Lotka-Volterra Work-Precision Diagrams

Lotka-Volterra

The purpose of this problem is to test the performance on easy problems. Since it's periodic, the error is naturally low, and so most of the difference will come down to startup times and, when measuring the interpolations, the algorithm choices.

using OrdinaryDiffEq, ParameterizedFunctions, ODE, ODEInterfaceDiffEq, LSODA,
      Sundials, DiffEqDevTools, StaticArrays

f = @ode_def LotkaVolterra begin
  dx = a*x - b*x*y
  dy = -c*y + d*x*y
end a b c d

p = SA[1.5,1.0,3.0,1.0]
prob = ODEProblem{true, SciMLBase.FullSpecialize}(f,[1.0;1.0],(0.0,10.0),p)
probstatic = ODEProblem{false}(f,SA[1.0;1.0],(0.0,10.0),p)

abstols = 1.0 ./ 10.0 .^ (6:13)
reltols = 1.0 ./ 10.0 .^ (3:10);
sol = solve(prob,Vern7(),abstol=1/10^14,reltol=1/10^14)
sol2 = solve(probstatic,Vern7(),abstol=1/10^14,reltol=1/10^14)
probs = [prob,probstatic]
test_sol = [sol,sol2];
using Plots; gr()

Plots.GRBackend()

plot(sol)

Low Order

setups = [Dict(:alg=>DP5())
          #Dict(:alg=>ode45()) # fail
          Dict(:alg=>dopri5())
          Dict(:alg=>Tsit5())
          Dict(:alg=>Vern6())
          Dict(:alg=>Tsit5(), :prob_choice => 2)
          Dict(:alg=>Vern6(), :prob_choice => 2)
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,save_everystep=false,maxiters=10000,numruns=100)
plot(wp)

Here we see the OrdinaryDiffEq.jl algorithms once again far in the lead.

Interpolation Error

Since the problem is periodic, the real measure of error is the error throughout the solution.

setups = [Dict(:alg=>DP5())
          #Dict(:alg=>ode45())
          Dict(:alg=>Tsit5())
          Dict(:alg=>Vern6())
          Dict(:alg=>Tsit5(), :prob_choice => 2)
          Dict(:alg=>Vern6(), :prob_choice => 2)
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,maxiters=10000,error_estimate=:L2,dense_errors=true,numruns=100)
plot(wp)

Here we see the power of algorithm specific interpolations. The ODE.jl algorithm is only able to reach $10^{-7}$ error even at a tolerance of $10^{-13}$, while the DifferentialEquations.jl algorithms are below $10^{-10}$

Higher Order

setups = [Dict(:alg=>DP8())
          Dict(:alg=>dop853())
          #Dict(:alg=>ode78()) # fails
          Dict(:alg=>Vern6())
          Dict(:alg=>Vern7())
          Dict(:alg=>Vern8())
          Dict(:alg=>Vern9())
          Dict(:alg=>Vern6(), :prob_choice => 2)
          Dict(:alg=>Vern7(), :prob_choice => 2)
          Dict(:alg=>Vern8(), :prob_choice => 2)
          Dict(:alg=>Vern9(), :prob_choice => 2)
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,save_everystep=false,maxiters=1000,numruns=100)
plot(wp)

setups = [Dict(:alg=>odex())
          Dict(:alg=>ddeabm())
          Dict(:alg=>Vern6())
          Dict(:alg=>Vern7())
          Dict(:alg=>Vern8())
          Dict(:alg=>Vern9())
          Dict(:alg=>Vern6(), :prob_choice => 2)
          Dict(:alg=>Vern7(), :prob_choice => 2)
          Dict(:alg=>Vern8(), :prob_choice => 2)
          Dict(:alg=>Vern9(), :prob_choice => 2)
          Dict(:alg=>CVODE_Adams())
          Dict(:alg=>lsoda())
          Dict(:alg=>ARKODE(Sundials.Explicit(),order=6))
          ]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,save_everystep=false,maxiters=1000,numruns=100)
plot(wp)

Again we look at interpolations:

setups = [Dict(:alg=>DP8())
          #Dict(:alg=>ode78())
          Dict(:alg=>Vern6())
          Dict(:alg=>Vern7())
          Dict(:alg=>Vern8())
          Dict(:alg=>Vern9())
          Dict(:alg=>Vern6(), :prob_choice => 2)
          Dict(:alg=>Vern7(), :prob_choice => 2)
          Dict(:alg=>Vern8(), :prob_choice => 2)
          Dict(:alg=>Vern9(), :prob_choice => 2)
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,dense=true,maxiters=1000,error_estimate=:L2,numruns=100)
plot(wp)

Again, the ODE.jl algorithms suffer when measuring the interpolations due to relying on an order 3 Hermite polynomial instead of an algorithm-specific order matching interpolation which uses the timesteps.

