Here, we collect some simple benchmarks of BSeries.jl. Take them with a grain of salt since they run on virtual machines in the cloud to generate the documentation automatically. You can of course also copy the code and run the benchmarks locally yourself.
Symbolic computations of modified_equations and
modifying_integrators in
BSeries.jl
support
- SymEngine.jl,
- SymPy.jl, and
- Symbolics.jl
as symbolic backends. Here, we compare them in the context of the explicit midpoint method and the nonlinear oscillator ODE
This particular combination of explicit Runge-Kutta method and ODE is special since the explicit midpoint method is unconditionally energy-conserving for this problem1.
First, we set up some code to perform the benchmarks. Here, we use a very naive
approach, run the code twice (to see the effect of compilation) and use @time
to print the runtime. More sophisticated approaches should of course use
something like @benchmark from
BenchmarkTools.jl.
However, this simple and cheap version suffices to compare the orders of
magnitude.
using BSeries, StaticArrays
function benchmark(u, dt, subs, order)
# explicit midpoint method
A = @SArray [0 0; 1//2 0]
b = @SArray [0, 1//1]
c = @SArray [0, 1//2]
# nonlinear oscillator
f = [-u[2], u[1]] / (u[1]^2 + u[2]^2)
println("\n Computing the series coefficients:")
@time coefficients = modifying_integrator(A, b, c, order)
@time coefficients = modifying_integrator(A, b, c, order)
println("\n Computing the series including elementary differentials:")
@time series = modifying_integrator(f, u, dt, A, b, c, order)
@time series = modifying_integrator(f, u, dt, A, b, c, order)
substitution_variables = Dict(u[1] => 1//1, u[2] => 0//1)
println("\n Substituting the initial condition:")
@time subs.(series, (substitution_variables, ))
@time subs.(series, (substitution_variables, ))
println("\n")
end
Next, we load the symbolic packages and run the benchmarks.
using SymPy # generates annoying output online when conda installs sympy
using SymEngine: SymEngine
using SymPy: SymPy
using Symbolics: Symbolics
println("SymEngine")
dt = SymEngine.symbols("dt")
u = SymEngine.symbols("u1, u2")
subs = SymEngine.subs
benchmark(u, dt, subs, 8)
println("SymPy")
dt = SymPy.symbols("dt")
u = SymPy.symbols("u1, u2")
subs = SymPy.subs
benchmark(u, dt, subs, 8)
println("Symbolics")
Symbolics.@variables dt
u = Symbolics.@variables u1 u2
subs = Symbolics.substitute
benchmark(u, dt, subs, 8)
These results were obtained using the following versions.
using InteractiveUtils
versioninfo()
using Pkg
Pkg.status(["BSeries", "RootedTrees", "SymEngine", "SymPy", "Symbolics"],
mode=PKGMODE_MANIFEST)
nothing # hide
There are also other open source packages for B-series. Currently, we are aware of the Python packages
If you know about similar open source packages out there, please inform us, e.g., by creating an issue on GitHub.
The packages listed above and BSeries.jl
all use different approaches and have different features. Thus, comparisons
must be restricted to their common subset of features. Here, we present some
simple performance comparisons. Again, we just use (the equivalent of) @time
twice to get an idea of the performance after compilation, allowing us to
compare orders of magnitude.
First, we start with the Python package
BSeries
and the following benchmark script.
using BSeries, Markdown # hide
filename = joinpath(pathof(BSeries) |> dirname |> dirname, "docs", "src", # hide
"benchmark_python_bseries.py") # hide
script = "```python\n" # hide
for line in Iterators.drop(readlines(filename), 4) # hide
startswith(line, "with") && continue # hide
line = replace(line, " print" => "print") # hide
global script # hide
script = script * replace(line, ", file=io" => "") * "\n" # hide
end # hide
script = script * "```\n" # hide
Markdown.parse(script) # hide
The results are as follows.
