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XM_40017/notebooks/jacobi_2D.ipynb
Gelieza K d13ea2408a Proof read Jacobi method
2023-08-14 13:04:06 +02:00

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{
"cells": [
{
"cell_type": "markdown",
"id": "25dd0180",
"metadata": {},
"source": [
"# Successive Over-relaxation (SOR)\n",
"\n",
"In this section, we want to examine another parallel algorithm called Successive Over-relaxation (SOR). \n",
"\n",
"The SOR algorithm is an iterative method used to solve Laplace equations. The underlying data structure of the SOR algorithm is a two-dimensional grid, whose elements are updated iteratively through some weighted function that considers the old value as well as the values of neighbouring cells. \n",
"\n",
"![Grid](images/SOR_grid_painted.png)\n",
"\n",
"This algorithm is applied, for instance, in physics to simulate the climate or temperature of some object.\n",
"\n",
"\n",
"Climate Simulation | Temperature of a metal plate\n",
":------:|:----:\n",
"![](images/SOR_climate_simulation.png) | ![](images/SOR_temperature_metal_plate.png)\n",
"\n",
"\n",
"## Sequential algorithm\n",
"\n",
"The sequential SOR algorithm is as follows (where `f` is some update function and `N`,`M` are the grid sizes):\n",
"\n",
"```julia\n",
"grid = zeros(N, M)\n",
"grid_new = zeros(N, M)\n",
"for step in 1:NSTEPS\n",
" for i in 2:N #update grid\n",
" for j in 2:M\n",
" grid_new[i,j] = f(grid[i,j], grid[i-1,j], grid[i+1,j], grid[i,j-1], grid[i,j+1])\n",
" end\n",
" end\n",
"end\n",
"```\n",
"\n",
"## Diffusion equation on grid \n",
"\n",
"Consider the diffusion of a chemical substance on a two-dimensional grid. The concentration of the chemical is given as $c(x,y)$, a function of the coordinates $x$ and $y$. We will consider a square grid with $0 \\leq x,y \\leq 1$ and the boundary conditions $c(x,y=1) = 1$ and $c(x,y=0) = 0$. That is, the concentration at the top of the grid is always 1 and the concentration at the very bottom is always 0. Furthermore, in the x-direction we will assume periodic boundary conditions, i.e. $c(x=0, y) = c(x=1,y)$. \n",
"\n",
"![diffusion](images/SOR_time_dep_diffusion.png)\n",
"\n",
"We will take the initial condition $c(x,y) = 0$ for $0 \\leq x \\leq 1, 0 \\leq y < 1$.\n",
"\n",
"The stable state of the diffusion, that is, when the concentraion does not change anymore, can be described by the Laplace equation\n",
"\n",
"$$\n",
"\\nabla^2 c = 0.\n",
"$$\n",
"\n",
"Numerically, we can approximate the solution with the Jacobi iteration \n",
"\n",
"$$\n",
"\\frac{1}{4}(c^k_{i,j+1} + c^k_{i,j-1} + c^k_{i+1,j} + c^k_{i-1,j}) = c^{k+1}_{i,j}.\n",
"$$\n",
"\n",
