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SDE Basic Work-Precision Diagrams

SDE Work-Precision Diagrams

In this notebook we will run some simple work-precision diagrams for the SDE integrators. These problems are additive and diagonal noise SDEs which can utilize the specialized Rossler methods. These problems are very well-behaved, meaning that adaptive timestepping should not be a significant advantage (unlike more difficult and realistic problems). Thus these tests will measure both the efficiency gains of the Rossler methods along with the overhead of adaptivity.

using StochasticDiffEq, Plots, DiffEqDevTools, SDEProblemLibrary
import SDEProblemLibrary: prob_sde_additivesystem,
            prob_sde_additive, prob_sde_2Dlinear, prob_sde_linear, prob_sde_wave
gr()
const N = 1000
1000

In this notebook, the error that will be measured is the strong error. The strong error is defined as

$

E = \mathbb{E}[Y_\delta(t) - Y(t)] $

where $Y_\delta$ is the numerical approximation to $Y$. This is the same as saying, for a given Wiener trajectory $W(t)$, how well does the numerical trajectory match the real trajectory? Note that this is not how well the mean or other moments match the true mean/variance/etc. (that's the weak error), this is how close the trajectory is to the true trajectory which is a stronger notion. In a sense, this is measuring convergence, rather than just convergence in distribution.

Additive Noise Problem

\begin{equation} dX{t}=\left(\frac{\beta}{\sqrt{1+t}}-\frac{1}{2\left(1+t\right)}X{t}\right)dt+\frac{\alpha\beta}{\sqrt{1+t}}dW{t},\thinspace\thinspace\thinspace X{0}=\frac{1}{2} \end{equation} where $\alpha=\frac{1}{10}$ and $\beta=\frac{1}{20}$. Actual Solution: \begin{equation} X{t}=\frac{1}{\sqrt{1+t}}X{0}+\frac{\beta}{\sqrt{1+t}}\left(t+\alpha W_{t}\right). \end{equation}

First let's solve this using a system of SDEs, repeating this same problem 4 times.

prob = prob_sde_additivesystem
prob = remake(prob,tspan=(0.0,1.0))

reltols = 1.0 ./ 10.0 .^ (1:5)
abstols = reltols#[0.0 for i in eachindex(reltols)]
setups = [Dict(:alg=>SRIW1())
          Dict(:alg=>EM(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1))
          Dict(:alg=>RKMil(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          Dict(:alg=>SRIW1(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          Dict(:alg=>SRA1(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          Dict(:alg=>SRA1())
          ]
names = ["SRIW1","EM","RKMil","SRIW1 Fixed","SRA1 Fixed","SRA1"]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;numruns=N,names=names,maxiters=1e7,error_estimate=:l2)
plot(wp)

prob = prob_sde_additivesystem
prob = remake(prob,tspan=(0.0,1.0))

reltols = 1.0 ./ 10.0 .^ (1:5)
abstols = reltols#[0.0 for i in eachindex(reltols)]
setups = [
          Dict(:alg=>SRA1())
          Dict(:alg=>SRA2())
          Dict(:alg=>SRA3())
          Dict(:alg=>SOSRA())
          Dict(:alg=>SOSRA2())
          ]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;numruns=N,maxiters=1e7,error_estimate=:l2)
plot(wp)

Now as a scalar SDE.

prob = prob_sde_additive
prob = remake(prob,tspan=(0.0,1.0))

reltols = 1.0 ./ 10.0 .^ (1:5)
abstols = reltols#[0.0 for i in eachindex(reltols)]



setups = [Dict(:alg=>SRIW1())
          Dict(:alg=>EM(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1))
          Dict(:alg=>RKMil(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          Dict(:alg=>SRIW1(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          Dict(:alg=>SRA1(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          Dict(:alg=>SRA1())
          ]
names = ["SRIW1","EM","RKMil","SRIW1 Fixed","SRA1 Fixed","SRA1"]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;numruns=N,names=names,maxiters=1e7,error_estimate=:l2)
plot(wp)

prob = prob_sde_additive
prob = remake(prob,tspan=(0.0,1.0))

reltols = 1.0 ./ 10.0 .^ (1:5)
abstols = reltols#[0.0 for i in eachindex(reltols)]
setups = [
          Dict(:alg=>SRA1())
          Dict(:alg=>SRA2())
          Dict(:alg=>SRA3())
          Dict(:alg=>SOSRA())
          Dict(:alg=>SOSRA2())
          ]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;numruns=N,error_estimate=:l2)
plot(wp)

