 1.5 M
- kernel SpMV
- Source NASA structural analysis problem

Example The Difficulty of Tuning

- n 21200
- nnz 1.5 M
- kernel SpMV
- Source NASA structural analysis problem

- 8x8 dense substructure exploit this to limit

mem_refs

Speedups on Itanium 2 The Need for Search

Mflop/s

Mflop/s

Register Profile Itanium 2

1190 Mflop/s

190 Mflop/s

Register Profiles IBM and Intel IA-64

Power3 - 17

Power4 - 16

252 Mflop/s

820 Mflop/s

122 Mflop/s

459 Mflop/s

Itanium 2 - 33

Itanium 1 - 8

247 Mflop/s

1.2 Gflop/s

107 Mflop/s

190 Mflop/s

Register Profiles Sun and Intel x86

Ultra 2i - 11

Ultra 3 - 5

72 Mflop/s

90 Mflop/s

35 Mflop/s

50 Mflop/s

Pentium III-M - 15

Pentium III - 21

108 Mflop/s

122 Mflop/s

42 Mflop/s

58 Mflop/s

Another example of tuning challenges

- More complicated non-zero structure in general
- N 16614
- NNZ 1.1M

Zoom in to top corner

- More complicated non-zero structure in general
- N 16614
- NNZ 1.1M

3x3 blocks look natural, but

- More complicated non-zero structure in general
- Example 3x3 blocking
- Logical grid of 3x3 cells
- But would lead to lots of fill-in

Extra Work Can Improve Efficiency!

- More complicated non-zero structure in general
- Example 3x3 blocking
- Logical grid of 3x3 cells
- Fill-in explicit zeros
- Unroll 3x3 block multiplies
- Fill ratio 1.5
- On Pentium III 1.5x speedup!
- Actual mflop rate 1.52 2.25 higher

Automatic Register Block Size Selection

- Selecting the r x c block size
- Off-line benchmark
- Precompute Mflops(r,c) using dense A for each r x

c - Once per machine/architecture
- Run-time search
- Sample A to estimate Fill(r,c) for each r x c
- Run-time heuristic model
- Choose r, c to minimize time Fill(r,c) /

Mflops(r,c)

Accurate and Efficient Adaptive Fill Estimation

- Idea Sample matrix
- Fraction of matrix to sample s Î 0,1
- Control cost O(s nnz ) by controlling s
- Search at run-time the constant matters!
- Control s automatically by computing statistical

confidence intervals, by monitoring variance - Cost of tuning
- Lower bound convert matrix in 5 to 40 unblocked

SpMVs - Heuristic 1 to 11 SpMVs
- Tuning only useful when we do many SpMVs
- Common case, eg in sparse solvers

Accuracy of the Tuning Heuristics (1/4)

See p. 375 of Vuducs thesis for matrices

NOTE Fair flops used (ops on explicit zeros

not counted as work)

Accuracy of the Tuning Heuristics (2/4)

Accuracy of the Tuning Heuristics (2/4)

DGEMV

Upper Bounds on Performance for blocked SpMV

- P (flops) / (time)
- Flops 2 nnz(A)
- Upper bound on speed Two main assumptions
- 1. Count memory ops only (streaming)
- 2. Count only compulsory, capacity misses ignore

conflicts - Account for line sizes
- Account for matrix size and nnz
- Charge minimum access latency ai at Li cache

amem - e.g., Saavedra-Barrera and PMaC MAPS benchmarks

- Upper bound on time assume all references to

x( ) miss

Example Bounds on Itanium 2

Example Bounds on Itanium 2

Example Bounds on Itanium 2

Summary of Other Performance Optimizations

- Optimizations for SpMV
- Register blocking (RB) up to 4x over CSR
- Variable block splitting 2.1x over CSR, 1.8x

over RB - Diagonals 2x over CSR
- Reordering to create dense structure splitting

2x over CSR - Symmetry 2.8x over CSR, 2.6x over RB
- Cache blocking 2.8x over CSR
- Multiple vectors (SpMM) 7x over CSR
- And combinations
- Sparse triangular solve
- Hybrid sparse/dense data structure 1.8x over CSR
- Higher-level kernels
- AATx, ATAx 4x over CSR, 1.8x over RB
- A2x 2x over CSR, 1.5x over RB
- Ax, A2x, A3x, .. , Akx
. more to say

later

Potential Impact on Applications Omega3P

- Application accelerator cavity design Ko
- Relevant optimization techniques
- Symmetric storage
- Register blocking
- Reordering, to create more dense blocks
- Reverse Cuthill-McKee ordering to reduce

bandwidth - Do Breadth-First-Search, number nodes in reverse

order visited - Traveling Salesman Problem-based ordering to

create blocks - Nodes columns of A
- Weights(u, v) no. of nz u, v have in common
- Tour ordering of columns
- Choose maximum weight tour
- See Pinar Heath 97
- 2.1x speedup on IBM Power 4

Source Accelerator Cavity Design Problem (Ko via

Husbands)

Post-RCM Reordering

100x100 Submatrix Along Diagonal

Microscopic Effect of RCM Reordering

Before Green Red After Green Blue

Microscopic Effect of Combined RCMTSP

Reordering

Before Green Red After Green Blue

(Omega3P)

Optimized Sparse Kernel Interface - OSKI

- Provides sparse kernels automatically tuned for

users matrix machine - BLAS-style functionality SpMV, Ax ATy, TrSV
- Hides complexity of run-time tuning
- Includes new, faster locality-aware kernels

ATAx, Akx - Faster than standard implementations
- Up to 4x faster matvec, 1.8x trisolve, 4x ATAx
- For advanced users solver library writers
- Available as stand-alone library (OSKI 1.0.1h,

6/07) - Available as PETSc extension (OSKI-PETSc .1d,

3/06) - Bebop.cs.berkeley.edu/oski

How the OSKI Tunes (Overview)

Application Run-Time

Library Install-Time (offline)

1. Build for Target Arch.

2. Benchmark

Workload from program monitoring

History

Matrix

Benchmark data

Heuristic models

1. Evaluate Models

Generated code variants

2. Select Data Struct. Code

To user Matrix handle for kernel calls

Extensibility Advanced users may write

dynamically add Code variants and Heuristic

models to system.

