# Parallel Approximations

A subset of the approximation algorithms are implemented in parallel in the LazySets.Parallel module. In order to use parallel versions of the algorithms, you can write:

using LazySets
import LazySets.Parallel

# call a method implemented in parallel, for example:
S = Ball2(ones(100), 1.0)
Parallel.box_approximation(S)
Hyperrectangle{Float64, Vector{Float64}, Vector{Float64}}([1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0], [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0  …  1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0])

Note that after importing or using LazySets.Parallel, the version of the function used must be fully qualified, eg. LazySets.Approximations.box_approximation for the sequential version or LazySets.Parallel.box_approximation for the parallel version.

The parallelization strategy that is available uses processes. To set the number of processes N, use the flag -p N at julia startup. For example, do

$julia -p 4 to launch 4 additional local worker julia processes. Use the keyword auto, as in $ julia -p auto

to launch as many workers as the number of local CPU cores.

## Parallel interval hulls

As an illustration of the symmetric interval hull approximation of a nested lazy set computed in parallel, consider the following calculation. It arises in the discretization of set-based ODEs, and is defined below for an artificial example of a tridiagonal matrix of order n, where n is a positive integer.

using LazySets, Expokit
using SparseArrays, LinearAlgebra

# define an nxn tridiagonal matrix
A(n) = sparse(diagm(0 => fill(0.05, n), -1 => fill(-1, n-1), 1 => fill(-1, n-1)))

# step size and initial set
δ = 0.1
X0(n) = Ball2(ones(n), 0.1)

# input coefficients matrix (nx2 matrix with coefficients from -1 to 1)
b(n) = vcat(range(-1, stop=1, length=n))
B(n) = [b(n) b(n)]
U = BallInf(zeros(2), 1.2)

# lazy matrix exponential
eAδ(n) = SparseMatrixExp(A(n) * δ)

# set that we want to overapproximate with an interval hull
Y(n) = ConvexHull(eAδ(n) * X0(n) ⊕ (δ * B(n) * U), X0(n))

The set Y(n) is parametric in the system's dimension n, to facilitate benchmarking. We will explore the computational cost as the dimension n increases, and compare the sequential algorithm with the parallel algorithm.

Given the lazy set Y(n), we want to calculate the symmetric interval hull, which corresponds to finding the smallest n-dimensional hyperrectangle that contains the set Y(n) and is symmetric with respect to the origin. Notice that this operation is inherently parallel, since one can evaluate the support function of Y independently in each dimension from 1 to n.

The sequential algorithm returns the following execution times. We use the @btime macro from the BenchmarkTools package to have a more accurate timing than @time; the $n argument is used for interpolation of the arguments (if you are not behchmarking, pass n to symmetric_interval_hull, as usual). using BenchmarkTools for n in [50, 100, 500, 1000] @btime res = Approximations.symmetric_interval_hull(Y($n));
end

59.103 ms (11554 allocations: 25.89 MiB)
129.453 ms (23118 allocations: 54.16 MiB)
1.943 s (115530 allocations: 381.26 MiB)
10.017 s (232506 allocations: 1.01 GiB)

For the parallel benchmark, we start Julia with 4 processes with the command $julia -p 4 and call LazySets.Parallel.symmetric_interval_hull(Y(n)). import LazySets.Parallel for n in [50, 100, 500, 1000] @btime LazySets.Parallel.symmetric_interval_hull($Y(\$n));
end

6.846 ms (2550 allocations: 160.59 KiB)
13.544 ms (3528 allocations: 271.94 KiB)
387.556 ms (11155 allocations: 2.51 MiB)
2.638 s (22156 allocations: 8.77 MiB)

In the following table we summarize the speedup.

nSequential (s)Parallel p=4 (s)Speedup
500.0590.0078.42
1000.1290.0139.92
5001.940.3874.96
100010.02.643.79

The results in this section were obtained with a standard MacBook Pro laptop with the following specifications:

julia> versioninfo()
Julia Version 1.0.2
Commit d789231e99 (2018-11-08 20:11 UTC)
Platform Info:
OS: macOS (x86_64-apple-darwin14.5.0)
CPU: Intel(R) Core(TM) i7-4770HQ CPU @ 2.20GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-6.0.0 (ORCJIT, haswell)