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Parallel Spectral MethodsSolving Elliptic

Problems with FFTs

- Kathy Yelick
- www.cs.berkeley.edu/yelick/cs267_s07

References

- Previous CS267 lectures
- Lecture by Geoffrey Fox
- http//grids.ucs.indiana.edu/ptliupages/presentati

ons/PC2007/cps615fft00.ppt - FFTW project
- http//www.fftw.org
- Spiral project
- http//www.spiral.net

Poissons equation arises in many models

3D ?2u/?x2 ?2u/?y2 ?2u/?z2 f(x,y,z)

f represents the sources also need boundary

conditions

2D ?2u/?x2 ?2u/?y2 f(x,y)

1D d2u/dx2 f(x)

- Electrostatic or Gravitational Potential

Potential(position) - Heat flow Temperature(position, time)
- Diffusion Concentration(position, time)
- Fluid flow Velocity,Pressure,Density(position,tim

e) - Elasticity Stress,Strain(position,time)
- Variations of Poisson have variable coefficients

Algorithms for 2D (3D) Poisson Equation (N n2

(n3) vars)

- Algorithm Serial PRAM Memory Procs
- Dense LU N3 N N2 N2
- Band LU N2 (N7/3) N N3/2 (N5/3) N (N4/3)
- Jacobi N2 (N5/3) N (N2/3) N N
- Explicit Inv. N2 log N N2 N2
- Conj.Gradients N3/2 (N4/3) N1/2(1/3) log N N N
- Red/Black SOR N3/2 (N4/3) N1/2 (N1/3) N N
- Sparse LU N3/2 (N2) N1/2 Nlog N (N4/3) N
- FFT Nlog N log N N N
- Multigrid N log2 N N N
- Lower bound N log N N
- PRAM is an idealized parallel model with zero

cost communication - Reference James Demmel, Applied Numerical

Linear Algebra, SIAM, 1997.

Solving Poissons Equation with the FFT

- Express any 2D function defined in 0 ? x,y ? 1 as

a series ?(x,y) Sj Sk ?jk sin(p jx) sin(p

ky) - Here ?jk are called Fourier coefficient of ?(x,y)

- The inverse of this is ?jk 4

?(x,y) sin(p jx) sin(p ky) - Poissons equation ?2 ? /? x2 ? 2 ? /? y2

f(x,y) becomes - Sj Sk (-p2j2 - p2k2) ?jk sin(p jx) sin(p ky)
- Sj Sk fjk sin(p jx) sin(p ky)
- where fjk are Fourier coefficients of f(x,y)
- and f(x,y) Sj Sk fjk sin(p jx) sin(p ky)
- This implies PDE can be solved exactly

algebraically, ?jk fjk / (-p2j2 - p2k2)

Solving Poissons Equation with the FFT

- So solution of Poissons equation involves the

following steps - 1) Find the Fourier coefficients fjk of f(x,y) by

performing integral - 2) Form the Fourier coefficients of ? by
- ?jk fjk / (-p2j2 - p2k2)
- 3) Construct the solution by performing sum

?(x,y) - There is another version of this (Discrete

Fourier Transform) which deals with functions

defined at grid points and not directly the

continuous integral - Also the simplest (mathematically) transform uses

exp(-2pijx) not sin(p jx) - Let us first consider 1D discrete version of this

case - PDE case normally deals with discretized

functions as these needed for other parts of

problem

Serial FFT

- Let isqrt(-1) and index matrices and vectors

from 0. - The Discrete Fourier Transform of an m-element

vector v is - Fv
- Where F is the mm matrix defined as
- Fj,k v (jk)
- Where v is
- v e (2pi/m) cos(2p/m)

isin(2p/m) - v is a complex number with whose mth power vm 1

and is therefore called an mth root of unity - E.g., for m 4
- v i, v2 -1, v3 -i, v4

1,

Using the 1D FFT for filtering

- Signal sin(7t) .5 sin(5t) at 128 points
- Noise random number bounded by .75
- Filter by zeroing out FFT components lt .25