"The superscript $k$ denotes the $k$-th iteration. The algorithm stepwise updates the cells of the grid, until a steady state is reached. To determine the end of the algorithm, we use the stopping condition \n",
"\n",
"$$\n",
"\\max_{i,j} \\lvert c^{k+1}_{i,j} - c^{k}_{i,j} \\rvert < \\epsilon.\n",
"$$\n",
"\n",
"That is, we stop when all changes to cell values are smaller than some small number, say $\\epsilon = 10^{-5}$.\n",
"\n",
"Furthermore, for this set of initial and boundary conditions, there exists an analytical solution for the stable state, namely \n",
"$$\n",
"c(x,y) = y.\n",
"$$\n",
"That is, the concentration profile is the identity function of the y-coordinate. "
]
},
{
"cell_type": "markdown",
"id": "84167bc6",
"metadata": {},
"source": [
"## Serial Program"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "b082bce8",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"SOR! (generic function with 1 method)"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Update grid until threshold reached\n",
"function update_grid!(new_grid, grid, M, N)\n",
" @assert grid[1,:] == grid[M-1,:]\n",
" @assert grid[M,:] == grid[2,:] \n",
" @assert grid[:,1] == fill(0,M) == new_grid[:,1]\n",
" @assert grid[:,N] == fill(1,M) == new_grid[:,N]\n",
" # Remove ghost cells\n",
" g_left = grid[1:M-2, 2:N-1]\n",
" g_right = grid[3:M, 2:N-1]\n",
" g_up = grid[2:M-1, 3:N]\n",
" g_down = grid[2:M-1, 1:N-2]\n",
" # Jacobi iteration\n",
" new_grid[2:M-1,2:N-1] = 0.25 * (g_up + g_down + g_left + g_right)\n",
" # Update ghost cells\n",
" new_grid[1,:] = new_grid[M-1,:]\n",
" new_grid[M,:] = new_grid[2,:]\n",
" return new_grid\n",
"end\n",
"\n",
"function SOR!(new_grid, grid, ϵ, M, N)\n",
" grid_old = true\n",
" while true\n",
" if grid_old\n",
" update_grid!(new_grid, grid, M, N)\n",
" else \n",
" update_grid!(grid, new_grid, M, N)\n",
" end\n",
" # Flip boolean value\n",
" grid_old = !grid_old\n",
" diffs = abs.(new_grid[2:M-1, :] - grid[2:M-1, :])\n",
" if maximum(diffs) < ϵ\n",
" break\n",
" end\n",
" end \n",
"end\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "e280865a",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Test, Plots\n",
"N = 100\n",
"M = N + 2\n",
"ϵ = 10^-5\n",
"\n",
"# initialize grid\n",
"grid = zeros(M,N)\n",
"grid[:,N] .= 1\n",
"new_grid = zeros(M,N)\n",
"new_grid[:,N] .= 1\n",
"\n",
"update_grid!(new_grid, grid, M, N)\n",
"\n",
"# Test if first iteration successful\n",
"@test all(new_grid[:,N-1] .≈ fill(0.25, M))\n",
"@test all(new_grid[:,N-2] .≈ fill(0,M))\n",
"@test all(new_grid[:,N] == fill(1,M))\n",
"@test all(new_grid[:,1] == fill(0,M))"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "f2f49b93",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