How to Call OSKI Basic Usage

- May gradually migrate existing apps
- Step 1 Wrap existing data structures
- Step 2 Make BLAS-like kernel calls

int ptr , ind double val /

Matrix, in CSR format / double x , y

/ Let x and y be two dense vectors / /

Compute y ?y ?Ax, 500 times / for( i 0

i lt 500 i ) my_matmult( ptr, ind, val, ?, x,

b, y )

How to Call OSKI Basic Usage

- May gradually migrate existing apps
- Step 1 Wrap existing data structures
- Step 2 Make BLAS-like kernel calls

int ptr , ind double val /

Matrix, in CSR format / double x , y

/ Let x and y be two dense vectors / / Step 1

Create OSKI wrappers around this data

/ oski_matrix_t A_tunable oski_CreateMatCSR(ptr

, ind, val, num_rows, num_cols, SHARE_INPUTMAT,

) oski_vecview_t x_view oski_CreateVecView(x,

num_cols, UNIT_STRIDE) oski_vecview_t y_view

oski_CreateVecView(y, num_rows, UNIT_STRIDE) /

Compute y ?y ?Ax, 500 times / for( i 0

i lt 500 i ) my_matmult( ptr, ind, val, ?, x,

b, y )

How to Call OSKI Basic Usage

- May gradually migrate existing apps
- Step 1 Wrap existing data structures
- Step 2 Make BLAS-like kernel calls

int ptr , ind double val /

Matrix, in CSR format / double x , y

/ Let x and y be two dense vectors / / Step 1

Create OSKI wrappers around this data

/ oski_matrix_t A_tunable oski_CreateMatCSR(ptr

, ind, val, num_rows, num_cols, SHARE_INPUTMAT,

) oski_vecview_t x_view oski_CreateVecView(x,

num_cols, UNIT_STRIDE) oski_vecview_t y_view

oski_CreateVecView(y, num_rows, UNIT_STRIDE) /

Compute y ?y ?Ax, 500 times / for( i 0

i lt 500 i ) oski_MatMult(A_tunable,

OP_NORMAL, ?, x_view, ?, y_view)/ Step 2 /

How to Call OSKI Tune with Explicit Hints

- User calls tune routine
- May provide explicit tuning hints (OPTIONAL)

oski_matrix_t A_tunable oski_CreateMatCSR(

) / / / Tell OSKI we will call SpMV 500

times (workload hint) / oski_SetHintMatMult(A_tun

able, OP_NORMAL, ?, x_view, ?, y_view, 500) /

Tell OSKI we think the matrix has 8x8 blocks

(structural hint) / oski_SetHint(A_tunable,

HINT_SINGLE_BLOCKSIZE, 8, 8) oski_TuneMat(A_tuna

ble) / Ask OSKI to tune / for( i 0 i lt

500 i ) oski_MatMult(A_tunable, OP_NORMAL, ?,

x_view, ?, y_view)

How the User Calls OSKI Implicit Tuning

- Ask library to infer workload
- Library profiles all kernel calls
- May periodically re-tune

oski_matrix_t A_tunable oski_CreateMatCSR(

) / / for( i 0 i lt 500 i )

oski_MatMult(A_tunable, OP_NORMAL, ?, x_view,

?, y_view) oski_TuneMat(A_tunable) / Ask OSKI

to tune /

Multicore SMPs Used for Tuning SpMV

AMD Opteron 2356 (Barcelona)

Intel Xeon E5345 (Clovertown)

IBM QS20 Cell Blade

Sun T2 T5140 (Victoria Falls)

Multicore SMPs with Conventional cache-based

memory hierarchy

AMD Opteron 2356 (Barcelona)

Intel Xeon E5345 (Clovertown)

Sun T2 T5140 (Victoria Falls)

IBM QS20 Cell Blade

Multicore SMPs with local store-based memory

hierarchy

AMD Opteron 2356 (Barcelona)

Intel Xeon E5345 (Clovertown)

Sun T2 T5140 (Victoria Falls)

IBM QS20 Cell Blade

Multicore SMPs with CMT Chip-MultiThreading

AMD Opteron 2356 (Barcelona)

Intel Xeon E5345 (Clovertown)

IBM QS20 Cell Blade

Sun T2 T5140 (Victoria Falls)

Multicore SMPs Number of threads

AMD Opteron 2356 (Barcelona)

Intel Xeon E5345 (Clovertown)

8 threads

8 threads

IBM QS20 Cell Blade

Sun T2 T5140 (Victoria Falls)

16 threads

128 threads

SPEs only

Multicore SMPs peak double precision flops

AMD Opteron 2356 (Barcelona)

Intel Xeon E5345 (Clovertown)

75 GFlop/s

74 Gflop/s

IBM QS20 Cell Blade

Sun T2 T5140 (Victoria Falls)

29 GFlop/s

19 GFlop/s

SPEs only

Multicore SMPs total DRAM bandwidth

AMD Opteron 2356 (Barcelona)

Intel Xeon E5345 (Clovertown)

21 GB/s (read) 10 GB/s (write)

21 GB/s

IBM QS20 Cell Blade

Sun T2 T5140 (Victoria Falls)

51 GB/s

42 GB/s (read) 21 GB/s (write)

SPEs only

Multicore SMPs with Non-Uniform Memory Access -

NUMA

AMD Opteron 2356 (Barcelona)

Intel Xeon E5345 (Clovertown)

IBM QS20 Cell Blade

Sun T2 T5140 (Victoria Falls)

