Programming large-scale parallel systems

Jacobi method

Contents

In this notebook, we will learn

• How to paralleize a Jacobi method
• How the data partition can impact the performance of a distributed algorithm
• How to use latency hiding

The Jacobi method

The 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.

Serial implementation

function jacobi(n,niters)
    u = zeros(n+2)
    u[1] = -1
    u[end] = 1
    u_new = copy(u)
    for t in 1:niters
        for i in 2:(n+1)
            u_new[i] = 0.5*(u[i-1]+u[i+1])
        end
        u, u_new = u_new, u
    end
    u
end

jacobi(5,1000)

Note: The values computed by the Jacobi method are linearly increasing from -1 to 1. It is possible to show mathematically that the method we implemented in the function above approximates a 1D Laplace equation via a finite difference method and the solution of this equation is a linear function.

Note: In our version of the jacobi method, we return after a given number of iterations. Other stopping criteria are possible. For instance, iterate until the difference between u and u_new is below a tolerance.

Where do we can exploit parallelism?

Look at the two nested loops in the sequential implementation:

for t in 1:nsteps
    for i in 2:(n+1)
        u_new[i] = 0.5*(u[i-1]+u[i+1])
    end
    u, u_new = u_new, u
end

• The outer loop cannot be parallelized. The value of u at step t+1 depends on the value at the previous step t.
• The inner loop can be parallelized

The Gauss-Seidel method

The usage of u_new seems a bit unnecessary at first sight, right?. If we remove it, we get another method called Gauss-Seidel.

function gauss_seidel(n,niters)
    u = zeros(n+2)
    u[1] = -1
    u[end] = 1
    for t in 1:niters
        for i in 2:(n+1)
            u[i] = 0.5*(u[i-1]+u[i+1])
        end
    end
    u
end

Note that the final solution is the same (up to machine precision).

gauss_seidel(5,1000)

Question: Which of the two loops in the Gauss-Seidel method are trivially parallelizable?

for t in 1:niters
    for i in 2:(n+1)
        u[i] = 0.5*(u[i-1]+u[i+1])
    end
end

a) Both of them
b) The outer, but not the inner
c) None of them
d) The inner, but not the outer

#TODO answer (c)

Parallelization of the Jacobi method

Parallelization strategy

• Each worker updates a consecutive section of the array u_new

Data dependencies

Recall:

u_new[i] = 0.5*(u[i-1]+u[i+1])

Thus, each process will need values from the neighboring processes to perform the update of its boundary values.

Ghost (aka halo) cells

A usual way of handling this type of data dependencies is using so-called ghost cells. Ghost cells represent the missing data dependencies in the data owned by each process. After importing the appropriate values from the neighbor processes one can perform the usual sequential jacoby update locally in the processes.

Question: Which is the communication and computation complexity in each process? N is the length of the vector and P the number of processes.

#TODO give multiple options

Implementation

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.

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.

] add MPI MPIClusterManagers

using MPIClusterManagers
using Distributed

if procs() == workers()
    nw = 3
    manager = MPIWorkerManager(nw)
    addprocs(manager)
end

@everywhere workers() begin
    using MPI
    MPI.Initialized() || MPI.Init()
    comm = MPI.Comm_dup(MPI.COMM_WORLD)
    nw = MPI.Comm_size(comm)
    iw = MPI.Comm_rank(comm)+1
    function jacobi_mpi(n,niters)
        if mod(n,nw) != 0
            println("n must be a multiple of nw")
            MPI.Abort(comm,1)
        end
        n_own = div(n,nw)
        u = zeros(n_own+2)
        u[1] = -1
        u[end] = 1
        u_new = copy(u)
        for t in 1:niters
            reqs = MPI.Request[]
            if iw != 1
                neig_rank = (iw-1)-1
                req = MPI.Isend(view(u,2:2),comm,dest=neig_rank,tag=0)
                push!(reqs,req)
                req = MPI.Irecv!(view(u,1:1),comm,source=neig_rank,tag=0)
                push!(reqs,req)
            end
            if iw != nw
                neig_rank = (iw+1)-1
                s = n_own-1
                r = n_own
                req = MPI.Isend(view(u,s:s),comm,dest=neig_rank,tag=0)
                push!(reqs,req)
                req = MPI.Irecv!(view(u,r:r),comm,source=neig_rank,tag=0)
                push!(reqs,req)
            end
            MPI.Waitall(reqs)
            for i in 2:(n_own+1)
                u_new[i] = 0.5*(u[i-1]+u[i+1])
            end
            u, u_new = u_new, u
        end
        u
        @show u
    end
    niters = 100
    load = 4
    n = load*nw
    jacobi_mpi(n,niters)
end

Question: How many messages per iteration are sent from a process away from the boundary?

a) 1
b) 2
c) 3
d) 4

# TODO (b) 2 mesajes. Add another question if you find it useful

Latency hiding

Note that we only need communications to update the values at the boundary of the portion owned by each process. The other values (the one in green in the figure below) can be updated without communications. This provides the opportunity of overlapping the computation of the interior values (green cells in the figure) with the communication of the ghost values. This technique is called latency hiding, since we are hiding communication latency by overlapping it with communications that we need to do anyway.