using BSeries, Markdown # hide
filename = joinpath(pathof(BSeries) |> dirname |> dirname, "docs", "src", # hide
"benchmark_python_bseries.txt") # hide
try # hide
results = "```\n" * read(filename, String) * "```\n" # hide
Markdown.parse(results) # hide
catch # hide
ci = get(ENV, "CI", nothing) == "true" && # hide
get(ENV, "GITHUB_REPOSITORY", nothing) == "ranocha/Bseries.jl" # hide
if ci # hide
rethrow() # hide
else # hide
println("We are not running CI so we do not show results here.") # hide
end # hide
end # hide
Next, we look at the Python package
pybs
and the following benchmark script. Note that this package does not provide
functionality for modifying integrators.
using BSeries, Markdown # hide
filename = joinpath(pathof(BSeries) |> dirname |> dirname, "docs", "src", # hide
"benchmark_python_pybs.py") # hide
script = "```python\n" # hide
for line in Iterators.drop(readlines(filename), 4) # hide
startswith(line, "with") && continue # hide
line = replace(line, " print" => "print") # hide
global script # hide
script = script * replace(line, ", file=io" => "") * "\n" # hide
end # hide
script = script * "```\n" # hide
Markdown.parse(script) # hide
The results are as follows.
using BSeries, Markdown # hide
filename = joinpath(pathof(BSeries) |> dirname |> dirname, "docs", "src", # hide
"benchmark_python_pybs.txt") # hide
try # hide
results = "```\n" * read(filename, String) * "```\n" # hide
Markdown.parse(results) # hide
catch # hide
ci = get(ENV, "CI", nothing) == "true" && # hide
get(ENV, "GITHUB_REPOSITORY", nothing) == "ranocha/Bseries.jl" # hide
if ci # hide
rethrow() # hide
else # hide
println("We are not running CI so we do not show results here.") # hide
end # hide
end # hide
Next, we look at the Python package
orderconditions
of Valentin Dallerit
and the following benchmark script.
using BSeries, Markdown # hide
filename = joinpath(pathof(BSeries) |> dirname |> dirname, "docs", "src", # hide
"benchmark_python_orderconditions.py") # hide
script = "```python\n" # hide
for line in Iterators.drop(readlines(filename), 4) # hide
startswith(line, "with") && continue # hide
line = replace(line, " print" => "print") # hide
global script # hide
script = script * replace(line, ", file=io" => "") * "\n" # hide
end # hide
script = script * "```\n" # hide
Markdown.parse(script) # hide
The results are as follows.
using BSeries, Markdown # hide
filename = joinpath(pathof(BSeries) |> dirname |> dirname, "docs", "src", # hide
"benchmark_python_orderconditions.txt") # hide
try # hide
results = "```\n" * read(filename, String) * "```\n" # hide
Markdown.parse(results) # hide
catch # hide
ci = get(ENV, "CI", nothing) == "true" && # hide
get(ENV, "GITHUB_REPOSITORY", nothing) == "ranocha/Bseries.jl" # hide
if ci # hide
rethrow() # hide
else # hide
println("We are not running CI so we do not show results here.") # hide
end # hide
end # hide
Finally, we perform the same task using BSeries.jl in Julia.
using BSeries, StaticArrays
A = @SArray [0 0; 1//2 0]
b = @SArray [0, 1//1]
c = @SArray [0, 1//2]
up_to_order = 9
println("Modified equation")
@time begin
series = modified_equation(A, b, c, up_to_order)
println(sum(values(series)))
end
@time begin
series = modified_equation(A, b, c, up_to_order)
println(sum(values(series)))
end
println("\nModifying integrator")
@time begin
series = modifying_integrator(A, b, c, up_to_order)
println(sum(values(series)))
end
@time begin
series = modifying_integrator(A, b, c, up_to_order)
println(sum(values(series)))
end
println("\nEnergy preservation")
@time begin
series = bseries(AverageVectorFieldMethod(), up_to_order)
println(is_energy_preserving(series))
end
@time begin
series = bseries(AverageVectorFieldMethod(), up_to_order)
println(is_energy_preserving(series))
end
Hendrik Ranocha and David Ketcheson (2020) Energy Stability of Explicit Runge-Kutta Methods for Nonautonomous or Nonlinear Problems. SIAM Journal on Numerical Analysis DOI: 10.1137/19M1290346