SPEs only

Set of 14 test matrices

- All bigger than the caches of our SMPs

2K x 2K Dense matrix stored in sparse format

Dense

Well Structured (sorted by nonzeros/row)

Protein

FEM / Spheres

FEM / Cantilever

Wind Tunnel

FEM / Harbor

QCD

FEM / Ship

Economics

Epidemiology

Poorly Structured hodgepodge

FEM / Accelerator

Circuit

webbase

Extreme Aspect Ratio (linear programming)

LP

SpMV Performance Naive parallelization

- Out-of-the box SpMV performance on a suite of 14

matrices - Scalability isnt great
- Compare to threads
- 8 8
- 128 16

Naïve Pthreads

Naïve

SpMV Performance NUMA and Software Prefetching

- NUMA-aware allocation is essential on NUMA SMPs.
- Explicit software prefetching can boost bandwidth

and change cache replacement policies - used exhaustive search

SpMV Performance Matrix Compression

- Compression includes
- register blocking
- other formats
- smaller indices
- Use heuristic rather than search

SpMV Performance cache and TLB blocking

Cache/LS/TLB Blocking

Matrix Compression

SW Prefetching

NUMA/Affinity

Naïve Pthreads

Naïve

SpMV Performance Architecture specific

optimizations

Cache/LS/TLB Blocking

Matrix Compression

SW Prefetching

NUMA/Affinity

Naïve Pthreads

Naïve

SpMV Performance max speedup

- Fully auto-tuned SpMV performance across the

suite of matrices - Included SPE/local store optimized version
- Why do some optimizations work better on some

architectures?

2.7x

4.0x

2.9x

35x

Cache/LS/TLB Blocking

Matrix Compression

SW Prefetching

NUMA/Affinity

Naïve Pthreads

Naïve

Avoiding Communication in Sparse Linear Algebra

- k-steps of typical iterative solver for Axb or

Ax?x - Does k SpMVs with starting vector (eg with b, if

solving Axb) - Finds best solution among all linear

combinations of these k1 vectors - Many such Krylov Subspace Methods
- Conjugate Gradients, GMRES, Lanczos, Arnoldi,
- Goal minimize communication in Krylov Subspace

Methods - Assume matrix well-partitioned, with modest

surface-to-volume ratio - Parallel implementation
- Conventional O(k log p) messages, because k

calls to SpMV - New O(log p) messages - optimal
- Serial implementation
- Conventional O(k) moves of data from slow to

fast memory - New O(1) moves of data optimal
- Lots of speed up possible (modeled and measured)
- Price some redundant computation

Locally Dependent Entries for x,Ax, A

tridiagonal, 2 processors

Proc 1

Proc 2

Can be computed without communication

Locally Dependent Entries for x,Ax,A2x, A

tridiagonal, 2 processors

Proc 1

Proc 2

Can be computed without communication

Locally Dependent Entries for x,Ax, ,A3x, A

tridiagonal, 2 processors

Proc 1

Proc 2

Can be computed without communication

Locally Dependent Entries for x,Ax, ,A4x, A

tridiagonal, 2 processors

Proc 1

Proc 2

Can be computed without communication

Locally Dependent Entries for x,Ax, ,A8x, A

tridiagonal, 2 processors

Proc 1

Proc 2

Can be computed without communication k8 fold

reuse of A

Remotely Dependent Entries for x,Ax, ,A8x, A

tridiagonal, 2 processors

Proc 1

Proc 2

One message to get data needed to compute

remotely dependent entries, not k8 Minimizes

number of messages latency cost Price

redundant work ? surface/volume ratio

Remotely Dependent Entries for x,Ax,A2x,A3x, A

irregular, multiple processors

Sequential x,Ax, ,A4x, with memory hierarchy

v

One read of matrix from slow memory, not

k4 Minimizes words moved bandwidth cost No

redundant work

Performance results on 8-Core Clovertown

Optimizing Communication Complexity of Sparse

Solvers

- Example GMRES for Axb on 2D Mesh
- x lives on n-by-n mesh
- Partitioned on p½ -by- p½ grid of p processors
- A has 5 point stencil (Laplacian)
- (Ax)(i,j) linear_combination(x(i,j), x(i,j1),

x(i1,j)) - Ex 18-by-18 mesh on 3-by-3 grid of 9 processors

Minimizing Communication of GMRES

- What is the cost (flops, words, mess)

of k steps of standard GMRES?

GMRES, ver.1 for i1 to k w A v(i-1)

MGS(w, v(0), ,v(i-1)) update v(i), H

endfor solve LSQ problem with H

n/p½

n/p½

- Cost(A v) k (9n2 /p, 4n / p½ , 4 )
- Cost(MGS) k2/2 ( 4n2 /p , log p , log p )
- Total cost Cost( A v ) Cost (MGS)
- Can we reduce the latency?

Minimizing Communication of GMRES

- Cost(GMRES, ver.1) Cost(Av) Cost(MGS)

( 9kn2 /p, 4kn / p½ , 4k ) ( 2k2n2 /p , k2

log p / 2 , k2 log p / 2 )

- How much latency cost from Av can you avoid?

Almost all

Minimizing Communication of GMRES

- Cost(GMRES, ver. 2) Cost(W) Cost(MGS)

( 9kn2 /p, 4kn / p½ , 8 ) ( 2k2n2 /p , k2

log p / 2 , k2 log p / 2 )

- How much latency cost from MGS can you avoid?

Almost all

Minimizing Communication of GMRES

- Cost(GMRES, ver. 2) Cost(W) Cost(MGS)

( 9kn2 /p, 4kn / p½ , 8 ) ( 2k2n2 /p , k2

log p / 2 , k2 log p / 2 )

- How much latency cost from MGS can you avoid?

