Quadratic Stiffness Benchmarks

Chris Rackauckas

Quadratic Stiffness

In this notebook we will explore the quadratic stiffness problem. References:

The composite Euler method for stiff stochastic differential equations

Kevin Burrage, Tianhai Tian




This is a scalar SDE with two arguments. The first controls the deterministic stiffness and the later controls the diffusion stiffness.

using DiffEqProblemLibrary, StochasticDiffEq, DiffEqDevTools
using DiffEqProblemLibrary.SDEProblemLibrary: importsdeproblems; importsdeproblems()
import DiffEqProblemLibrary.SDEProblemLibrary: prob_sde_stiffquadito
using Plots; gr()
const N = 10
prob = remake(prob_sde_stiffquadito,p=(50.0,1.0))
sol = solve(prob,SRIW1())
prob = remake(prob_sde_stiffquadito,p=(500.0,1.0))
sol = solve(prob,SRIW1())

Top dts

Let's first determine the maximum dts which are allowed. Anything higher is mostly unstable.

Deterministic Stiffness Mild

prob = remake(prob_sde_stiffquadito,p=(50.0,1.0))
@time sol = solve(prob,SRIW1())
@time sol = solve(prob,SRIW1(),adaptive=false,dt=0.01)
@time sol = solve(prob,ImplicitRKMil(),dt=0.005)
@time sol = solve(prob,EM(),dt=0.01);
0.000161 seconds (1.35 k allocations: 80.969 KiB)
  0.169792 seconds (622.19 k allocations: 35.992 MiB, 99.89% compilation ti
  1.752111 seconds (4.75 M allocations: 259.676 MiB, 5.16% gc time, 99.99% 
compilation time)
  1.116955 seconds (2.83 M allocations: 146.269 MiB, 99.98% compilation tim

Deterministic Stiffness High

prob = remake(prob_sde_stiffquadito,p=(500.0,1.0))
@time sol = solve(prob,SRIW1())
@time sol = solve(prob,SRIW1(),adaptive=false,dt=0.002)
@time sol = solve(prob,ImplicitRKMil(),dt=0.001)
@time sol = solve(prob,EM(),dt=0.002);
0.000922 seconds (9.85 k allocations: 571.938 KiB)
  0.000575 seconds (7.58 k allocations: 360.828 KiB)
  0.000105 seconds (392 allocations: 21.719 KiB)
  0.000545 seconds (7.58 k allocations: 338.906 KiB)

Mixed Stiffness

prob = remake(prob_sde_stiffquadito,p=(5000.0,70.0))
@time sol = solve(prob,SRIW1(),dt=0.0001)
@time sol = solve(prob,SRIW1(),adaptive=false,dt=0.00001)
@time sol = solve(prob,ImplicitRKMil(),dt=0.00001)
@time sol = solve(prob,EM(),dt=0.00001);
0.076964 seconds (315.25 k allocations: 19.839 MiB, 98.02% compilation ti
  0.154182 seconds (1.50 M allocations: 62.111 MiB, 36.84% gc time)
  0.298183 seconds (228 allocations: 39.479 MiB)
  0.093673 seconds (1.50 M allocations: 56.884 MiB)

Notice that in this problem, the stiffness in the noise term still prevents the semi-implicit integrator to do well. In that case, the advantage of implicitness does not take effect, and thus explicit methods do well. When we don't care about the error, Euler-Maruyama is fastest. When there's mixed stiffness, the adaptive algorithm is unstable.

Work-Precision Diagrams

prob = remake(prob_sde_stiffquadito,p=(50.0,1.0))

reltols = 1.0 ./ 10.0 .^ (3:5)
abstols = reltols#[0.0 for i in eachindex(reltols)]
setups = [Dict(:alg=>SRIW1()),
          Dict(:alg=>EM(),:dts=>1.0./8.0.^((1:length(reltols)) .+ 1)),
          Dict(:alg=>SRIW1(),:dts=>1.0./8.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          #Dict(:alg=>RKMil(),:dts=>1.0./8.0.^((1:length(reltols)) .+ 1),:adaptive=>false),
names = ["SRIW1","EM","SRIW1 Fixed"] #"RKMil",
wp = WorkPrecisionSet(prob,abstols,reltols,setups;numruns=N,names=names,error_estimate=:l2)
prob = remake(prob_sde_stiffquadito,p=(500.0,1.0))

reltols = 1.0 ./ 10.0 .^ (3:5)
abstols = reltols#[0.0 for i in eachindex(reltols)]
setups = [Dict(:alg=>SRIW1()),
          Dict(:alg=>EM(),:dts=>1.0./8.0.^((1:length(reltols)) .+ 2)),
          Dict(:alg=>SRIW1(),:dts=>1.0./8.0.^((1:length(reltols)) .+ 2),:adaptive=>false)
          #Dict(:alg=>RKMil(),:dts=>1.0./8.0.^((1:length(reltols)) .+ 2),:adaptive=>false),
names = ["SRIW1","EM","SRIW1 Fixed"] #"RKMil",
wp = WorkPrecisionSet(prob,abstols,reltols,setups;numruns=N,names=names,error_estimate=:l2,print_names=true)


Noise stiffness is tough. Right now the best solution is to run an explicit integrator with a low enough dt. Adaptivity does have a cost in this case, likely due to memory management.


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

Computer Information:

Julia Version 1.7.3
Commit 742b9abb4d (2022-05-06 12:58 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, znver2)
  BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953

Package Information:

      Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/StiffSDE/Project.toml`
  [f3b72e0c] DiffEqDevTools v2.30.0
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  [a077e3f3] DiffEqProblemLibrary v4.17.0
  [91a5bcdd] Plots v1.31.4
  [31c91b34] SciMLBenchmarks v0.1.0
  [789caeaf] StochasticDiffEq v6.50.1
  [37e2e46d] LinearAlgebra
  [9a3f8284] Random
  [10745b16] Statistics

And the full manifest:

      Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/StiffSDE/Manifest.toml`
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  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [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 v1.8.0
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.0
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v0.5.2+0
  [deac9b47] LibCURL_jll v7.81.0+0
  [29816b5a] LibSSH2_jll v1.10.2+0
  [c8ffd9c3] MbedTLS_jll v2.28.0+0
  [14a3606d] MozillaCACerts_jll v2022.2.1
  [4536629a] OpenBLAS_jll v0.3.20+0
  [05823500] OpenLibm_jll v0.8.1+0
  [efcefdf7] PCRE2_jll v10.40.0+0
  [bea87d4a] SuiteSparse_jll v5.10.1+0
  [83775a58] Zlib_jll v1.2.12+3
  [8e850b90] libblastrampoline_jll v5.1.0+0
  [8e850ede] nghttp2_jll v1.41.0+1
  [3f19e933] p7zip_jll v17.4.0+0