Parallel ProcessingParallel ProcessingFinal ProjectFinal Project
Parallel FFT using to solve Parallel FFT using to solve Poisson’s EquationPoisson’s Equation
Parallel ProcessingParallel ProcessingFinal ProjectFinal Project
Parallel FFT using to solve Parallel FFT using to solve Poisson’s EquationPoisson’s Equation
Amir TorjemanAmir Torjeman
Nitay ShiranNitay Shiran
Poisson’s EquationPoisson’s Equation
j k
kj ikyijxyx 2exp2exp, ,
The Fourier coefficients for function Φ:
Solving the Equation by DFTSolving the Equation by DFT
Perform 2D DFT on both sides of the equation becomes:
2222
,,
,
,2222
2exp2exp
2exp2exp
kj
f
ikyijxf
ikyijxkj
kjkj
j kkj
j kkj
The DFTThe DFTThe DFTThe DFT
fWg kl
1
0
2expN
k N
klikflg
The problem: huge number of The problem: huge number of calculations: O(Ncalculations: O(N^2)^2)
The solution: FFT: Fast DFT algorithmThe solution: FFT: Fast DFT algorithm
FFT: Decimation in time: FFT: Decimation in time: RADIX2RADIX2
Assume: N=2^d
Use: Recursive formula:
1- divide series into 2 series: fodd,feven
2- perform FFT to each serie.(recursive part)
3- F= Feven+Fodd*exp(-2πi k/N) *(-1)^kd-1
FFT:FFT: cont.cont.
The Butterfly:The Butterfly:
2D DFT2D DFT2D DFT2D DFT
2 dimensional transform:2 dimensional transform:
1) Transform each row
2) Replace each row with its transform
3) Transform each column
4) Replace each column with its transform
2D DFT example2D DFT example
FFT
Square cube sinc
Parallel 2D DFTParallel 2D DFT::
Process 0
Process 1
Process 2
Process 3
.
.
.
.
.
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Step 1: transform rows:
Divide rows to num
of process
Parallel 2D DFT: contParallel 2D DFT: cont..
Pro
cess
0
Pro
cess
1
Pro
cess
2
Pro
cess
3
. . . . . .
Step 2: transform columns:
Divide columns to num
of process
Our WorkOur WorkOur WorkOur Work
1) Syntsize the source function f(x,y) in Matlab, and save in file.
2) Perform 2D parallel FFT in MPI on the source file.
Find the solution to Poisson’s equation.
save solution in file.
3) Load file in MATLAB and display solution.
Syntsize MATLAB
Parallel ComputingMPI
DisplayMATLAB
2D FFT example2D FFT example::Before:After:
THANK YOU!THANK YOU!
ANY ANY QUESTIONSQUESTIONS
??
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