Almost all

Minimizing Communication of GMRES

- Cost(GMRES, ver. 2) Cost(W) Cost(MGS)

( 9kn2 /p, 4kn / p½ , 8 ) ( 2k2n2 /p , k2

log p / 2 , k2 log p / 2 )

- How much latency cost from MGS can you avoid?

Almost all

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Minimizing Communication of GMRES

- Cost(GMRES, ver. 3) Cost(W) Cost(TSQR)

( 9kn2 /p, 4kn / p½ , 8 ) ( 2k2n2 /p , k2

log p / 2 , log p )

- Latency cost independent of k, just log p

optimal - Oops W from power method, so precision lost

What to do?

- Use a different polynomial basis
- Not Monomial basis W v, Av, A2v,
, instead

- Newton Basis WN v, (A ?1 I)v , (A ?2 I)(A

?1 I)v, or - Chebyshev Basis WC v, T1(v), T2(v),

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Speed ups on 8-core Clovertown

Conclusions

- Fast code must minimize communication
- Especially for sparse matrix computations because

communication dominates - Generating fast code for a single SpMV
- Design space of possible algorithms must be

searched at run-time, when sparse matrix

available - Design space should be searched automatically
- Biggest speedups from minimizing communication in

an entire sparse solver - Many more opportunities to minimize communication

in multiple SpMVs than in one - Requires transforming entire algorithm
- Lots of open problems

Extra slides

Quick-and-dirty Parallelism OSKI-PETSc

- Extend PETScs distributed memory SpMV (MATMPIAIJ)

- PETSc
- Each process stores diag (all-local) and off-diag

submatrices - OSKI-PETSc
- Add OSKI wrappers
- Each submatrix tuned independently

p0

p1

p2

p3

OSKI-PETSc Proof-of-Concept Results

- Matrix 1 Accelerator cavity design (R. Lee _at_

SLAC) - N 1 M, 40 M non-zeros
- 2x2 dense block substructure
- Symmetric
- Matrix 2 Linear programming (Italian Railways)
- Short-and-fat 4k x 1M, 11M non-zeros
- Highly unstructured
- Big speedup from cache-blocking no native PETSc

format - Evaluation machine Xeon cluster
- Peak 4.8 Gflop/s per node

Accelerator Cavity Matrix

OSKI-PETSc Performance Accel. Cavity

Linear Programming Matrix

OSKI-PETSc Performance LP Matrix

Performance Results

- Measured Multicore (Clovertown) speedups up to

6.4x - Measured/Modeled sequential OOC speedup up to 3x
- Modeled parallel Petascale speedup up to 6.9x
- Modeled parallel Grid speedup up to 22x
- Sequential speedup due to bandwidth, works for

many problem sizes - Parallel speedup due to latency, works for

smaller problems on many processors

Speedups on Intel Clovertown (8 core)

Extensions

- Other Krylov methods
- Arnoldi, CG, Lanczos,
- Preconditioning
- Solve MAxMb where preconditioning matrix M

chosen to make system easier - M approximates A-1 somehow, but cheaply, to

accelerate convergence - Cheap as long as contributions from distant

parts of the system can be compressed - Sparsity
- Low rank

Design Space for x,Ax, ,Akx (1/3)

- Mathematical Operation
- Keep all vectors
- Krylov Subspace Methods
- Improving conditioning of basis
- W x, p1(A)x, p2(A)x, ,pk(A)x
- pi(A) degree i polynomial chosen to reduce

cond(W) - Preconditioning (Ayb ? MAyMb)
- x,Ax,MAx,AMAx,MAMAx, ,(MA)kx
- Keep last vector Akx only
- Jacobi, Gauss Seidel

Design Space for x,Ax, ,Akx (2/3)

- Representation of sparse A
- Zero pattern may be explicit or implicit
- Nonzero entries may be explicit or implicit
- Implicit ? save memory, communication

- Representation of dense preconditioners M
- Low rank off-diagonal blocks (semiseparable)

Design Space for x,Ax, ,Akx (3/3)

- Parallel implementation
- From simple indexing, with redundant flops ?

surface/volume ratio - To complicated indexing, with fewer redundant

flops - Sequential implementation
- Depends on whether vectors fit in fast memory
- Reordering rows, columns of A
- Important in parallel and sequential cases
- Can be reduced to pair of Traveling Salesmen

Problems - Plus all the optimizations for one SpMV!

Summary

- Communication-Avoiding Linear Algebra (CALA)
- Lots of related work
- Some going back to 1960s
- Reports discuss this comprehensively, not here
- Our contributions
- Several new algorithms, improvements on old ones
- Unifying parallel and sequential approaches to

avoiding communication - Time for these algorithms has come, because of

growing communication costs - Why avoid communication just for linear algebra

motifs?

Possible Class Projects

- Come to BEBOP meetings (T 9 1030, 606 Soda)
- Incorporate multicore optimizations into OSKI
- Experiment with SpMV on GPU
- Which optimizations are most effective?
- Try to speed up particular matrices of interest
- Data mining
- Experiment with new x,Ax, ,Akx kernel
- GPU, multicore, distributed memory
- On matrices of interest
- Experiment with new solvers using this kernel

Extra Slides

Tuning Higher Level Algorithms

- So far we have tuned a single sparse matrix

kernel - y ATAx motivated by higher level algorithm

(SVD) - What can we do by extending tuning to a higher

level? - Consider Krylov subspace methods for Axb, Ax

lx - Conjugate Gradients (CG), GMRES, Lanczos,
- Inner loop does yAx, dot products, saxpys,

scalar ops - Inner loop costs at least O(1) messages
- k iterations cost at least O(k) messages
- Our goal show how to do k iterations with O(1)

messages - Possible payoff make Krylov subspace methods

much - faster on machines with slow networks
- Memory bandwidth improvements too (not

discussed) - Obstacles numerical stability, preconditioning,

Krylov Subspace Methods for Solving Axb

- Compute a basis for a subspace V by doing y Ax

k times - Find best solution in that Krylov subspace V
- Given starting vector x1, V spanned by x2 Ax1,

x3 Ax2 , , xk Axk-1 - GMRES finds an orthogonal basis of V by

Gram-Schmidt, so it actually does a different

set of SpMVs than in last bullet

Example Standard GMRES

- r b - Ax1, b length(r), v1 r / b

length(r) sqrt(S ri2 ) - for m 1 to k do
- w Avm at least O(1) messages
- for i 1 to m do Gram-Schmidt
- him dotproduct(vi , w )
at least