Using the 2D FFT for image compression

- Image 200x320 matrix of values
- Compress by keeping largest 2.5 of FFT

components - Similar idea used by jpeg

Related Transforms

- Most applications require multiplication by both

F and inverse(F). - Multiplying by F and inverse(F) are essentially

the same. (inverse(F) is the complex conjugate

of F divided by n.) - For solving the Poisson equation and various

other applications, we use variations on the FFT - The sin transform -- imaginary part of F
- The cos transform -- real part of F
- Algorithms are similar, so we will focus on the

forward FFT.

Serial Algorithm for the FFT

- Compute the FFT of an m-element vector v, Fv
- (Fv)j S F(j,k) v(k)
- S v (jk) v(k)
- S (v j)k v(k)
- V(v j)
- Where V is defined as the polynomial
- V(x) S xk v(k)

m-1 k 0

m-1 k 0

m-1 k 0

m-1 k 0

Divide and Conquer FFT

- V can be evaluated using divide-and-conquer
- V(x) S (x)k v(k)
- v0 x2v2

x4v4 - x(v1 x2v3

x4v5 ) - Veven(x2) xVodd(x2)
- V has degree m-1, so Veven and Vodd are

polynomials of degree m/2-1 - We evaluate these at points (v j)2 for 0ltjltm-1
- But this is really just m/2 different points,

since - (v (jm/2) )2 (v j v m/2) )2 v 2j v m

(v j)2 - So FFT on m points reduced to 2 FFTs on m/2

points - Divide and conquer!

m-1 k 0

Divide-and-Conquer FFT

- FFT(v, v, m)
- if m 1 return v0
- else
- veven FFT(v02m-2, v 2, m/2)
- vodd FFT(v12m-1, v 2, m/2)
- v-vec v0, v1, v (m/2-1)
- return veven (v-vec . vodd),
- veven - (v-vec . vodd)
- The . above is component-wise multiply.
- The , is construction an m-element vector

from 2 m/2 element vectors - This results in an O(m log m) algorithm.

precomputed

An Iterative Algorithm

- The call tree of the dc FFT algorithm is a

complete binary tree of log m levels - An iterative algorithm that uses loops rather

than recursion, goes each level in the tree

starting at the bottom - Algorithm overwrites vi by (Fv)bitreverse(i)
- Practical algorithms combine recursion (for

memory hiearchy) and iteration (to avoid function

call overhead)

FFT(0,1,2,3,,15) FFT(xxxx)

even

odd

FFT(1,3,,15) FFT(xxx1)

FFT(0,2,,14) FFT(xxx0)

FFT(xx10)

FFT(xx01)

FFT(xx11)

FFT(xx00)

FFT(x100)

FFT(x010)

FFT(x110)

FFT(x001)

FFT(x101)

FFT(x011)

FFT(x111)

FFT(x000)

FFT(0) FFT(8) FFT(4) FFT(12) FFT(2) FFT(10)

FFT(6) FFT(14) FFT(1) FFT(9) FFT(5) FFT(13)

FFT(3) FFT(11) FFT(7) FFT(15)

Parallel 1D FFT

- Data dependencies in 1D FFT
- Butterfly pattern
- A PRAM algorithm takes O(log m) time
- each step to right is parallel
- there are log m steps
- What about communication cost?
- See LogP paper for details

Block Layout of 1D FFT

- Using a block layout (m/p contiguous elts per

processor) - No communication in last log m/p steps
- Each step requires fine-grained communication in

first log p steps

Cyclic Layout of 1D FFT

- Cyclic layout (only 1 element per processor,

wrapped) - No communication in first log(m/p) steps
- Communication in last log(p) steps

Parallel Complexity

- m vector size, p number of processors
- f time per flop 1
- a startup for message (in f units)
- b time per word in a message (in f units)
- Time(blockFFT) Time(cyclicFFT)
- 2mlog(m)/p
- log(p) a
- mlog(p)/p b

FFT With Transpose

- If we start with a cyclic layout for first log(p)

steps, there is no communication - Then transpose the vector for last log(m/p) steps
- All communication is in the transpose
- Note This example has log(m/p) log(p)
- If log(m/p) gt log(p) more phases/layouts will be

needed - We will work with this assumption for simplicity

Why is the Communication Step Called a Transpose?

