Parallel Processing Final Project Parallel FFT using to solve Poisson’s Equation

Parallel ProcessingFinal ProjectParallel FFT using to solve Poisson’s Equation Amir Torjeman Nitay Shiran

Poisson’s Equation The Fourier coefficients for function Φ:

Solving the Equation by DFT Perform 2D DFT on both sides of the equation becomes:

The DFT The problem: huge number of calculations: O(N^2) The solution: FFT: Fast DFT algorithm

FFT: Decimation in time: RADIX2 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:cont. The Butterfly:

2D DFT • 2 dimensional transform: • Transform each row • Replace each row with its transform • Transform each column • Replace each column with its transform

2D DFT example FFT sinc Square cube

Parallel 2D DFT: Step 1: transform rows: Divide rows to num of process Process 0  Process 1  Process 2  Process 3  . . . . . .

Parallel 2D DFT: cont. Step 2: transform columns: Divide columns to num of process Process 0  Process 1  Process 2  Process 3  . . . . . .

Parallel Computing MPI Display MATLAB Syntsize MATLAB Our Work • Syntsize the source function f(x,y) in Matlab, and save in file. • 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.

2D FFT example: Before: After:

THANK YOU! ANY QUESTIONS?

Parallel Processing Final Project Parallel FFT using to solve Poisson’s Equation