O(1) messages, or log(p) - w w h im vi
- end for
- hm1,m length(w)
at least O(1) messages,

or log(p) - vm1 w / hm1,m
- end for
- find y minimizing length( Hk y be1 )

small, local problem - new x x1 Vk y
Vk v1 , v2 ,
,

vk

O(k2), or O(k2 log p), messages altogether

Latency-Avoiding GMRES (1)

- r b - Ax1, b length(r), v1 r / b

O(log p) messages - Wk1 v1 , A v1 , A2 v1 ,
, Ak v1

O(1) messages - Q, R qr(Wk1)
QR decomposition, O(log

p) messages - Hk R(, 2k1) (R(1k,1k))-1
small, local

problem - find y minimizing length( Hk y be1 )

small, local problem - new x x1 Qk y local problem

O(log p) messages altogether Independent of k

Latency-Avoiding GMRES (2)

- Q, R qr(Wk1)
QR decomposition, O(log

p) messages - Easy, but not so stable way to do it
- X(myproc) Wk1T(myproc) Wk1 (myproc)
- local computation
- Y sum_reduction(X(myproc))
O(log p)

messages -

Y Wk1T Wk1 - R (cholesky(Y))T
small, local

computation - Q(myproc) Wk1 (myproc) R-1
local

computation

Numerical example (1)

Diagonal matrix with n1000, Aii from 1 down to

10-5 Instability as k grows, after many iterations

Numerical Example (2)

Partial remedy restarting periodically (every

120 iterations) Other remedies high precision,

different basis than v , A v , , Ak v

Operation Counts for Ax,A2x,A3x, ,Akx on p procs

Summary and Future Work

- Dense
- LAPACK
- ScaLAPACK
- Communication primitives
- Sparse
- Kernels, Stencils
- Higher level algorithms
- All of the above on new architectures
- Vector, SMPs, multicore, Cell,
- High level support for tuning
- Specification language
- Integration into compilers

Extra Slides

A Sparse Matrix You Encounter Every Day

Who am I?

I am a Big Repository Of useful And useless Facts

alike. Who am I? (Hint Not your e-mail inbox.)

What about the Google Matrix?

- Google approach
- Approx. once a month rank all pages using

connectivity structure - Find dominant eigenvector of a matrix
- At query-time return list of pages ordered by

rank - Matrix A aG (1-a)(1/n)uuT
- Markov model Surfer follows link with

probability a, jumps to a random page with

probability 1-a - G is n x n connectivity matrix n billions
- gij is non-zero if page i links to page j
- Normalized so each column sums to 1
- Very sparse about 78 non-zeros per row (power

law dist.) - u is a vector of all 1 values
- Steady-state probability xi of landing on page i

is solution to x Ax - Approximate x by power method x Akx0
- In practice, k 25

Current Work

- Public software release
- Impact on library designs Sparse BLAS, Trilinos,

PETSc, - Integration in large-scale applications
- DOE Accelerator design plasma physics
- Geophysical simulation based on Block Lanczos

(ATAX LBL) - Systematic heuristics for data structure

selection? - Evaluation of emerging architectures
- Revisiting vector micros
- Other sparse kernels
- Matrix triple products, Akx
- Parallelism
- Sparse benchmarks (with UTK) Gahvari Hoemmen
- Automatic tuning of MPI collective ops Nishtala,

et al.

Summary of High-Level Themes

- Kernel-centric optimization
- Vs. basic block, trace, path optimization, for

instance - Aggressive use of domain-specific knowledge
- Performance bounds modeling
- Evaluating software quality
- Architectural characterizations and consequences
- Empirical search
- Hybrid on-line/run-time models
- Statistical performance models
- Exploit information from sampling, measuring

Related Work

- My bibliography 337 entries so far
- Sample area 1 Code generation
- Generative generic programming
- Sparse compilers
- Domain-specific generators
- Sample area 2 Empirical search-based tuning
- Kernel-centric
- linear algebra, signal processing, sorting, MPI,

- Compiler-centric
- profiling FDO, iterative compilation,

superoptimizers, self-tuning compilers,

continuous program optimization

Future Directions (A Bag of Flaky Ideas)

- Composable code generators and search spaces
- New application domains
- PageRank multilevel block algorithms for

topic-sensitive search? - New kernels cryptokernels
- rich mathematical structure germane to

performance lots of hardware - New tuning environments
- Parallel, Grid, whole systems
- Statistical models of application performance
- Statistical learning of concise parametric models

from traces for architectural evaluation - Compiler/automatic derivation of parametric models

Possible Future Work

- Different Architectures
- New FP instruction sets (SSE2)
- SMP / multicore platforms
- Vector architectures
- Different Kernels
- Higher Level Algorithms
- Parallel versions of kenels, with optimized

communication - Block algorithms (eg Lanczos)
- XBLAS (extra precision)
- Produce Benchmarks
- Augment HPCC Benchmark
- Make it possible to combine optimizations easily

for any kernel - Related tuning activities (LAPACK ScaLAPACK)

Review of Tuning by Illustration

- (Extra Slides)

Splitting for Variable Blocks and Diagonals

- Decompose A A1 A2 At
- Detect canonical structures (sampling)
- Split
- Tune each Ai
- Improve performance and save storage
- New data structures
- Unaligned block CSR
- Relax alignment in rows columns
- Row-segmented diagonals

Example Variable Block Row (Matrix 12)

2.1x over CSR 1.8x over RB

Example Row-Segmented Diagonals

2x over CSR

Mixed Diagonal and Block Structure

Summary

- Automated block size selection
- Empirical modeling and search
- Register blocking for SpMV, triangular solve,

ATAx - Not fully automated
- Given a matrix, select splittings and

transformations - Lots of combinatorial problems
- TSP reordering to create dense blocks (Pinar 97

Moon, et al. 04)

Extra Slides

A Sparse Matrix You Encounter Every Day

Who am I?