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nRu8DDY5YKxcFLaoUBU0GuZTzTgEAuuH2CQCAdmT1Izy9T1UhEAIANCPXvgvq9Q6EstlOLrXTtw8AAM1YMSLUe46Q+TkAgE0jNQoA0I4Vq0ZtPDUKAKhTJIOQVJ4XJut8vpjec4QmO7mUYAwA0A1BCACgGVk2yCpHeJXey3s/EAgBANqphatG9Q6EZiOpUQCoOyRR6+YI2T4BALBpjMYAAJqRZaMsqRtiyXpPEhIIAQDasWL7hI3PEUomO4kj1gAA+mFECADQjmxQv/jFtjfUAwDqElkSsurUKKtGAQDQj95HrJmNsolRKQDUFbJRqFw1qvs+QoIQAEA7kkFtalTtkWyaIxACALRjxWIZGz9rVDbbSRyxBgDQD0EIAKAZWTaoPllG7Zyi1giEAADtWHMx711+n5aW9sEHH9y4cWPw4MExMTHlfvvbb799+eWXZWtGjx7duHHjqj+f7RMAgJorJyfn4YcfNpvNHTp0GD9+/KZNm8o1KCkpyf2v48ePv/LKK87OzqoewRFrAADNyLLGq0ZXr1794IMPLlmyRAjh4ODw5ptvDhs2rGyDsLCwhQsXKuXnnntu0KBBPj4+qjrAiBAAoBlZNqp9Vb7K9ODBg71791bKvXv3PnLkSHFxcYUtTSZTcnLyhAkT1PaZQAgA0I7032nCqr8qDYQZGRne3t5K2cfHR5blzMzMClt+/fXXdnZ2ffv2Vdtl3bdPGCVOlgEA23bw4MHo6OiyNQMGDJgxY4YQwsnJqaSkRKlUCneaAly1atUTTzxhZ6d6uo0gBADQjGzFyTKSoUWLFrNmzSpb2aJFC6UQGBh46dIlpZyenu7o6FjhFGBmZubOnTuXLVtmRZ8JhAAADRmFrHbSzeDr6xsVFVXh74YMGTJnzpzExERnZ+d169YNGjTIaDQKIfbv3x8QENC8eXOlWVJSUteuXUNCQqzqMQAANdXgwYObNGnSuXPnwYMHr1q1at68eUr9c889t3XrVkuzTz75xIplMgq9t0+Y7cwmSd8+AAA0Y1VqtJJBmb29/Y4dOw4fPpyXl/fJJ5+4u7sr9evXr69fv75SNpvN//rXv1Rtov8/j7DubQAA3M6ai3nvdki30Wjs0qVLucqgoCBL2c7OzpIjtQKBEACgodp3+wRzhAAAm6b7HCH7CAGg7pAlo+rbJ1SvMtUYQQgAoBlZNqi9cV4mNQoAgI50T43amU36dgEAoBkrTpZRvbhGa6RGAQDakQ1C7Y3zagOn1giEAADtWDFHKHQOhMwRAgBsWk24hknfLgAANGPd7RP3qDNVRGoUAKAZWahOjeqO1CgAwKbpvX1CMppNtezfDgCAO5Elg+qTZdSuMtUaqVEAgGasOVlG71WjBEIAgGas2VCv96W0zBECAGyaziNCs9nObNa3CwAADZEaBQDYMsmg+sg0vfcRkhoFANg0RoQAAM3IskHtRbu6b8CvAfsIzewjBIA6gu0TAACbJstsnwAAoFbRffuE0WQiGANAHUFqFABg02RZSGoDoY0vlgEAoHLnzp378MMPc3Nz4+LiBg4cWGGb/fv3f/HFFzdv3uzYseOTTz6p6vNJSwIANCPLRllS+ap0RJidnd2lSxcHB4cePXpMnjz5iy++uL3NypUrhw0bFhgY2KtXr7Nnz6rtc03YPmGnbx8AAFqxZo6w0vZJSUkRERELFiwQQhgMhjfffHP48OFlG+Tl5c2YMWPXrl0PP/ywEGL8+PFq+1zdEeFnn33WoUMHb2/vzp07nz9/Xqn897//3b59ez8/v1GjRuXm5lbzEQAAm3Xo0KFevXop5V69ev3888/FxcXlGjRs2NDe3v7VV19999138/Pz1T6iWoFw7dq1L7744uLFi3/99de3337b1dVVCHH9+vWEhIR58+adOXPGbDb//e9/r84jAAC1iDIiVPUScmUfmJGR4e3trZR9fHxkWc7MzCzbIC0tLT8//9lnn/X19f3Pf/7TqVOnW7duqepztVKjc+fOXbx4cVRUlBDC0tHk5OROnToNGTJECDF//vz27du/8847bm5uFX6CJNmRGgWAOkOWhNoN9bJs2LNnT8eOHctWjhgx4vnnnxdCODs7l5SUKJXKWNDZ2blsS0dHx+vXr584ccLf33/q1KlhYWGbNm0aN25c1TtgfSAsKCg4d+5cUVFRr169SkpKxo4dO23aNIPB8Msvv7Rv315p06pVK4PBkJaWFh4ebvWDAAC1hlVzhO3bt1+yZEnZykaNGimFwMDA9PR0pZyenu7k5OTj41O2ZWBgoIuLi7+/vxDCYDAEBwdnZGSo6oD1gfDKlStCiKSkpBUrVmRnZyckJLi5uY0bNy4nJ6dhw4aWZu7u7tevX6/wE7KysvLy8kSDBlb3AQBw35SWljo4ONyLT/bw8IiIiKjwV3FxcbNnz05MTHzggQeSk5MHDx5sNBqFEN9//33jxo1DQkJ69uzp4uJy5MiRTp063bhx48iRI9OnT1f1dOvnCD09PYUQL730UsuWLSMjI6dOnbpp0yYhhJeX1x9//GFplp+f3+AOoc7Pz8/Dw8PqDgAA7qeqREFZqJ8jrFRsbGxISEinTp369+//6aefzps3T6mfPXv2tm3bhBDOzs7Lli0bOHDg8OHD27dvP2DAgJiYGFV/LutHhD4+Pp6enpZcrbOzc2lpqRAiJCRk9+7dSmVqaqrZbG7atOmdPsQsG00Sc4QAUEfIslH9NUyVtbe3t9+