The modification of the implementation above to include latency hiding is leaved as an exercise (see below).

Extension to 2D

Serial implementation

function jacobi_2d(n,niters)
    u = zeros(n+2,n+2)
    u[1,:] = u[end,:] = u[:,1] = u[:,end] .= 1
    heater = 1/n^2
    u_new = copy(u)
    for t in 1:niters
        for j in 2:(n+1)
            for i in 2:(n+1)
                north = u[i,j+1]
                south = u[i,j-1]
                east = u[i+1,j]
                west = u[i-1,j]
                u_new[i,j] = 0.25*(north+south+east+west) + heater
            end
        end
        u, u_new = u_new, u
    end
    u
end

u = jacobi_2d(100,1000)

Exercises

Exercise 1

Transform the following parallel implementation of the 1d Jacobi method (it is copied from above) to use latency hiding (overlap between computation of interior values and communication)

@everywhere workers() begin
    using MPI
    MPI.Initialized() || MPI.Init()
    comm = MPI.Comm_dup(MPI.COMM_WORLD)
    nw = MPI.Comm_size(comm)
    iw = MPI.Comm_rank(comm)+1
    function jacobi_mpi(n,niters)
        if mod(n,nw) != 0
            println("n must be a multiple of nw")
            MPI.Abort(comm,1)
        end
        n_own = div(n,nw)
        u = zeros(n_own+2)
        u[1] = -1
        u[end] = 1
        u_new = copy(u)
        for t in 1:niters
            reqs = MPI.Request[]
            if iw != 1
                neig_rank = (iw-1)-1
                req = MPI.Isend(view(u,2:2),comm,dest=neig_rank,tag=0)
                push!(reqs,req)
                req = MPI.Irecv!(view(u,1:1),comm,source=neig_rank,tag=0)
                push!(reqs,req)
            end
            if iw != nw
                neig_rank = (iw+1)-1
                s = n_own-1
                r = n_own
                req = MPI.Isend(view(u,s:s),comm,dest=neig_rank,tag=0)
                push!(reqs,req)
                req = MPI.Irecv!(view(u,r:r),comm,source=neig_rank,tag=0)
                push!(reqs,req)
            end
            MPI.Waitall(reqs)
            for i in 2:(n_own+1)
                u_new[i] = 0.5*(u[i-1]+u[i+1])
            end
            u, u_new = u_new, u
        end
        u
        @show u
    end
    niters = 100
    load = 4
    n = load*nw
    jacobi_mpi(n,niters)
end

## TODO move the following solution to its appropiate place:

@everywhere workers() begin
    using MPI
    MPI.Initialized() || MPI.Init()
    comm = MPI.Comm_dup(MPI.COMM_WORLD)
    nw = MPI.Comm_size(comm)
    iw = MPI.Comm_rank(comm)+1
    function jacobi_mpi(n,niters)
        if mod(n,nw) != 0
            println("n must be a multiple of nw")
            MPI.Abort(comm,1)
        end
        n_own = div(n,nw)
        u = zeros(n_own+2)
        u[1] = -1
        u[end] = 1
        u_new = copy(u)
        for t in 1:niters
            reqs_snd = MPI.Request[]
            reqs_rcv = MPI.Request[]
            if iw != 1
                neig_rank = (iw-1)-1
                req = MPI.Isend(view(u,2:2),comm,dest=neig_rank,tag=0)
                push!(reqs_snd,req)
                req = MPI.Irecv!(view(u,1:1),comm,source=neig_rank,tag=0)
                push!(reqs_rcv,req)
            end
            if iw != nw
                neig_rank = (iw+1)-1
                s = n_own-1
                r = n_own
                req = MPI.Isend(view(u,s:s),comm,dest=neig_rank,tag=0)
                push!(reqs_snd,req)
                req = MPI.Irecv!(view(u,r:r),comm,source=neig_rank,tag=0)
                push!(reqs_rcv,req)
            end
            for i in 3:n_own
                u_new[i] = 0.5*(u[i-1]+u[i+1])
            end
            MPI.Waitall(reqs_rcv)
            for i in (2,n_own+1)
                u_new[i] = 0.5*(u[i-1]+u[i+1])
            end
            MPI.Waitall(reqs_snd)
            u, u_new = u_new, u
        end
        u
    end
    niters = 100
    load = 4
    n = load*nw
    jacobi_mpi(n,niters)
end