I am a Big Repository Of useful And useless Facts

alike. Who am I? (Hint Not your e-mail inbox.)

Problem Context

- Sparse kernels abound
- Models of buildings, cars, bridges, economies,
- Google PageRank algorithm
- Historical trends
- Sparse matrix-vector multiply (SpMV) 10 of peak
- 2x faster with hand-tuning
- Tuning becoming more difficult over time
- Promise of automatic tuning PHiPAC/ATLAS, FFTW,

- Challenges to high-performance
- Not dense linear algebra!
- Complex data structures indirect, irregular

memory access - Performance depends strongly on run-time inputs

Key Questions, Ideas, Conclusions

- How to tune basic sparse kernels automatically?
- Empirical modeling and search
- Up to 4x speedups for SpMV
- 1.8x for triangular solve
- 4x for ATAx 2x for A2x
- 7x for multiple vectors
- What are the fundamental limits on performance?
- Kernel-, machine-, and matrix-specific upper

bounds - Achieve 75 or more for SpMV, limiting low-level

tuning - Consequences for architecture?
- General techniques for empirical search-based

tuning? - Statistical models of performance

Road Map

- Sparse matrix-vector multiply (SpMV) in a

nutshell - Historical trends and the need for search
- Automatic tuning techniques
- Upper bounds on performance
- Statistical models of performance

Compressed Sparse Row (CSR) Storage

Matrix-vector multiply kernel y(i) ? y(i)

A(i,j)x(j)

Matrix-vector multiply kernel y(i) ? y(i)

A(i,j)x(j) for each row i for kptri to

ptri1 do yi yi valkxindk

Matrix-vector multiply kernel y(i) ? y(i)

A(i,j)x(j) for each row i for kptri to

ptri1 do yi yi valkxindk

Road Map

- Sparse matrix-vector multiply (SpMV) in a

nutshell - Historical trends and the need for search
- Automatic tuning techniques
- Upper bounds on performance
- Statistical models of performance

Historical Trends in SpMV Performance

- The Data
- Uniprocessor SpMV performance since 1987
- Untuned and Tuned implementations
- Cache-based superscalar micros some vectors
- LINPACK

SpMV Historical Trends Mflop/s

Example The Difficulty of Tuning

- n 21216
- nnz 1.5 M
- kernel SpMV
- Source NASA structural analysis problem

Still More Surprises

- More complicated non-zero structure in general

Still More Surprises

- More complicated non-zero structure in general
- Example 3x3 blocking
- Logical grid of 3x3 cells

Historical Trends Mixed News

- Observations
- Good news Moores law like behavior
- Bad news Untuned is 10 peak or less,

worsening - Good news Tuned roughly 2x better today, and

improving - Bad news Tuning is complex
- (Not really news SpMV is not LINPACK)
- Questions
- Application Automatic tuning?
- Architect What machines are good for SpMV?

Road Map

- Sparse matrix-vector multiply (SpMV) in a

nutshell - Historical trends and the need for search
- Automatic tuning techniques
- SpMV SC02 IJHPCA 04b
- Sparse triangular solve (SpTS) ICS/POHLL 02
- ATAx ICCS/WoPLA 03
- Upper bounds on performance
- Statistical models of performance

SPARSITY Framework for Tuning SpMV

- SPARSITY Automatic tuning for SpMV Im Yelick

99 - General approach
- Identify and generate implementation space
- Search space using empirical models experiments
- Prototype library and heuristic for choosing

register block size - Also cache-level blocking, multiple vectors
- Whats new?
- New block size selection heuristic
- Within 10 of optimal replaces previous version
- Expanded implementation space
- Variable block splitting, diagonals, combinations
- New kernels sparse triangular solve, ATAx, Arx

Automatic Register Block Size Selection

- Selecting the r x c block size
- Off-line benchmark characterize the machine
- Precompute Mflops(r,c) using dense matrix for

each r x c - Once per machine/architecture
- Run-time search characterize the matrix
- Sample A to estimate Fill(r,c) for each r x c
- Run-time heuristic model
- Choose r, c to maximize Mflops(r,c) / Fill(r,c)
- Run-time costs
- Up to 40 SpMVs (empirical worst case)

Accuracy of the Tuning Heuristics (1/4)

DGEMV

NOTE Fair flops used (ops on explicit zeros

not counted as work)

Accuracy of the Tuning Heuristics (2/4)

DGEMV

Accuracy of the Tuning Heuristics (3/4)

DGEMV

Accuracy of the Tuning Heuristics (4/4)

DGEMV

Road Map

- Sparse matrix-vector multiply (SpMV) in a

nutshell - Historical trends and the need for search
- Automatic tuning techniques
- Upper bounds on performance
- SC02
- Statistical models of performance

Motivation for Upper Bounds Model

- Questions
- Speedups are good, but what is the speed limit?
- Independent of instruction scheduling, selection
- What machines are good for SpMV?

