mirror of
https://github.com/fverdugo/XM_40017.git
synced 2025-11-09 00:34:24 +01:00
Proof read Jacobi method
This commit is contained in:
Gelieza K
parent
1c3062f249
commit
d13ea2408a
@ -2110,7 +2110,7 @@
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Julia 1.9.0",
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"display_name": "Julia 1.9.1",
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"language": "julia",
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"name": "julia-1.9"
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},
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@ -2118,7 +2118,7 @@
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"file_extension": ".jl",
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"mimetype": "application/julia",
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"name": "julia",
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"version": "1.9.0"
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"version": "1.9.1"
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}
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},
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"nbformat": 4,
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@ -35,17 +35,17 @@
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"execution_count": 2,
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"id": "1dc78750",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"jacobi_2_check (generic function with 1 method)"
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"jacobi_4_check (generic function with 1 method)"
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]
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},
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"execution_count": 5,
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -62,7 +62,9 @@
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"end\n",
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"gauss_seidel_1_check(answer) = answer_checker(answer,\"c\")\n",
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"jacobi_1_check(answer) = answer_checker(answer, \"d\")\n",
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"jacobi_2_check(answer) = answer_checker(answer, \"b\")"
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"jacobi_2_check(answer) = answer_checker(answer, \"b\")\n",
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"jacobi_3_check(answer) = answer_checker(answer, \"c\")\n",
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"jacobi_4_check(anwswer) = answer_checker(answer, \"d\")"
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]
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},
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{
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@ -73,7 +75,7 @@
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"## The Jacobi method\n",
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"\n",
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"\n",
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"The [Jacobi method](https://en.wikipedia.org/wiki/Jacobi_method) is a numerical tool to solve systems of linear algebraic equations. One of the main applications of the method is to solve boundary value problems (BVPs). I.e., given the values at the boundary (of a grid), the Jacoby method will find the interior values that fulfill a certain equation.\n",
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"The [Jacobi method](https://en.wikipedia.org/wiki/Jacobi_method) is a numerical tool to solve systems of linear algebraic equations. One of the main applications of the method is to solve boundary value problems (BVPs). I.e., given the values at the boundary (of a grid), the Jacobi method will find the interior values that fulfill a certain equation.\n",
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"\n"
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]
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},
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@ -147,7 +149,7 @@
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"metadata": {},
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"source": [
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"\n",
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"### Where do we can exploit parallelism?\n",
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"### Where can we exploit parallelism?\n",
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"\n",
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"Look at the two nested loops in the sequential implementation:\n",
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"\n",
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@ -161,7 +163,7 @@
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"```\n",
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"\n",
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"- The outer loop cannot be parallelized. The value of `u` at step `t+1` depends on the value at the previous step `t`.\n",
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"- The inner loop can be parallelized\n",
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"- The inner loop can be parallelized.\n",
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"\n"
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]
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},
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@ -239,20 +241,12 @@
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"execution_count": null,
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"id": "4edad93f",
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"It's not correct. Keep trying! 💪\n"
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]
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}
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],
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"outputs": [],
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"source": [
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"answer = \"x\" # replace x with a, b, c or d\n",
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"gauss_seidel_1_check(answer)"
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@ -340,18 +334,10 @@
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"execution_count": null,
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"id": "3a03fc4c",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"It's not correct. Keep trying! 💪\n"
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]
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}
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],
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"outputs": [],
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"source": [
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"answer = \"x\" # replace x with a, b, c or d\n",
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"jacobi_1_check(answer)"
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@ -366,7 +352,7 @@
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"\n",
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"We consider the implementation using MPI. The programming model of MPI is generally better suited for data-parallel algorithms like this one than the task-based model provided by Distributed.jl. In any case, one can also implement it using Distributed, but it requires some extra effort to setup remote channels right for the communication between neighbor processes.\n",
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"\n",
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"Take a look at the implementation below and try to understand it. Note that we have used MPIClustermanagers and Distributed just to run the MPI code on the notebook. When running it on a cluster MPIClustermanagers and Distributed are not needed.\n"
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"Take a look at the implementation below and try to understand it. Note that we have used MPIClustermanagers and Distributed just to run the MPI code on the notebook. When running it on a cluster, MPIClustermanagers and Distributed are not needed.\n"
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]
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},
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{
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@ -386,7 +372,7 @@
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"metadata": {},
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"outputs": [],
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"source": [
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"using MPIClusterManagers\n",
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"using MPIClusterManagers \n",
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"using Distributed"
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]
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},
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@ -407,13 +393,26 @@
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "68851107",
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"id": "a0923606",