2bdvRo0fz8vIiIyPr1aun1G/cuNGy+mTMmDGPPPLIyZMnX3vttZYtW6rts/UjQqPROGHChHfffbekpOT69eurV6/u16+f0qH9+/fv37+/tLT0tddei4uLc3d3t/opAAAbZzAYOnXqFB0dbYmCQojGjRuXzSk2atQoJibGiigoqrl94tVXX3VycvL19W3Xrl1sbOy0adOEEP7+/klJSWPGjPH09Lx8+fLy5cur8wgAQC0iSQa1L9XXNmmtWtsnXF1dk5OTb69PSEhISEiozicDAGolWfUh2pVuI7wfdL+GyWBSmU0GANRYymIZdW/R+/YJghAAwKZxDRMAQDOaH7p9H9SA2ydktk8AQB0hyQbVF/Peo65UGalRAIBNIzUKANAMqVEAgG1Tv31C2HggNMtGE3OEAFBXWHPEmmD7BAAA+iE1CgDQjCQbJLUX89bqI9aqzywZTYxKAaCusGaxzD3qSpURhAAANo3UKABAO2yfAADYMkkWqk+WsfFAyPYJAKhLrLl94h51pcqYIwQA2DRSowAAzcjCoH6DvG2nRgEAdYms/vYJte01p/c1TLLBJJGeBQDohhEhAEAzVmyot/VDtwEAdYkV2yckvZeN6r19QjKaSY0CQF1h1RFr3D4BAIB+SI0CADRjxarRu44gz549+9577+Xl5cXFxcXFxd32dvnFF1+0/BgZGTl48GBVHWBECADQjHKyjMpXZR947dq1rl27enh4PProo9OnT1+3bt3tbRYtWuTs7Ozp6enp6eni4qK2z2yfAADUXElJSZ06dfrHP/4hhJAkafHixaNGjbq92TPPPOPp6WndI0iNAgA0Y832iUoXyxw+fLhXr15KuWfPnuPHjy8qKnJ2di7X7I033nBwcOjevXu/fv1UPp3UKABAO7IwSOpflXxgZmZmgwYNlLK3t7csy5mZmeXajBo1KjAw0NnZedKkSTNnzlTbZ/23T5AaBYA6Q5bVX6skiz179nTs2LFsXXx8/OzZs4UQzs7OJSUlSmVxcbEQotwsoMFgWLt2rVIePHhwhw4dXn75ZS8vr6o/n9QoAEBn7du3X7JkSdmagIAApRAYGHjx4kWlfPHiRWdnZ29v7zt9Ttu2be3s7DIyMgiEAAB9SFYduu3h4REREVHhb4cNG/b8888nJia6uLh8+umncXFxRqNRCLFr166mTZu2bNkyNzfXzc3N3t5eCJGcnOzq6hocHKyqAwRCAIBmZFlUvh1CrYEDByYlJT300EOBgYFnzpzZvXu3Uj9nzpxRo0a1bNly+/btzz33XGhoaEFBQWpq6urVq29fSlM5vecIZYNJZo4QAFAxOzu7LVu2HD9+PD8//y9/+csDDzyg1G/ZssXV1VUIMWbMmK5du6amptarV69Nmzb16tVT+whGhAAAzcjintxH2L59+3I1/v7+lnJQUFBQUJCqh5ZFIAQAaMaqG+p1RloSAGDTdJ8jZB8hANQd1q0avUedqSJSowAAzVixoV73VCqBEACgGfluR6ZV9Bad6Z4aZfsEAEBPjAgBAJqxYkO9thvwrUAgBABoxorFMuqvbdIYaUkAgE2rAXOEUi3begkAuBNZqL6Yl1WjAIC6w6pVowRCAEBdIcmqN8jb+mIZs2ww6T1NCgCwZYwIAQCaYfsEAMCm1cY5QrZPAABsWg2YI+T2CQCoK2SZ7RMAABsmCSGpf4u+CIQAAO2oHxHqfv0EaUkAgE2rAXOE7CMEgLqCG+oBADbNijlCvTOjpEYBALZN5xGhJAtunwCAOsOa2ydIjQIA6gyZ1CgAwJbJ8p976lW87vaZp0+fnjZt2ujRo7/44otKmp05c2b27NnHjh1T22cCIQCg5rp69Wr37t0bNWo0ZMiQmTNnfvbZZxU2M5vNEydOXLFixalTp9Q+gu0TAADNWLF9ovI5wlWrVkVGRiYmJgohTCbT4sWLx44de3uzZcuWdenSpbCwUNWjFRqMCNPT0x9++OF58+ZZar766quWLVu6u7sPHjz4+vXr1X8EAKC2kNW/KvHjjz/27NlTKffo0ePEiRNFRUXl2qSmpq5evXru3LnWdViDQDh16lRJktLS0pQfs7Kyxo4d+95772VmZrq7u8+YMaP6jwAA2KaMjAwvLy+l7O3tLctyZmZm2QaSJD3xxBPLli1zdXW17hHVTY2uWbPG19c3PDzc0rPk5OTIyMi+ffsKIV555ZW2bdu+//779evXr/DtZsnA9gkAqDOsS43u2bOnY8eOZSuHDRv24osvCiFcXFyKi4uVSmU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"</svg>\n"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"N = 100\n",
"M = N + 2\n",
"ϵ = 10^-5\n",
"\n",
"# initialize grid\n",
"grid = zeros(M,N)\n",
"grid[:,N] .= 1\n",
"new_grid = zeros(M,N)\n",
"new_grid[:,N] .= 1\n",
"\n",
"SOR!(new_grid, grid, ϵ, M, N)\n",
"plt = heatmap(transpose(new_grid[2:M-1,:]))\n",
"display(plt)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "4aefd11d",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# analytical solution\n",