- Analogous to transposing an array
- View as a 2D array of n/p by p
- Note same idea is useful for uniprocessor caches

Complexity of the FFT with Transpose

- If no communication is pipelined (overestimate!)
- Time(transposeFFT)
- 2mlog(m)/p

same as before - (p-1) a

was log(p) a - m(p-1)/p2 b

was m log(p)/p b - If communication is pipelined, so we do not pay

for p-1 messages, the second term becomes simply

a, rather than (p-1)a. - This is close to optimal. See LogP paper for

details. - See also following papers on class resource page
- A. Sahai, Hiding Communication Costs in

Bandwidth Limited FFT - R. Nishtala et al, Optimizing bandwidth limited

problems using one-sided communication

Comment on the 1D Parallel FFT

- The above algorithm leaves data in bit-reversed

order - Some applications can use it this way, like

Poisson - Others require another transpose-like operation
- Other parallel algorithms also exist
- A very different 1D FFT is due to Edelman (see

http//www-math.mit.edu/edelman) - Based on the Fast Multipole algorithm
- Less communication for non-bit-reversed algorithm

Higher Dimension FFTs

- FFTs on 2 or 3 dimensions are define as 1D FFTs

on vectors in all dimensions. - E.g., a 2D FFT does 1D FFTs on all rows and then

all columns - There are 3 obvious possibilities for the 2D FFT
- (1) 2D blocked layout for matrix, using 1D

algorithms for each row and column - (2) Block row layout for matrix, using serial 1D

FFTs on rows, followed by a transpose, then more

serial 1D FFTs - (3) Block row layout for matrix, using serial 1D

FFTs on rows, followed by parallel 1D FFTs on

columns - Option 2 is best, if we overlap communication and

computation - For a 3D FFT the options are similar
- 2 phases done with serial FFTs, followed by a

transpose for 3rd - can overlap communication with 2nd phase in

practice

FFTW Fastest Fourier Transform in the West

- www.fftw.org
- Produces FFT implementation optimized for
- Your version of FFT (complex, real,)
- Your value of n (arbitrary, possibly prime)
- Your architecture
- Close to optimal for serial, can be improved for

parallel - Similar in spirit to PHIPAC/ATLAS/Sparsity
- Won 1999 Wilkinson Prize for Numerical Software
- Widely used for serial FFTs
- Had parallel FFTs in version 2, but no longer

supporting them - Layout constraints from users/apps network

differences are hard to support

Bisection Bandwidth

- FFT requires one (or more) transpose operations
- Ever processor send 1/P of its data to each other

one - Bisection Bandwidth limits this performance
- Bisection bandwidth is the bandwidth across the

narrowest part of the network - Important in global transpose operations,

all-to-all, etc. - Full bisection bandwidth is expensive
- Fraction of machine cost in the network is

increasing - Fat-tree and full crossbar topologies may be too

expensive - Especially on machines with 100K and more

processors - SMP clusters often limit bandwidth at the node

level

Modified LogGP Model

- LogGP no overlap

- LogGP no overlap

P0

g

P1

EEL end to end latency (1/2 roundtrip) g

minimum time between small message sends G

additional gap per byte for larger messages

Historical Perspective

½ round-trip latency

- Potential performance advantage for fine-grained,

one-sided programs - Potential productivity advantage for irregular

applications

General Observations

- The overlap potential is the difference between

the gap and overhead - No potential if CPU is tied up throughout message

send - E.g., no send size DMA
- Grows with message size for machines with DMA

(per byte cost is handled by network) - Because per-Byte cost is handled by NIC
- Grows with amount of network congestion
- Because gap grows as network becomes saturated
- Remote overhead is 0 for machine with RDMA