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"metadata": {},
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"outputs": [],
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"source": [
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"@everywhere workers() begin\n",
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"# Test cell, remove me\n",
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"u = [-1, 0, 0, 0, 0, 1]\n",
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"view(u, 6:6)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "68851107",
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"metadata": {
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"code_folding": []
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},
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"outputs": [],
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"source": [
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"@mpi_do manager begin\n",
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" using MPI\n",
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" MPI.Initialized() || MPI.Init()\n",
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" comm = MPI.Comm_dup(MPI.COMM_WORLD)\n",
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" nw = MPI.Comm_size(comm)\n",
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" iw = MPI.Comm_rank(comm)+1\n",
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@ -429,6 +428,7 @@
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" u_new = copy(u)\n",
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" for t in 1:niters\n",
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" reqs = MPI.Request[]\n",
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" # Exchange cell values with neighbors\n",
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" if iw != 1\n",
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" neig_rank = (iw-1)-1\n",
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" req = MPI.Isend(view(u,2:2),comm,dest=neig_rank,tag=0)\n",
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@ -438,8 +438,8 @@
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" end\n",
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" if iw != nw\n",
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" neig_rank = (iw+1)-1\n",
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" s = n_own-1\n",
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" r = n_own\n",
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" s = n_own+1\n",
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" r = n_own+2\n",
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" req = MPI.Isend(view(u,s:s),comm,dest=neig_rank,tag=0)\n",
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" push!(reqs,req)\n",
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" req = MPI.Irecv!(view(u,r:r),comm,source=neig_rank,tag=0)\n",
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@ -453,6 +453,14 @@
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" end\n",
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" u\n",
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" @show u\n",
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" # Gather results in root process\n",
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" results = zeros(n+2)\n",
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" results[1] = -1\n",
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" results[n+2] = 1\n",
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" MPI.Gather!(view(u,2:n_own+1), view(results, 2:n+1), root=0, comm)\n",
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" if iw == 1\n",
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" @show results\n",
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" end \n",
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" end\n",
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" niters = 100\n",
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" load = 4\n",
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@ -485,8 +493,60 @@
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"outputs": [],
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"source": [
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"answer = \"x\" # replace x with a, b, c or d\n",
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"jacobi_2_check(answer)\n",
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"# TODO: think of more questions"
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"jacobi_2_check(answer)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "075dd6d8",
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"metadata": {},
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"source": [
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"<div class=\"alert alert-block alert-success\">\n",
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"<b>Question:</b> After the end of the for-loop (line 43), ...\n",
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"</div>\n",
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"\n",
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" a) each worker holds the complete solution.\n",
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" b) the root process holds the solution. \n",
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" c) the ghost cells contain redundant values. \n",
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" d) all ghost cells contain the initial values -1 and 1. "
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "c3b58002",
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"metadata": {},
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"outputs": [],
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"source": [
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"answer = \"x\" # replace x with a, b, c or d\n",
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"jacobi_3_check(answer)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "4537661d",
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"metadata": {},
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"source": [
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"<div class=\"alert alert-block alert-success\">\n",
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"<b>Question:</b> In line 35 of the code, we wait for all receive and send requests. Is it possible to instead wait for just the receive requests?\n",
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"</div>\n",
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"\n",
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" \n",
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" a) No, because the send buffer might be overwritten if we don't wait for send requests.\n",
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" b) No, because MPI does not allow an asynchronous send without a Wait().\n",
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" c) Yes, because each send has a matching receive, so all requests are done when the receive requests return. \n",
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" d) Yes, because there are no writes to the send buffer in this iteration."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "e16ea5eb",
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"metadata": {},
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"outputs": [],
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"source": [
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"answer = \"x\" # replace x with a, b, c or d.\n",
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"jacobi_4_check(answer)"
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]
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},
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{
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@ -595,9 +655,8 @@
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"metadata": {},
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"outputs": [],
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"source": [
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"@everywhere workers() begin\n",
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"@mpi_do manager begin\n",
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" using MPI\n",
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" MPI.Initialized() || MPI.Init()\n",
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" comm = MPI.Comm_dup(MPI.COMM_WORLD)\n",
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" nw = MPI.Comm_size(comm)\n",
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" iw = MPI.Comm_rank(comm)+1\n",
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@ -613,6 +672,7 @@
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" u_new = copy(u)\n",
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" for t in 1:niters\n",
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" reqs = MPI.Request[]\n",
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" # Exchange cell values with neighbors\n",
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" if iw != 1\n",