"function analytical_solution(N)\n",
" # Returns the analytical solution as a square grid of size N\n",
" grid = zeros(N,N)\n",
" for i in 2:N\n",
" grid[:,i] .= (i-1)/(N-1)\n",
" end\n",
" return grid\n",
"end\n",
"\n",
"# Test if solution is identical with analytical solution\n",
"sol = analytical_solution(N)\n",
"@test maximum(abs.(sol - new_grid[2:M-1,:])) < 0.01 * N"
]
},
{
"cell_type": "markdown",
"id": "f9f090f1",
"metadata": {},
"source": [
"## Parallel Program with MPI"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "49c4e9b4",
"metadata": {},
"outputs": [],
"source": [
"#] add MPIClusterManagers DelimitedFiles"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "6ec8e2eb",
"metadata": {
"code_folding": []
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" From worker 5:\tProc 4 gets rows 17:33 and columns 2:17\n",
" From worker 8:\tProc 7 gets rows 34:50 and columns 2:17\n",
" From worker 3:\tProc 2 gets rows 1:16 and columns 18:34\n",
" From worker 6:\tProc 5 gets rows 17:33 and columns 18:34\n",
" From worker 9:\tProc 8 gets rows 34:50 and columns 18:34\n",
" From worker 4:\tProc 3 gets rows 1:16 and columns 35:51\n",
" From worker 10:\tProc 9 gets rows 34:50 and columns 35:51\n",
" From worker 7:\tProc 6 gets rows 17:33 and columns 35:51\n",
" From worker 2:\tProc 1 gets rows 1:16 and columns 2:17\n",
" From worker 2:\tTHRESHOLD SUBCEEDED - TERMINATE SOR\n"
]
}
],
"source": [
"# to import MPIManager\n",
"using MPIClusterManagers\n",
"\n",
"# need to also import Distributed to use addprocs()\n",
"using Distributed\n",
"\n",
"# specify, number of mpi workers, launch cmd, etc.\n",
"manager=MPIWorkerManager(9)\n",
"\n",
"# start mpi workers and add them as julia workers too.\n",
"addprocs(manager)\n",
"\n",
"@mpi_do manager begin\n",
" \n",
"function calculate_partition(p, N, nrows, ncols)\n",
" # Calculates the row and column indices of this processor\n",
" # Get row and column number for processor p \n",
" if mod(p,ncols) == 0\n",
" i = div(p,ncols)\n",
" else\n",
" i = floor(div(p,ncols)) + 1\n",
" end\n",
" j = p - (i-1)*ncols\n",
" # Rows\n",
" if mod(N,nrows) == 0\n",
" prows = div(N, nrows)\n",
" row_range =((i-1) * prows + 1) : (i*prows)\n",
" else\n",
" # nlower processors get the smaller partition\n",
" nlower = nrows - (mod(N,nrows)) \n",
" n_floor = floor(div(N,nrows))\n",
" if i <= nlower\n",
" row_range =((i-1) * n_floor + 1) : (i*n_floor)\n",
" else\n",
" row_range = ((i-1) * n_floor + (i - nlower)) : (i*n_floor + (i-nlower))\n",
" end\n",
" end\n",
" # Columns\n",
" if mod(N,ncols) == 0\n",
" prows = div(N, ncols)\n",
" col_range =((j-1) * prows + 1) : (j*prows)\n",
