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

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);
11.436644 seconds (4.99 M allocations: 868.432 MiB, 1.42% gc time, 16.57% 
compilation time)

Highest dt

Let's try to find the highest possible dt:

@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),SRIW1());
1.758056 seconds (1.11 M allocations: 145.601 MiB, 20.25% compilation tim
e)
@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),SRIW1(),progress=true,adaptive=false,dt=0.00005);
1.009126 seconds (438.75 k allocations: 79.354 MiB, 9.67% gc time, 20.97%
 compilation time)
@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),EM(),progress=true,adaptive=false,dt=0.00005);
1.877603 seconds (3.29 M allocations: 217.135 MiB, 77.77% compilation tim
e)
@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),ImplicitRKMil(),progress=true,dt=0.1);
3.318805 seconds (8.16 M allocations: 543.324 MiB, 4.49% gc time, 99.74% 
compilation time)
@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),ImplicitRKMil(),progress=true,dt=0.01);
0.004084 seconds (3.06 k allocations: 655.219 KiB)
@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),ImplicitRKMil(),progress=true,dt=0.001);
0.004095 seconds (3.11 k allocations: 658.406 KiB)
@time sol = solve(generate_stiff_stoch_heat(1.0,1.0),ImplicitEM(),progress=true,dt=0.001);
2.902864 seconds (7.59 M allocations: 507.324 MiB, 2.82% gc time, 99.80% 
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...);
    sum(abs2,sol[end] - exp(A*sol.t[end]+sol.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)
3.1775927041537334e-9
mean(simple_error(ImplicitRKMil(interpretation=:Stratanovich),dt=0.1) for i in 1:400)
Error: Alg interpretation invalid. Use either :Ito or :Stratonovich
mean(simple_error(ImplicitRKMil(interpretation=:Stratanovich),dt=0.01) for i in 1:400)
Error: Alg interpretation invalid. Use either :Ito or :Stratonovich
mean(simple_error(ImplicitRKMil(interpretation=:Stratanovich),dt=0.001) for i in 1:400)
Error: Alg interpretation invalid. Use either :Ito or :Stratonovich
mean(simple_error(ImplicitEulerHeun(),dt=0.001) for i in 1:400)
0.0011785552520833247
mean(simple_error(ImplicitEulerHeun(),dt=0.01) for i in 1:400)
0.001041197390281144
mean(simple_error(ImplicitEulerHeun(),dt=0.1) for i in 1:400)
0.0009875023106019514

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 1.7.3
Commit 742b9abb4d (2022-05-06 12:58 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, znver2)
Environment:
  JULIA_CPU_THREADS = 128
  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
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  [789caeaf] StochasticDiffEq v6.50.1
  [37e2e46d] LinearAlgebra
  [9a3f8284] Random
  [10745b16] Statistics

And the full manifest:

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  [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 v0.9.1+5
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [b27032c2] LibCURL v0.6.3
  [76f85450] LibGit2
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  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
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  [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
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  [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