GASNet Communications System

- GASNet offers put/get communication
- One-sided no remote CPU involvement required in

API (key difference with MPI) - Message contains remote address
- No need to match with a receive
- No implicit ordering required

Compiler-generated code

- Used in language runtimes (UPC, etc.)
- Fine-grained and bulk xfers
- Split-phase communication

Language-specific runtime

GASNet

Network Hardware

Performance of 1-Sided vs 2-sided Communication

GASNet vs MPI

- Comparison on Opteron/InfiniBand GASNets

vapi-conduit and OSU MPI 0.9.5 - Up to large message size (gt 256 Kb), GASNet

provides up to 2.2X improvement in streaming

bandwidth - Half power point (N/2) differs by one order of

magnitude

GASNet Performance for mid-range message sizes

GASNet usually reaches saturation bandwidth

before MPI - fewer costs to amortize Usually

outperform MPI at medium message sizes - often by

a large margin

NAS FT Case Study

- Performance of Exchange (Alltoall) is critical
- Communication to computation ratio increases with

faster, more optimized 1-D FFTs - Determined by available bisection bandwidth
- Between 30-40 of the applications total runtime
- Two ways to reduce Exchange cost
- 1. Use a better network (higher Bisection BW)
- 2. Overlap the all-to-all with communication

(where possible) break up the exchange - Default NAS FT Fortran/MPI relies on 1
- Our approach uses UPC/GASNet and builds on 2
- Started as CS267 project
- 1D partition of 3D grid is a limitation
- At most N processors for N3 grid
- HPC Challenge benchmark has large 1D FFT (can be

viewed as 3D or more with proper roots of unity)

3D FFT Operation with Global Exchange

1D-FFT Columns

Transpose 1D-FFT (Rows)

1D-FFT (Columns)

Cachelines

1D-FFT Rows

Exchange (Alltoall)

send to Thread 0

send to Thread 1

Transpose 1D-FFT

Divide rows among threads

send to Thread 2

Last 1D-FFT (Thread 0s view)

- Single Communication Operation (Global Exchange)

sends THREADS large messages - Separate computation and communication phases

Communication Strategies for 3D FFT

chunk all rows with same destination

- Three approaches
- Chunk
- Wait for 2nd dim FFTs to finish
- Minimize messages
- Slab
- Wait for chunk of rows destined for 1 proc to

finish - Overlap with computation
- Pencil
- Send each row as it completes
- Maximize overlap and
- Match natural layout

pencil 1 row

slab all rows in a single plane with same

destination

Joint work with Chris Bell, Rajesh Nishtala, Dan

Bonachea

Decomposing NAS FT Exchange into Smaller Messages

- Three approaches
- Chunk
- Wait for 2nd dim FFTs to finish
- Slab
- Wait for chunk of rows destined for 1 proc to

finish - Pencil
- Send each row as it completes
- Example Message Size Breakdown for
- Class D (2048 x 1024 x 1024)
- at 256 processors

Overlapping Communication

- Goal make use of all the wires
- Distributed memory machines allow for

asynchronous communication - Berkeley Non-blocking extensions expose GASNets

non-blocking operations - Approach Break all-to-all communication
- Interleave row computations and row

communications since 1D-FFT is independent across

rows - Decomposition can be into slabs (contiguous sets

of rows) or pencils (individual row) - Pencils allow
- Earlier start for communication phase and

improved local cache use - But more smaller messages (same total volume)

NAS FT UPC Non-blocking MFlops

- Berkeley UPC compiler support non-blocking UPC

extensions - Produce 15-45 speedup over best UPC Blocking

version - Non-blocking version requires about 30 extra

lines of UPC code

NAS FT Variants Performance Summary

- Shown are the largest classes/configurations

possible on each test machine - MPI not particularly tuned for many small/medium

size messages in flight (long message matching

queue depths)