" else\n",
" nlower = ncols - (mod(N,ncols)) \n",
" n_floor = floor(div(N,ncols))\n",
" if j <= nlower\n",
" col_range =((j-1) * n_floor + 1) : (j*n_floor)\n",
" else\n",
" col_range = ((j-1) * n_floor + (j - nlower)) : (j*n_floor + (j-nlower))\n",
" end\n",
" end\n",
" # Add 1 to each column index because of ghost cells\n",
" col_range = col_range .+ 1\n",
" return row_range, col_range\n",
"end\n",
"\n",
" \n",
"function update_grid(grid)\n",
" # Returns the updated grid as M-2 x N-2 matrix where M and N are sizes of grid\n",
" M = size(grid, 1)\n",
" N = size(grid, 2) \n",
" # Remove ghost cells\n",
" g_left = grid[2:M - 1,1:N-2]\n",
" g_right = grid[2:M - 1, 3:N]\n",
" g_up = grid[1:M-2, 2:N-1]\n",
" g_down = grid[3:M, 2:N-1]\n",
" # Jacobi iteration\n",
" return 0.25 * (g_up + g_down + g_left + g_right)\n",
"end\n",
"\n",
" using MPI\n",
" comm=MPI.COMM_WORLD\n",
" id = MPI.Comm_rank(comm) + 1\n",
" \n",
"\n",
" M = 50\n",
" N = M + 2\n",
" ϵ = 10^-5 # Stopping threshold\n",
" nrows = 3 # Number of grid rows\n",
" ncols = 3 # Number of grid columns\n",
" n_procs = nrows * ncols\n",
" @assert n_procs == MPI.Comm_size(comm)\n",
" max_diffs = ones(n_procs) # Differences between iterations\n",
" max_diff_buf = MPI.UBuffer(max_diffs,1) # Buffer to store maximum differences\n",
" \n",
"\n",
" # initialize grid\n",
" if id == 1\n",
" grid_a = zeros(M,N)\n",
" grid_a[1,:] .= 1\n",
" grid_b = zeros(M,N)\n",
" grid_b[1,:] .= 1\n",
" else\n",
" grid_a = nothing\n",
" grid_b = nothing\n",
" end\n",
" \n",
" # Broadcast matrix to other processors\n",
" grid_a = MPI.bcast(grid_a, 0, comm)\n",
" grid_b = MPI.bcast(grid_b, 0, comm)\n",
"\n",
" # Determine if processor is in top or bottom row of grid\n",
" top_pos = id <= ncols\n",
" bottom_pos = id > ((nrows-1) * ncols)\n",
" local grid_a_old = false # Grid a is the source grid for the first update\n",
"\n",
" # Get local partition\n",
" ind_rows, ind_cols = calculate_partition(id, M, nrows, ncols)\n",
" println(\"Proc $(id) gets rows $(ind_rows) and columns $(ind_cols)\")\n",
"\n",
" # Determine neighbors\n",
" n_left = id - 1\n",
" n_right = id + 1\n",
" n_down = id + ncols\n",
" n_up = id - ncols\n",
" if mod(id, ncols) == 1\n",
" # Left neighbor is last in row\n",
" n_left = id + ncols - 1\n",
" end \n",
" if mod(id, ncols) == 0\n",
" # Right neighbor is first in row\n",
" n_right = id - ncols + 1\n",
" end\n",
"\n",
" #println(\"Proc $(id) has neighbors left $(n_left) and right $(n_right) and up $(n_up) and down $(n_down)\")\n",
"\n",
" local finished = false\n",
"\n",
" #Perform SOR\n",
" while !finished\n",
" # Flip old and new grid\n",
" grid_a_old = !grid_a_old \n",
" \n",
" # Determine which grid is updated\n",
" if grid_a_old \n",