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" neig_rank = (iw-1)-1\n",
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" req = MPI.Isend(view(u,2:2),comm,dest=neig_rank,tag=0)\n",
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@ -622,8 +682,8 @@
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" end\n",
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" if iw != nw\n",
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" neig_rank = (iw+1)-1\n",
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" s = n_own-1\n",
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" r = n_own\n",
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" s = n_own+1\n",
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" r = n_own+2\n",
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" req = MPI.Isend(view(u,s:s),comm,dest=neig_rank,tag=0)\n",
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" push!(reqs,req)\n",
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" req = MPI.Irecv!(view(u,r:r),comm,source=neig_rank,tag=0)\n",
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@ -637,77 +697,14 @@
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" end\n",
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" u\n",
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" @show u\n",
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" end\n",
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" niters = 100\n",
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" load = 4\n",
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" n = load*nw\n",
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" jacobi_mpi(n,niters)\n",
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"end"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "f302cce2",
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"metadata": {},
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"outputs": [],
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"source": [
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"## TODO move the following solution to its appropiate place:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "4fa7fad3",
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"metadata": {},
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"outputs": [],
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"source": [
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"@everywhere workers() begin\n",
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" using MPI\n",
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" MPI.Initialized() || MPI.Init()\n",
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" comm = MPI.Comm_dup(MPI.COMM_WORLD)\n",
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" nw = MPI.Comm_size(comm)\n",
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" iw = MPI.Comm_rank(comm)+1\n",
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" function jacobi_mpi(n,niters)\n",
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" if mod(n,nw) != 0\n",
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" println(\"n must be a multiple of nw\")\n",
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" MPI.Abort(comm,1)\n",
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" end\n",
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" n_own = div(n,nw)\n",
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" u = zeros(n_own+2)\n",
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" u[1] = -1\n",
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" u[end] = 1\n",
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" u_new = copy(u)\n",
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" for t in 1:niters\n",
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" reqs_snd = MPI.Request[]\n",
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" reqs_rcv = MPI.Request[]\n",
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" if iw != 1\n",
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" neig_rank = (iw-1)-1\n",
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" req = MPI.Isend(view(u,2:2),comm,dest=neig_rank,tag=0)\n",
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" push!(reqs_snd,req)\n",
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" req = MPI.Irecv!(view(u,1:1),comm,source=neig_rank,tag=0)\n",
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" push!(reqs_rcv,req)\n",
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" end\n",
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" if iw != nw\n",
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" neig_rank = (iw+1)-1\n",
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" s = n_own-1\n",
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" r = n_own\n",
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" req = MPI.Isend(view(u,s:s),comm,dest=neig_rank,tag=0)\n",
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" push!(reqs_snd,req)\n",
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" req = MPI.Irecv!(view(u,r:r),comm,source=neig_rank,tag=0)\n",
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" push!(reqs_rcv,req)\n",
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" end\n",
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" for i in 3:n_own\n",
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" u_new[i] = 0.5*(u[i-1]+u[i+1])\n",
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" end\n",
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" MPI.Waitall(reqs_rcv)\n",
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" for i in (2,n_own+1)\n",
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" u_new[i] = 0.5*(u[i-1]+u[i+1])\n",
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" end\n",
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" MPI.Waitall(reqs_snd)\n",
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" u, u_new = u_new, u\n",
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" end\n",
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" u\n",
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" # Gather results in root process\n",
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" results = zeros(n+2)\n",
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" results[1] = -1\n",
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" results[n+2] = 1\n",
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" MPI.Gather!(view(u,2:n_own+1), view(results, 2:n+1), root=0, comm)\n",
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" if iw == 1\n",
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" @show results\n",
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" end \n",
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" end\n",
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" niters = 100\n",
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" load = 4\n",
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@ -56,9 +56,8 @@
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"metadata": {},
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"outputs": [],
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"source": [
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"@everywhere workers() begin\n",
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"@mpi_do manager begin\n",
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" using MPI\n",
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" MPI.Initialized() || MPI.Init()\n",
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" comm = MPI.Comm_dup(MPI.COMM_WORLD)\n",
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" nw = MPI.Comm_size(comm)\n",
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" iw = MPI.Comm_rank(comm)+1\n",
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@ -84,8 +83,8 @@
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" end\n",
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" if iw != nw\n",
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" neig_rank = (iw+1)-1\n",
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" s = n_own-1\n",
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" r = n_own\n",
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" s = n_own+1\n",
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" r = n_own+2\n",
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" req = MPI.Isend(view(u,s:s),comm,dest=neig_rank,tag=0)\n",
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" push!(reqs_snd,req)\n",
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" req = MPI.Irecv!(view(u,r:r),comm,source=neig_rank,tag=0)\n",
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@ -102,6 +101,7 @@
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" u, u_new = u_new, u\n",
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" end\n",
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" u\n",
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" @show u\n",
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" end\n",
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" niters = 100\n",
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" load = 4\n",
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