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    "# 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. "
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   "source": [
    "## Serial Program"
   ]
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      "text/plain": [
       "SOR! (generic function with 1 method)"
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     "output_type": "execute_result"
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    "# 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"
   ]
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    {
     "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))"
   ]
  },
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      ]
     },
     "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": [
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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": []
  }
 ],
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   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
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