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Parallel Spectral Methods: Solving Elliptic Problems with FFTs

Parallel Spectral Methods: Solving Elliptic Problems with FFTs. Horst Simon hdsimon@eecs.berkeley.edu www.cs.berkeley.edu/~demmel/cs267_Spr09. Motifs. The Motifs (formerly “Dwarfs”) from “ The Berkeley View” ( Asanovic et al.) Motifs form key computational patterns.

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Parallel Spectral Methods: Solving Elliptic Problems with FFTs

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  1. Parallel Spectral Methods:Solving Elliptic Problems with FFTs Horst Simon hdsimon@eecs.berkeley.edu www.cs.berkeley.edu/~demmel/cs267_Spr09 CS267 Lecture 20

  2. Motifs The Motifs (formerly “Dwarfs”) from “The Berkeley View” (Asanovic et al.) Motifs form key computational patterns Topic of this lecture CS267 Lecture 21 2

  3. References • Previous CS267 lectures • Lecture by Geoffrey Fox: http://grids.ucs.indiana.edu/ptliupages/presentations/PC2007/cps615fft00.ppt • FFTW project http://www.fftw.org • Spiral project http://www.spiral.net CS267 Lecture 21

  4. Poisson’s equation arises in many models 3D: 2u/x2 + 2u/y2 + 2u/z2 = f(x,y,z) • Electrostatic or Gravitational Potential: Potential(position) • Heat flow: Temperature(position, time) • Diffusion: Concentration(position, time) • Fluid flow: Velocity,Pressure,Density(position,time) • Elasticity: Stress,Strain(position,time) • Variations of Poisson have variable coefficients f represents the sources; also need boundary conditions 2D: 2u/x2 + 2u/y2 = f(x,y) 1D: d2u/dx2 = f(x) CS267 Lecture 21

  5. 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 N*log N (N4/3) N • FFT N*log 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. CS267 Lecture 21

  6. Solving Poisson’s Equation with the FFT • Express any 2D function defined in 0  x,y  1 as a series(x,y) = SjSk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) • Poisson’s equation 2  /x2 + 2  /y2 = f(x,y) becomes SjSk (-p2j2 - p2k2) jk sin(p jx) sin(p ky) = SjSk fjk sin(p jx) sin(p ky) • where fjk are Fourier coefficients of f(x,y) • andf(x,y) = SjSkfjk sin(p jx) sin(p ky) • This implies PDE can be solved exactly algebraically, jk = fjk /(-p2j2 - p2k2) CS267 Lecture 21

  7. Solving Poisson’s Equation with the FFT • So solution of Poisson’s 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 CS267 Lecture 21

  8. Serial FFT • Let i=sqrt(-1) and index matrices and vectors from 0. • The Discrete Fourier Transform of an m-element vector v is: • F*v • Where F is the m*m matrix defined as: • F[j,k] = v(j*k) • Where v is: • v = e (2pi/m) = cos(2p/m) + i*sin(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, CS267 Lecture 21

  9. 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 < .25 CS267 Lecture 21

  10. 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 CS267 Lecture 21

  11. 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. CS267 Lecture 21

  12. Serial Algorithm for the FFT • Compute the FFT of an m-element vector v, F*v (F*v)[j] = S F(j,k) * v(k) = Sv(j*k) * v(k) = S (vj)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 CS267 Lecture 21

  13. Divide and Conquer FFT • V can be evaluated using divide-and-conquer V(x) = S (x)k * v(k) = v[0] + x2*v[2] + x4*v[4] + … + x*(v[1] + x2*v[3] + x4*v[5] + … ) = Veven(x2) + x*Vodd(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 0<=j<=m-1 • But this is really just m/2 different points, since (v(j+m/2) )2 = (vj *v m/2) )2 = v2j *v m = (vj)2 • So FFT on m points reduced to 2 FFTs on m/2 points • Divide and conquer! m-1 k = 0 CS267 Lecture 21

  14. Divide-and-Conquer FFT FFT(v, v, m) if m = 1 return v[0] else veven = FFT(v[0:2:m-2], v 2, m/2) vodd = FFT(v[1:2:m-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 CS267 Lecture 21