" old_grid = grid_a\n",
" new_grid = grid_b\n",
" else\n",
" old_grid = grid_b\n",
" new_grid = grid_a\n",
" end \n",
" \n",
" # send left and right columns\n",
" left_ind = first(ind_cols)\n",
" right_ind = last(ind_cols)\n",
" left_col = old_grid[ind_rows, left_ind]\n",
" right_col = old_grid[ind_rows, right_ind]\n",
" slreq = MPI.Isend(left_col, comm; dest=n_left-1)\n",
" srreq = MPI.Isend(right_col, comm; dest=n_right-1)\n",
" \n",
" # Send bottom row if not bottom\n",
" bottom_ind = last(ind_rows)\n",
" if !bottom_pos\n",
" bottom_col = old_grid[bottom_ind, ind_cols]\n",
" sbreq = MPI.Isend(bottom_col, comm; dest=n_down -1)\n",
" end\n",
"\n",
" # Send top row if not at the top \n",
" top_ind = first(ind_rows)\n",
" if !top_pos\n",
" top_row = old_grid[top_ind, ind_cols]\n",
" streq = MPI.Isend(top_row, comm; dest = n_up -1)\n",
" end\n",
"\n",
" # Receive left and right column\n",
" left_buf = Array{Float64,1}(undef, length(ind_rows))\n",
" right_buf = Array{Float64,1}(undef, length(ind_rows))\n",
" rlreq = MPI.Irecv!(left_buf, comm; source=n_left -1)\n",
" rrreq = MPI.Irecv!(right_buf, comm; source=n_right -1)\n",
"\n",
" # Receive top row if not at the top\n",
" if !top_pos\n",
" top_buf = Array{Float64,1}(undef, length(ind_cols))\n",
" rtreq = MPI.Irecv!(top_buf, comm; source=n_up -1)\n",
" end\n",
"\n",
" # Receive bottom row if not at the bottom \n",
" if !bottom_pos\n",
" bottom_buf = Array{Float64,1}(undef, length(ind_cols))\n",
" rbreq = MPI.Irecv!(bottom_buf, comm; source=n_down -1)\n",
" end\n",
"\n",
" # Wait for results\n",
" statlr = MPI.Waitall([rlreq, rrreq], MPI.Status)\n",
" old_grid[ind_rows, left_ind - 1] = left_buf\n",
" old_grid[ind_rows, right_ind + 1] = right_buf\n",
" #println(\"Proc $(id) received left $(old_grid[ind_rows, left_ind - 1]) and right $(old_grid[ind_rows, right_ind + 1])\")\n",
" \n",
" if !top_pos\n",
" statt = MPI.Wait(rtreq)\n",
" old_grid[top_ind - 1, ind_cols] = top_buf\n",
" #println(\"Proc $(id) received top $(old_grid[top_ind - 1, ind_cols])\")\n",
" end\n",
"\n",
" if !bottom_pos\n",
" statb = MPI.Wait(rbreq)\n",
" old_grid[bottom_ind + 1, ind_cols] = bottom_buf\n",
" #println(\"Proc $(id) received bottom $(old_grid[bottom_ind + 1, ind_cols])\")\n",
" end \n",
"\n",
" # Get local subgrid\n",
" if !top_pos & !bottom_pos\n",
" local_with_ghosts = old_grid[top_ind - 1 : bottom_ind + 1, left_ind - 1 : right_ind + 1]\n",
" lb_row = top_ind\n",
" ub_row = bottom_ind\n",
" elseif top_pos \n",
" local_with_ghosts = old_grid[top_ind : bottom_ind + 1, left_ind - 1 : right_ind + 1]\n",
" lb_row = top_ind + 1\n",
" ub_row = bottom_ind\n",
" elseif bottom_pos\n",
" local_with_ghosts = old_grid[top_ind - 1 : bottom_ind, left_ind - 1 : right_ind + 1]\n",