Upper Bounds on Performance Blocked SpMV

- P (flops) / (time)
- Flops 2 nnz(A)
- Lower bound on time Two main assumptions
- 1. Count memory ops only (streaming)
- 2. Count only compulsory, capacity misses ignore

conflicts - Account for line sizes
- Account for matrix size and nnz
- Charge min access latency ai at Li cache amem
- e.g., Saavedra-Barrera and PMaC MAPS benchmarks

Example Bounds on Itanium 2

Example Bounds on Itanium 2

Example Bounds on Itanium 2

Fraction of Upper Bound Across Platforms

Achieved Performance and Machine Balance

- Machine balance Callahan 88 McCalpin 95
- Balance Peak Flop Rate / Bandwidth (flops /

double) - Ideal balance for mat-vec 2 flops / double
- For SpMV, even less
- SpMV streaming
- 1 / (avg load time to stream 1 array)

(bandwidth) - Sustained balance peak flops / model bandwidth

(No Transcript)

Where Does the Time Go?

- Most time assigned to memory
- Caches disappear when line sizes are equal
- Strictly increasing line sizes

Execution Time Breakdown Matrix 40

Speedups with Increasing Line Size

Summary Performance Upper Bounds

- What is the best we can do for SpMV?
- Limits to low-level tuning of blocked

implementations - Refinements?
- What machines are good for SpMV?
- Partial answer balance characterization
- Architectural consequences?
- Example Strictly increasing line sizes

Road Map

- Sparse matrix-vector multiply (SpMV) in a

nutshell - Historical trends and the need for search
- Automatic tuning techniques
- Upper bounds on performance
- Tuning other sparse kernels
- Statistical models of performance
- FDO 00 IJHPCA 04a

Statistical Models for Automatic Tuning

- Idea 1 Statistical criterion for stopping a

search - A general search model
- Generate implementation
- Measure performance
- Repeat
- Stop when probability of being within e of

optimal falls below threshold - Can estimate distribution on-line
- Idea 2 Statistical performance models
- Problem Choose 1 among m implementations at

run-time - Sample performance off-line, build statistical

model

Example Select a Matmul Implementation

Example Support Vector Classification

Road Map

- Sparse matrix-vector multiply (SpMV) in a

nutshell - Historical trends and the need for search
- Automatic tuning techniques
- Upper bounds on performance
- Tuning other sparse kernels
- Statistical models of performance
- Summary and Future Work

Summary of High-Level Themes

- Kernel-centric optimization
- Vs. basic block, trace, path optimization, for

instance - Aggressive use of domain-specific knowledge
- Performance bounds modeling
- Evaluating software quality
- Architectural characterizations and consequences
- Empirical search
- Hybrid on-line/run-time models
- Statistical performance models
- Exploit information from sampling, measuring

Related Work

- My bibliography 337 entries so far
- Sample area 1 Code generation
- Generative generic programming
- Sparse compilers
- Domain-specific generators
- Sample area 2 Empirical search-based tuning
- Kernel-centric
- linear algebra, signal processing, sorting, MPI,

- Compiler-centric
- profiling FDO, iterative compilation,

superoptimizers, self-tuning compilers,

continuous program optimization

Future Directions (A Bag of Flaky Ideas)

- Composable code generators and search spaces
- New application domains
- PageRank multilevel block algorithms for

topic-sensitive search? - New kernels cryptokernels
- rich mathematical structure germane to

performance lots of hardware - New tuning environments
- Parallel, Grid, whole systems
- Statistical models of application performance
- Statistical learning of concise parametric models

from traces for architectural evaluation - Compiler/automatic derivation of parametric models

Acknowledgements

- Super-advisors Jim and Kathy
- Undergraduate R.A.s Attila, Ben, Jen, Jin,

Michael, Rajesh, Shoaib, Sriram, Tuyet-Linh - See pages xvixvii of dissertation.

TSP-based Reordering Before

(Pinar 97 Moon, et al 04)

TSP-based Reordering After

(Pinar 97 Moon, et al 04) Up to

2x speedups over CSR

Example L2 Misses on Itanium 2

Misses measured using PAPI Browne 00

Example Distribution of Blocked Non-Zeros

Sparse/Dense Partitioning for SpTS

- Partition L into sparse (L1,L2) and dense LD

- Perform SpTS in three steps

- Sparsity optimizations for (1)(2) DTRSV for (3)
- Tuning parameters block size, size of dense

triangle

SpTS Performance Power3

(No Transcript)

Summary of SpTS and AATx Results

- SpTS Similar to SpMV
- 1.8x speedups limited benefit from low-level

tuning - AATx, ATAx
- Cache interleaving only up to 1.6x speedups
- Reg cache up to 4x speedups
- 1.8x speedup over register only
- Similar heuristic same accuracy ( 10 optimal)
- Further from upper bounds 6080
- Opportunity for better low-level tuning a la

PHiPAC/ATLAS - Matrix triple products? Akx?
- Preliminary work

Register Blocking Speedup

Register Blocking Performance

Register Blocking Fraction of Peak

Example Confidence Interval Estimation

Costs of Tuning

Splitting UBCSR Pentium III

Splitting UBCSR Power4

SplittingUBCSR Storage Power4

(No Transcript)

Example Variable Block Row (Matrix 13)

(No Transcript)

Preliminary Results (Matrix Set 2) Itanium 2

Dense

FEM

FEM (var)

Bio

LP

Econ

Stat

Multiple Vector Performance

(No Transcript)

What about the Google Matrix?

