Stochastic Heat Equation Benchmarks

Stochastic Heat Equation Benchmarks

In this notebook we will benchmark against the stochastic heat equation with Dirichlet BCs and scalar noise. The function for generating the problem is as follows:

Stochastic Heat Equation with scalar multiplicative noise

S-ROCK: CHEBYSHEV METHODS FOR STIFF STOCHASTIC DIFFERENTIAL EQUATIONS

ASSYR ABDULLE AND STEPHANE CIRILLI

Raising D or k increases stiffness

using StochasticDiffEq, DiffEqNoiseProcess, LinearAlgebra, Statistics, SciMLBase

function generate_stiff_stoch_heat(D=1,k=1;N = 100, t_end = 3.0, adaptivealg = :RSwM3)
    A = Array(Tridiagonal([1.0 for i in 1:N-1],[-2.0 for i in 1:N],[1.0 for i in 1:N-1]))
    dx = 1/N
    A = D/(dx^2) * A
    function f(du,u,p,t)
        mul!(du,A,u)
    end
    #=
    function f(::Type{Val{:analytic}},u0,p,t,W)
        exp((A-k/2)*t+W*I)*u0 # no -k/2 for Strat
    end
    =#
    function g(du,u,p,t)
        @. du = k*u
    end
    SDEProblem(f,g,ones(N),(0.0,t_end),noise=WienerProcess(0.0,0.0,0.0,rswm=RSWM(adaptivealg=adaptivealg)))
end

N = 100
D = 1; k = 1
    A = Array(Tridiagonal([1.0 for i in 1:N-1],[-2.0 for i in 1:N],[1.0 for i in 1:N-1]))
    dx = 1/N
    A = D/(dx^2) * A;

Now lets solve it with high accuracy.

prob = generate_stiff_stoch_heat(1.0,1.0)
@time sol = solve(prob,SRIW1(),progress=true,abstol=1e-6,reltol=1e-6);

7.651443 seconds (12.15 M allocations: 1.232 GiB, 8.59% gc time, 63.26% c
ompilation time)

Highest dt

Let's try to find the highest possible dt:

@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),SRIW1());

0.742404 seconds (687.16 k allocations: 122.374 MiB, 48.61% compilation t
ime)

@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),SRIW1(),progress=true,adaptive=false,dt=0.00005);

0.485169 seconds (686.81 k allocations: 97.997 MiB, 57.31% compilation ti
me)

@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),EM(),progress=true,adaptive=false,dt=0.00005);

1.523249 seconds (2.51 M allocations: 202.953 MiB, 14.34% gc time, 92.22%
 compilation time)

@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),ImplicitRKMil(),progress=true,dt=0.1);

5.723386 seconds (11.33 M allocations: 609.104 MiB, 11.81% gc time, 99.79
% compilation time)

@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),ImplicitRKMil(),progress=true,dt=0.01);

0.009396 seconds (1.03 k allocations: 688.500 KiB)

@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),ImplicitRKMil(),progress=true,dt=0.001);

0.009707 seconds (1.05 k allocations: 694.891 KiB)

@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),ImplicitEM(),progress=true,dt=0.001);

2.512927 seconds (5.74 M allocations: 321.901 MiB, 3.74% gc time, 99.79% 
compilation time)

Simple Error Analysis

Now let's check the error at an arbitrary timepoint in there. Our analytical solution only exists in the Stratanovich sense, so we are limited in the methods we can calculate errors for.

function simple_error(alg;kwargs...)
    sol = solve(generate_stiff_stoch_heat(1.0,1.0,t_end=0.25),alg;kwargs...);
    W_end = sol.W.u[end]
    sum(abs2, sol.u[end] .- exp(A * sol.t[end] + W_end * I) * prob.u0)
end

simple_error (generic function with 1 method)

mean(simple_error(EulerHeun(),dt=0.00005) for i in 1:400)

3.294274694655326e-9

mean(simple_error(ImplicitRKMil(interpretation=SciMLBase.AlgorithmInterpretation.Stratonovich),dt=0.1) for i in 1:400)

0.018185517528369

mean(simple_error(ImplicitRKMil(interpretation=SciMLBase.AlgorithmInterpretation.Stratonovich),dt=0.01) for i in 1:400)

0.018418466765605716

mean(simple_error(ImplicitRKMil(interpretation=SciMLBase.AlgorithmInterpretation.Stratonovich),dt=0.001) for i in 1:400)

0.018451613923516377

mean(simple_error(ImplicitEulerHeun(),dt=0.001) for i in 1:400)

0.007449617244075225

mean(simple_error(ImplicitEulerHeun(),dt=0.01) for i in 1:400)

0.007250951082943943

mean(simple_error(ImplicitEulerHeun(),dt=0.1) for i in 1:400)

0.009393747399189691

Interesting Property

Note that RSwM1 and RSwM2 are not stable on this problem.

sol = solve(generate_stiff_stoch_heat(1.0,1.0,adaptivealg=:RSwM1),SRIW1());

Conclusion

In this problem, the implicit methods do not have a stepsize limit. This is because the stiffness almost entirely deteriministic due to diffusion. In that case, if we do not care about the error too much, the implicit methods dominate. Of course, as the tolerance gets lower there is a tradeoff point where the higher order methods will become more efficient. The explicit methods are clearly stability-bound and thus unless we want an error of like 10^-10 we are better off using an implicit method here.

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/StiffSDE","StochasticHeat.jmd")

Computer Information:

Julia Version 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 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
  LLVM: libLLVM-16.0.6 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
  JULIA_DEPOT_PATH = /home/crackauc/github-runners/amdci8-1/.julia
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/StiffSDE/Project.toml`
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`

And the full manifest:

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  [1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`