Comparison with Non-RK methods

Now let's test Tsit5 and Vern9 against parallel extrapolation methods and an Adams-Bashforth-Moulton:

abstols = 1.0 ./ 10.0 .^ (8:13)
reltols = 1.0 ./ 10.0 .^ (8:13)
setups = [Dict(:alg=>Tsit5())
          Dict(:alg=>Vern9())
          Dict(:alg=>VCABM())
          Dict(:alg=>Vern9(), :prob_choice => 2)
          Dict(:alg=>VCABM(), :prob_choice => 2)
          Dict(:alg=>AitkenNeville(min_order=1, max_order=9, init_order=4, threading=true))
          Dict(:alg=>ExtrapolationMidpointDeuflhard(min_order=1, max_order=9, init_order=4, threading=true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(min_order=2, max_order=11, init_order=4, threading=true))]
solnames = ["Tsit5","Vern9","VCABM","Vern9 Static","VCABM Static","AitkenNeville","Midpoint Deuflhard","Midpoint Hairer Wanner"]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,names=solnames,
                      save_everystep=false,verbose=false,numruns=100)
plot(wp)

setups = [Dict(:alg=>ExtrapolationMidpointDeuflhard(min_order=1, max_order=9, init_order=9, threading=false))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(min_order=2, max_order=11, init_order=4, threading=false))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(min_order=2, max_order=11, init_order=4, threading=true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(min_order=2, max_order=11, init_order=4, sequence = :romberg, threading=true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(min_order=2, max_order=11, init_order=4, sequence = :bulirsch, threading=true))]
solnames = ["Deuflhard","No threads","standard","Romberg","Bulirsch"]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,names=solnames,
                      save_everystep=false,verbose=false,numruns=100)
plot(wp)

setups = [Dict(:alg=>ExtrapolationMidpointHairerWanner(min_order=2, max_order=11, init_order=10, threading=true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(min_order=2, max_order=11, init_order=4, threading=true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(min_order=5, max_order=11, init_order=10, threading=true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(min_order=2, max_order=15, init_order=10, threading=true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(min_order=5, max_order=7, init_order=6, threading=true))]
solnames = ["1","2","3","4","5"]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,names=solnames,
                      save_everystep=false,verbose=false,numruns=100)
plot(wp)

Conclusion

The OrdinaryDiffEq.jl are quicker and still solve to a much higher accuracy, especially when the interpolations are involved. ODE.jl errors a lot.

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/NonStiffODE","LotkaVolterra_wpd.jmd")

Computer Information:

Julia Version 1.9.3
Commit bed2cd540a1 (2023-08-24 14:43 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-14.0.6 (ORCJIT, znver2)
  Threads: 128 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/NonStiffODE/Project.toml`
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Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

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  [e7412a2a] Ogg_jll v1.3.5+1
⌅ [458c3c95] OpenSSL_jll v1.1.21+0
  [efe28fd5] OpenSpecFun_jll v0.5.5+0
  [91d4177d] Opus_jll v1.3.2+0
  [30392449] Pixman_jll v0.42.2+0
  [c0090381] Qt6Base_jll v6.4.2+3
  [f50d1b31] Rmath_jll v0.4.0+0
⌅ [fb77eaff] Sundials_jll v5.2.1+0
  [a2964d1f] Wayland_jll v1.21.0+0
  [2381bf8a] Wayland_protocols_jll v1.25.0+0
⌃ [02c8fc9c] XML2_jll v2.10.3+0
  [aed1982a] XSLT_jll v1.1.34+0
⌃ [ffd25f8a] XZ_jll v5.4.3+1
  [4f6342f7] Xorg_libX11_jll v1.8.6+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.11+0
  [935fb764] Xorg_libXcursor_jll v1.2.0+4
  [a3789734] Xorg_libXdmcp_jll v1.1.4+0
  [1082639a] Xorg_libXext_jll v1.3.4+4
  [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
  [a51aa0fd] Xorg_libXi_jll v1.7.10+4
  [d1454406] Xorg_libXinerama_jll v1.1.4+4
  [ec84b674] Xorg_libXrandr_jll v1.5.2+4
  [ea2f1a96] Xorg_libXrender_jll v0.9.10+4
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.1+0
  [c7cfdc94] Xorg_libxcb_jll v1.15.0+0
  [cc61e674] Xorg_libxkbfile_jll v1.1.2+0
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.6+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
  [c5fb5394] Xorg_xtrans_jll v1.5.0+0
  [8f1865be] ZeroMQ_jll v4.3.4+0
  [3161d3a3] Zstd_jll v1.5.5+0
⌅ [214eeab7] fzf_jll v0.29.0+0
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [b53b4c65] libpng_jll v1.6.38+0
  [a9144af2] libsodium_jll v1.0.20+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+0
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [b27032c2] LibCURL v0.6.3
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.9.0
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics v1.9.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.0.2+0
  [deac9b47] LibCURL_jll v7.84.0+0
  [29816b5a] LibSSH2_jll v1.10.2+0
  [c8ffd9c3] MbedTLS_jll v2.28.2+0
  [14a3606d] MozillaCACerts_jll v2022.10.11
  [4536629a] OpenBLAS_jll v0.3.21+4
  [05823500] OpenLibm_jll v0.8.1+0
  [efcefdf7] PCRE2_jll v10.42.0+0
  [bea87d4a] SuiteSparse_jll v5.10.1+6
  [83775a58] Zlib_jll v1.2.13+0
  [8e850b90] libblastrampoline_jll v5.8.0+0
  [8e850ede] nghttp2_jll v1.48.0+0
  [3f19e933] p7zip_jll v17.4.0+0
Info Packages marked with ⌃ and ⌅ have new versions available, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`