  15. An Iterative Algorithm • The call tree of the d&c 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 v[i] by (F*v)[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(0,2,…,14) = FFT(xxx0) FFT(1,3,…,15) = FFT(xxx1) FFT(xx00) FFT(xx10) FFT(xx01) FFT(xx11) FFT(x000) FFT(x100) FFT(x010) FFT(x110) FFT(x001) FFT(x101) FFT(x011) FFT(x111) 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) CS267 Lecture 21

  16. 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 CS267 Lecture 21

  17. 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 CS267 Lecture 21

  18. 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 CS267 Lecture 21

  19. 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) = 2*m*log(m)/p + log(p) * a + m*log(p)/p * b CS267 Lecture 21

  20. 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) > log(p) more phases/layouts will be needed • We will work with this assumption for simplicity CS267 Lecture 21

  21. 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 CS267 Lecture 21

  22. Complexity of the FFT with Transpose • If no communication is pipelined (overestimate!) • Time(transposeFFT) = 2*m*log(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” CS267 Lecture 21

  23. 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 CS267 Lecture 21

  24. 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 CS267 Lecture 21

  25. 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 CS267 Lecture 21

  26. 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 CS267 Lecture 21

  27. LogGP: no overlap P0 osend L orecv P1 Modified LogGP Model • 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 CS267 Lecture 21

  28. Historical Perspective ½ round-trip latency • Potential performance advantage for fine-grained, one-sided programs • Potential productivity advantage for irregular applications CS267 Lecture 21

  29. 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 CS267 Lecture 21

  30. 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 CS267 Lecture 21

  31. Performance of 1-Sided vs 2-sided Communication: GASNet vs MPI • Comparison on Opteron/InfiniBand – GASNet’s vapi-conduit and OSU MPI 0.9.5 • Up to large message size (> 256 Kb), GASNet provides up to 2.2X improvement in streaming bandwidth • Half power point (N/2) differs by one order of magnitude CS267 Lecture 21

  32. (up is good) 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 CS267 Lecture 21

  33. 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 N^3 grid • HPC Challenge benchmark has large 1D FFT (can be viewed as 3D or more with proper roots of unity) CS267 Lecture 21

  34. 3D FFT Operation with Global Exchange • Single Communication Operation (Global Exchange) sends THREADS large messages • Separate computation and communication phases 1D-FFT Columns Transpose + 1D-FFT (Rows) 1D-FFT (Columns) Cachelines 1D-FFT Rows send to Thread 0 Exchange (Alltoall) Transpose + 1D-FFT send to Thread 1 Divide rows among threads send to Thread 2 Last 1D-FFT (Thread 0’s view) CS267 Lecture 21

  35. 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 CS267 Lecture 21

  36. 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 CS267 Lecture 21

  37. Overlapping Communication • Goal: make use of “all the wires” • Distributed memory machines allow for asynchronous communication • Berkeley Non-blocking extensions expose GASNet’s 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) CS267 Lecture 21

  38. 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 CS267 Lecture 21

  39. 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) CS267 Lecture 21

  40. 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 CS267 Lecture 21

  41. 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 CS267 Lecture 21

  42. free software:http://www.fftw.org/ the “Fastest Fourier Tranform in the West” FFTW • 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 CS267 Lecture 21

  43. FFTW performancepower-of-two sizes, double precision 833 MHz Alpha EV6 2 GHz PowerPC G5 500 MHz Ultrasparc IIe 2 GHz AMD Opteron CS267 Lecture 21

  44. 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 CS267 Lecture 21

  45. FFTW performancedouble precision, 2.8GHz Pentium IV:2-way SIMD (SSE2) powers of two exploiting CPU-specific SIMD instructions (rewriting the code) is easy non-powers-of-two …because we let the code write itself CS267 Lecture 21

  46. 3 1 2 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” CS267 Lecture 21

  47. Key fact: usually, many transforms of same size are required. FFTW is easy to use { complex x[n]; plan p; p = plan_dft_1d(n, x, x, FORWARD, MEASURE); ... execute(p); /* repeat as needed */ ... destroy_plan(p); } CS267 Lecture 21

  48. 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 CS267 Lecture 21

  49. But traditional implementation is non-recursive, breadth-first traversal: log2n passes over whole array 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 CS267 Lecture 21

  50. 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 CS267 Lecture 21

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