Pencil/Slab optimizations UPC vs MPI

- Same data, viewed in the context of what MPI is

able to overlap - For the amount of time that MPI spends in

communication, how much of that time can UPC

effectively overlap with computation - On Infiniband, UPC overlaps almost all the time

the MPI spends in communication - On Elan3, UPC obtains more overlap than MPI as

the problem scales up

Summary of Overlap in FFTs

- One-sided communication has performance

advantages - Better match for most networking hardware
- Most cluster networks have RDMA support
- Machines with global address space support (X1,

Altix) shown elsewhere - Smaller messages may make better use of network
- Spread communication over longer period of time
- Postpone bisection bandwidth pain
- Smaller messages can also prevent cache thrashing

for packing - Avoid packing overheads if natural message size

is reasonable

FFTW

the Fastest Fourier Tranform in the West

C library for real complex FFTs (arbitrary

size/dimensionality)

( parallel versions for threads MPI)

Computational kernels (80 of code)

automatically generated

Self-optimizes for your hardware (picks best

composition of steps) portability performance

FFTW performancepower-of-two sizes, double

precision

833 MHz Alpha EV6

2 GHz PowerPC G5

500 MHz Ultrasparc IIe

2 GHz AMD Opteron

FFTW performancenon-power-of-two sizes, double

precision

unusual non-power-of-two sizes receive as much

optimization as powers of two

833 MHz Alpha EV6

2 GHz AMD Opteron

because we let the code do the optimizing

FFTW performancedouble precision, 2.8GHz Pentium

IV 2-way SIMD (SSE2)

powers of two

exploiting CPU-specific SIMD instructions (rewriti

ng the code) is easy

non-powers-of-two

because we let the code write itself

Why is FFTW fast?three unusual features

FFTW implements many FFT algorithms A planner

picks the best composition by measuring the speed

of different combinations.

The resulting plan is executed with explicit

recursion enhances locality

The base cases of the recursion are

codelets highly-optimized dense

code automatically generated by a special-purpose

compiler

FFTW is easy to use

complex xn plan p p plan_dft_1d(n, x,

x, FORWARD, MEASURE) ... execute(p) / repeat

as needed / ... destroy_plan(p)

Why is FFTW fast?three unusual features

FFTW implements many FFT algorithms A planner

picks the best composition by measuring the speed

of different combinations.

3

The resulting plan is executed with explicit

recursion enhances locality

1

The base cases of the recursion are

codelets highly-optimized dense

code automatically generated by a special-purpose

compiler

2

FFTW Uses Natural Recursion

Size 8 DFT

p 2 (radix 2)

Size 4 DFT

Size 4 DFT

Size 2 DFT

Size 2 DFT

Size 2 DFT

Size 2 DFT

Traditional cache solution Blocking

Size 8 DFT

p 2 (radix 2)

Size 4 DFT

Size 4 DFT

Size 2 DFT

Size 2 DFT

Size 2 DFT

Size 2 DFT

breadth-first, but with blocks of size cache

requires program specialized for cache size

Recursive Divide Conquer is Good

Singleton, 1967

(depth-first traversal)

Size 8 DFT

p 2 (radix 2)

Size 4 DFT

Size 4 DFT

Size 2 DFT

Size 2 DFT

Size 2 DFT

Size 2 DFT

Cache Obliviousness

A cache-oblivious algorithm does not know the

cache size it can be optimal for any machine

for all levels of cache simultaneously

Exist for many other algorithms, too Frigo et

al. 1999

all via the recursive divide conquer approach

Why is FFTW fast?three unusual features

FFTW implements many FFT algorithms A planner

picks the best composition by measuring the speed

of different combinations.

3

The resulting plan is executed with explicit

recursion enhances locality

1

The base cases of the recursion are

codelets highly-optimized dense

code automatically generated by a special-purpose

compiler

2