- Google approach
- Approx. once a month rank all pages using

connectivity structure - Find dominant eigenvector of a matrix
- At query-time return list of pages ordered by

rank - Matrix A aG (1-a)(1/n)uuT
- Markov model Surfer follows link with

probability a, jumps to a random page with

probability 1-a - G is n x n connectivity matrix n 3 billion
- gij is non-zero if page i links to page j
- Normalized so each column sums to 1
- Very sparse about 78 non-zeros per row (power

law dist.) - u is a vector of all 1 values
- Steady-state probability xi of landing on page i

is solution to x Ax - Approximate x by power method x Akx0
- In practice, k 25

(No Transcript)

MAPS Benchmark Example Power4

MAPS Benchmark Example Itanium 2

Saavedra-Barrera Example Ultra 2i

Execution Time Breakdown (PAPI) Matrix 40

Preliminary Results (Matrix Set 1) Itanium 2

LP

FEM

FEM (var)

Assorted

Dense

Tuning Sparse Triangular Solve (SpTS)

- Compute xL-1b where L sparse lower triangular,

x b dense - L from sparse LU has rich dense substructure
- Dense trailing triangle can account for 2090 of

matrix non-zeros - SpTS optimizations
- Split into sparse trapezoid and dense trailing

triangle - Use tuned dense BLAS (DTRSV) on dense triangle
- Use Sparsity register blocking on sparse part
- Tuning parameters
- Size of dense trailing triangle
- Register block size

Sparse Kernels and Optimizations

- Kernels
- Sparse matrix-vector multiply (SpMV) yAx
- Sparse triangular solve (SpTS) xT-1b
- yAATx, yATAx
- Powers (yAkx), sparse triple-product (RART),

- Optimization techniques (implementation space)
- Register blocking
- Cache blocking
- Multiple dense vectors (x)
- A has special structure (e.g., symmetric, banded,

) - Hybrid data structures (e.g., splitting,

switch-to-dense, ) - Matrix reordering
- How and when do we search?
- Off-line Benchmark implementations
- Run-time Estimate matrix properties, evaluate

performance models based on benchmark data

Cache Blocked SpMV on LSI Matrix Ultra 2i

A 10k x 255k 3.7M non-zeros Baseline 16

Mflop/s Best block size performance 16k x

64k 28 Mflop/s

Cache Blocking on LSI Matrix Pentium 4

A 10k x 255k 3.7M non-zeros Baseline 44

Mflop/s Best block size performance 16k x

16k 210 Mflop/s

Cache Blocked SpMV on LSI Matrix Itanium

A 10k x 255k 3.7M non-zeros Baseline 25

Mflop/s Best block size performance 16k x

32k 72 Mflop/s

Cache Blocked SpMV on LSI Matrix Itanium 2

A 10k x 255k 3.7M non-zeros Baseline 170

Mflop/s Best block size performance 16k x

65k 275 Mflop/s

Inter-Iteration Sparse Tiling (1/3)

- Strout, et al., 01
- Let A be 6x6 tridiagonal
- Consider yA2x
- tAx, yAt
- Nodes vector elements
- Edges matrix elements aij

Inter-Iteration Sparse Tiling (2/3)

- Strout, et al., 01
- Let A be 6x6 tridiagonal
- Consider yA2x
- tAx, yAt
- Nodes vector elements
- Edges matrix elements aij
- Orange everything needed to compute y1
- Reuse a11, a12

Inter-Iteration Sparse Tiling (3/3)

- Strout, et al., 01
- Let A be 6x6 tridiagonal
- Consider yA2x
- tAx, yAt
- Nodes vector elements
- Edges matrix elements aij
- Orange everything needed to compute y1
- Reuse a11, a12
- Grey y2, y3
- Reuse a23, a33, a43

Inter-Iteration Sparse Tiling Issues

- Tile sizes (colored regions) grow with no. of

iterations and increasing out-degree - G likely to have a few nodes with high out-degree

(e.g., Yahoo) - Mathematical tricks to limit tile size?
- Judicious dropping of edges Ng01

Summary and Questions

- Need to understand matrix structure and machine
- BeBOP suite of techniques to deal with different

sparse structures and architectures - Google matrix problem
- Established techniques within an iteration
- Ideas for inter-iteration optimizations
- Mathematical structure of problem may help
- Questions
- Structure of G?
- What are the computational bottlenecks?
- Enabling future computations?
- E.g., topic-sensitive PageRank ? multiple vector

version Haveliwala 02 - See www.cs.berkeley.edu/richie/bebop/intel/google

for more info, including more complete Itanium 2

results.

Exploiting Matrix Structure

- Symmetry (numerical or structural)
- Reuse matrix entries
- Can combine with register blocking, multiple

vectors, - Matrix splitting
- Split the matrix, e.g., into r x c and 1 x 1
- No fill overhead
- Large matrices with random structure
- E.g., Latent Semantic Indexing (LSI) matrices
- Technique cache blocking
- Store matrix as 2i x 2j sparse submatrices
- Effective when x vector is large
- Currently, search to find fastest size

Symmetric SpMV Performance Pentium 4

SpMV with Split Matrices Ultra 2i

Cache Blocking on Random Matrices Itanium

Speedup on four banded random matrices.

Sparse Kernels and Optimizations

- Kernels
- Sparse matrix-vector multiply (SpMV) yAx
- Sparse triangular solve (SpTS) xT-1b
- yAATx, yATAx
- Powers (yAkx), sparse triple-product (RART),

- Optimization techniques (implementation space)
- Register blocking
- Cache blocking
- Multiple dense vectors (x)
- A has special structure (e.g., symmetric, banded,

) - Hybrid data structures (e.g., splitting,

switch-to-dense, ) - Matrix reordering
- How and when do we search?
- Off-line Benchmark implementations
- Run-time Estimate matrix properties, evaluate

performance models based on benchmark data

Register Blocked SpMV Pentium III

Register Blocked SpMV Ultra 2i

Register Blocked SpMV Power3

Register Blocked SpMV Itanium

Possible Optimization Techniques

- Within an iteration, i.e., computing (GuuT)x

once - Cache block Gx
- On linear programming matrices and matrices with

random structure (e.g., LSI), 1.54x speedups - Best block size is matrix and machine dependent
- Reordering and/or splitting of G to separate

dense structure (rows, columns, blocks) - Between iterations, e.g., (GuuT)2x
- (GuuT)2x G2x (Gu)uTx u(uTG)x u(uTu)uTx
- Compute Gu, uTG, uTu once for all iterations
- G2x Inter-iteration tiling to read G only once

Multiple Vector Perform