" lb_row = top_ind\n",
" ub_row = bottom_ind - 1\n",
" end\n",
"\n",
" # Perform one step of Jacobi iteration\n",
" new_grid[lb_row : ub_row, left_ind : right_ind] = update_grid(local_with_ghosts)\n",
"\n",
" # Calculate max difference\n",
" diffs = abs.(new_grid[lb_row : ub_row, left_ind : right_ind] - old_grid[lb_row : ub_row, left_ind : right_ind])\n",
" maxdiff = maximum(diffs)\n",
" \n",
" # Gather maxdiffs in processor 1 \n",
" MPI.Gather!(maxdiff, max_diff_buf, comm; root=0)\n",
"\n",
" # First processor determines if threshold is exeeded\n",
" if id == 1\n",
" if all(max_diffs .< ϵ)\n",
" finished = true\n",
" println(\"THRESHOLD SUBCEEDED - TERMINATE SOR\") \n",
" end\n",
" end\n",
" \n",
" finished = MPI.bcast(finished, 0, comm)\n",
"\n",
" if finished\n",
" # Set ghost cells to zero again so MPI.Reduce gives correct output\n",
" new_grid[ind_rows, left_ind - 1] .= 0.0\n",
" new_grid[ind_rows, right_ind + 1] .= 0.0\n",
" if !bottom_pos\n",
" new_grid[bottom_ind + 1, ind_cols] .= 0.0\n",
" end\n",
" if !top_pos\n",
" new_grid[top_ind - 1, ind_cols] .= 0.0\n",
" end\n",
" end\n",
" end\n",
"\n",
" using DelimitedFiles\n",
"\n",
" # Reduce matrix & store result\n",
" if !grid_a_old\n",
" sor_result = grid_a\n",
" else \n",
" sor_result = grid_b\n",
" end\n",
"\n",
" MPI.Reduce!(sor_result, +, comm, root = 0)\n",
" sor_result[1,:] .= 1.0\n",
" if id == 1\n",
" writedlm(\"SOR_result.txt\", sor_result)\n",
" end \n",
"\n",
" MPI.Finalize()\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "9465e78a",
"metadata": {},
"outputs": [],
"source": [
"rmprocs(manager);"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "d0bbbb61",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
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"</svg>\n"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"using Plots, DelimitedFiles\n",
"final_grid = readdlm(\"SOR_result.txt\")\n",
"M = size(final_grid, 1)\n",
"N = size(final_grid, 2)\n",
"plt = heatmap(final_grid[:, 2:N-1])\n",
"display(plt)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "22b071c2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Test if solution is identical with analytical solution\n",
"sol = analytical_solution(M)\n",
"# Bring solution into correct form \n",
"sol = reverse(transpose(sol), dims = 1)\n",
"@test maximum(abs.(sol - final_grid[:,2:N-1])) < 0.01 * M"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3130c9fb",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.9.1",
"language": "julia",
"name": "julia-1.9"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.9.1"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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