SDE Lokta-Volterra Work-Precision Diagrams
using StochasticDiffEq, DiffEqDevTools, ParameterizedFunctions
using Plots;
gr()
const N = 100
f = @ode_def LotkaVolterraTest begin
dx = a*x - b*x*y
dy = -c*y + d*x*y
end a b c d
p = [1.5, 1.0, 3.0, 1.0]
function g(du, u, p, t)
du .= 0.1u
end
u0 = [1.0; 1.0]
tspan = (0.0, 10.0)
prob = SDEProblem(f, g, u0, tspan, p);sol = solve(prob, SRIW1(), abstol = 1e-4, reltol = 1e-4)
plot(sol)
Strong Error
The starting dts was chosen as the largest in the 1/4^i which were stable. All larger dts contained trajectories which would veer off to infinity.
reltols = 1.0 ./ 4.0 .^ (2:4)
abstols = reltols#[0.0 for i in eachindex(reltols)]
setups = [Dict(:alg=>SRIW1())
Dict(:alg=>EM(), :dts=>1.0 ./ 12.0 .^ ((1:length(reltols)) .+ 1.5))
Dict(:alg=>RKMil(), :dts=>1.0 ./ 12.0 .^ ((1:length(reltols)) .+ 1.5), :adaptive=>false)
Dict(:alg=>SRIW1(), :dts=>1.0 ./ 4.0 .^ ((1:length(reltols)) .+ 5), :adaptive=>false)
Dict(:alg=>SRIW2())
Dict(:alg=>SOSRI())
Dict(:alg=>SOSRI2())]
test_dt = 1/10^2
appxsol_setup = Dict(:alg=>SRIW1(), :abstol=>1e-4, :reltol=>1e-4)
wp = WorkPrecisionSet(prob, abstols, reltols, setups, test_dt;
maxiters = 1e7,
verbose = false, save_everystep = false,
parallel_type = :threads,
appxsol_setup = appxsol_setup,
numruns_error = N, error_estimate = :final)
plot(wp)
Weak Error
reltols = 1.0 ./ 4.0 .^ (2:4)
abstols = reltols#[0.0 for i in eachindex(reltols)]
setups = [Dict(:alg=>SRIW1())
Dict(:alg=>EM(), :dts=>1.0 ./ 12.0 .^ ((1:length(reltols)) .+ 1.5))
Dict(:alg=>RKMil(), :dts=>1.0 ./ 12.0 .^ ((1:length(reltols)) .+ 1.5), :adaptive=>false)
Dict(:alg=>SRIW1(), :dts=>1.0 ./ 4.0 .^ ((1:length(reltols)) .+ 5), :adaptive=>false)
Dict(:alg=>SRIW2())
Dict(:alg=>SOSRI())
Dict(:alg=>SOSRI2())]
test_dt = 1e-2
appxsol_setup = Dict(:alg=>SRIW1(), :abstol=>1e-4, :reltol=>1e-4)
wp = WorkPrecisionSet(prob, abstols, reltols, setups, test_dt;
maxiters = 1e7,
verbose = false, save_everystep = false,
parallel_type = :none,
appxsol_setup = appxsol_setup,
numruns_error = N, error_estimate = :weak_final)
plot(wp; legend = :topleft)
sample_size = Int[10; 1e2; 1e3]
se = get_sample_errors(prob, setups[6], test_dt, numruns = sample_size,
appxsol_setup = appxsol_setup,
sample_error_runs = 100_000, solution_runs = 20)3-element Vector{Float64}:
0.12397682065514846
0.010312550452511556
0.0010814155260340478plot(wp; legend = :topleft)
times = [wp[i].times for i in 1:length(wp)]
times = [minimum(minimum(t) for t in times), maximum(maximum(t) for t in times)]
plot!([se[end]; se[end]], times, color = :orange,
linestyle = :dash, label = "Sample Error: 1000", lw = 3)
Conclusion
These results show that in both strong and weak error, the high order method is more efficient. The strong and the weak are track each other well for the methods tested on this problem, with the strong error slightly higher than the weak error. To reach the sample error for a 100 trajectories, the higher order method is around 5x faster. To reach the sampling error for 10000 trajectories, the higher order method is nearly 100x faster.
Appendix
These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.
To locally run this benchmark, do the following commands:
using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/NonStiffSDE","LotkaVolterraSDE.jmd")Computer Information:
Julia Version 1.10.10
Commit 95f30e51f41 (2025-06-27 09:51 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 128 × AMD EPYC 7502 32-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
JULIA_CPU_THREADS = 128
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953
Package Information:
Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/NonStiffSDE/Project.toml`
[f3b72e0c] DiffEqDevTools v2.48.0
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[789caeaf] StochasticDiffEq v6.81.0And the full manifest:
Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/NonStiffSDE/Manifest.toml`
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[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[a63ad114] Mmap
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.10.0
[de0858da] Printf
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays v1.10.0
[10745b16] Statistics v1.10.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll v1.1.1+0
[deac9b47] LibCURL_jll v8.4.0+0
[e37daf67] LibGit2_jll v1.6.4+0
[29816b5a] LibSSH2_jll v1.11.0+1
[c8ffd9c3] MbedTLS_jll v2.28.2+1
[14a3606d] MozillaCACerts_jll v2023.1.10
[4536629a] OpenBLAS_jll v0.3.23+4
[05823500] OpenLibm_jll v0.8.5+0
[efcefdf7] PCRE2_jll v10.42.0+1
[bea87d4a] SuiteSparse_jll v7.2.1+1
[83775a58] Zlib_jll v1.2.13+1
[8e850b90] libblastrampoline_jll v5.11.0+0
[8e850ede] nghttp2_jll v1.52.0+1
[3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`