Implementation of Fast Fourier Transform for Three Dimensional Cubic Matrices

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Implementation of Fast Fourier Transform for
Three DimensionalCubic Matrices
Alex C. Szatmary Department of Mechanical
Engineering University of Maryland, Baltimore
County al1_at_umbc.edu December 18, 2006
Sponsored by NSF GRFP
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Purpose
Implement three dimensional fast Fourier
transform on Beowulf cluster Numerical PDEs
Alex C. Szatmary
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Introduction to FFT

• Fourier transform (FT) represent continuous
  functions as infinite summation of sinusoidal
  basis functions
• Discrete Fourier transform (DFT) represent
  discrete series of data as finite summation of
  basis functions
• Fast Fourier transform O(n log(n))
  implementation of DFT (Cooley-Tukey, etc.)
• Simplest implementation for n2m elements

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Introduction to FFT
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Terminology
n2m N n3 elements total A(i,j,k) First index
changes first (Fortran column-oriented)
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3DFFT Algorithm
Scale (cheap) do k1,n do j1,n FFT
in i-direction on A(,j,k) do k1,n do i1,n
FFT in j-direction on A(i,,k) do j1,n
do i1,n FFT in k-direction on A(i,j,)
O(n3 log(n))
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Parallelization

• Do 2 loops in parallel, 1 in serial
• Cost
• Floating point operations
• Transfer data between memory and cache
• Network Communications
• Array copy operations

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Machine Specifications

• Kali Beowulf Cluster, Math Department at UMBC
• 30 nodes
• 2 Intel Xeon 2.0 GHz processors per node
• 1 GB RAM each
• Fortran 90
• MPI
• http//www.math.umbc.edu/gobbert/kali/

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Optimal split of array

• Why split in k-direction?
• Natural
• Maintain element contiguity
• Use MPI_Scatter/MPI_Gather

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Tried k-direction split
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Serial Timing Results by Part
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Why did k-direction split fail?

• Same flops needed as i- or j-direction split
• Same amount of information to be communicated
  across network through MPI
• Fewer array copy operations
• Higher cost of memory access for FFT in j and k
  directions!

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Disadvantages of i-direction split

• Disrupt element contiguity
• Need to make copies of each block before sending
  with MPI
• Need MPI_Send, MPI_Recv for each process

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i-direction Split Results
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i-direction Split Results
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Conclusion

• A parallel 3DFFT algorithm was successfully
  implemented
• Speedup of up to 12
• Split in i-direction
• Nonsequential memory access during computation is
  expensive

Alex C. Szatmary