Diagonal Noise

We will use a 4x2 matrix of independent linear SDEs (also known as the Black-Scholes equation)

\begin{equation} dX{t}=\alpha X{t}dt+\beta X{t}dW{t},\thinspace\thinspace\thinspace X{0}=\frac{1}{2} \end{equation} where $\alpha=\frac{1}{10}$ and $\beta=\frac{1}{20}$. Actual Solution: \begin{equation} X{t}=X{0}e^{\left(\beta-\frac{\alpha^{2}}{2}\right)t+\alpha W{t}}. \end{equation}

prob = prob_sde_2Dlinear
prob = remake(prob,tspan=(0.0,1.0))

reltols = 1.0 ./ 10.0 .^ (1:5)
abstols = reltols#[0.0 for i in eachindex(reltols)]

setups = [Dict(:alg=>SRIW1())
          Dict(:alg=>EM(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1))
          Dict(:alg=>RKMil(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          Dict(:alg=>SRIW1(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          ]
names = ["SRIW1","EM","RKMil","SRIW1 Fixed"]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;numruns=N,names=names,maxiters=1e7,error_estimate=:l2)
plot(wp)
Error: BoundsError: attempt to access 4-element Vector{Float64} at index [5
]
prob = prob_sde_2Dlinear
prob = remake(prob,tspan=(0.0,1.0))

reltols = 1.0 ./ 10.0 .^ (1:5)
abstols = reltols#[0.0 for i in eachindex(reltols)]

setups = [Dict(:alg=>EM(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 2))
          Dict(:alg=>RKMil(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 2),:adaptive=>false)
          Dict(:alg=>SRI())
          Dict(:alg=>SRIW1())
          Dict(:alg=>SRIW2())
          Dict(:alg=>SOSRI())
          Dict(:alg=>SOSRI2())
          ]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;numruns=N,maxiters=1e7,error_estimate=:l2)
plot(wp)
Error: BoundsError: attempt to access 0-element Vector{Matrix{Float64}} at 
index [0]

Now just the scalar Black-Scholes

prob = prob_sde_linear
prob = remake(prob,tspan=(0.0,1.0))

reltols = 1.0 ./ 10.0 .^ (1:5)
abstols = reltols#[0.0 for i in eachindex(reltols)]

setups = [Dict(:alg=>SRIW1())
          Dict(:alg=>EM(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1))
          Dict(:alg=>RKMil(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          Dict(:alg=>SRIW1(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          ]
names = ["SRIW1","EM","RKMil","SRIW1 Fixed"]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;numruns=N,names=names,maxiters=1e7,error_estimate=:l2)
plot(wp)
Error: BoundsError: attempt to access 4-element Vector{Float64} at index [5
]
setups = [Dict(:alg=>EM(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 2))
          Dict(:alg=>RKMil(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 2),:adaptive=>false)
          Dict(:alg=>SRI())
          Dict(:alg=>SRIW1())
          Dict(:alg=>SRIW2())
          Dict(:alg=>SOSRI())
          Dict(:alg=>SOSRI2())
          ]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;numruns=N,maxiters=1e7,error_estimate=:l2)
plot(wp)

Now a scalar wave SDE:

\begin{equation} dX{t}=-\left(\frac{1}{10}\right)^{2}\sin\left(X{t}\right)\cos^{3}\left(X{t}\right)dt+\frac{1}{10}\cos^{2}\left(X{t}\right)dW{t},\thinspace\thinspace\thinspace X{0}=\frac{1}{2} \end{equation} Actual Solution: \begin{equation} X{t}=\arctan\left(\frac{1}{10}W{t}+\tan\left(X_{0}\right)\right). \end{equation}

prob = prob_sde_wave
prob = remake(prob,tspan=(0.0,1.0))

reltols = 1.0 ./ 10.0 .^ (1:5)
abstols = reltols#[0.0 for i in eachindex(reltols)]

setups = [Dict(:alg=>SRIW1())
          Dict(:alg=>EM(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1))
          Dict(:alg=>RKMil(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          Dict(:alg=>SRIW1(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 1),:adaptive=>false)
          ]
names = ["SRIW1","EM","RKMil","SRIW1 Fixed"]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;numruns=N,names=names,maxiters=1e7,error_estimate=:l2)
plot(wp)
Error: BoundsError: attempt to access 4-element Vector{Float64} at index [5
]

Note that in this last problem, the adaptivity algorithm accurately detects that the error is already low enough, and does not increase the number of steps as the tolerance drops further.

setups = [Dict(:alg=>EM(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 2))
          Dict(:alg=>RKMil(),:dts=>1.0./5.0.^((1:length(reltols)) .+ 2),:adaptive=>false)
          Dict(:alg=>SRI())
          Dict(:alg=>SRIW1())
          Dict(:alg=>SRIW2())
          Dict(:alg=>SOSRI())
          Dict(:alg=>SOSRI2())
          ]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;numruns=N,maxiters=1e7,error_estimate=:l2)
plot(wp)

Conclusion

The RSwM3 adaptivity algorithm does not appear to have any significant overhead even on problems which do not necessitate adaptive timestepping. The tolerance clearly In addition, the Rossler methods are shown to be orders of magnitude more efficient and should be used whenever applicable. The Oval2 tests show that these results are only magnified as the problem difficulty increases.

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","BasicSDEWorkPrecision.jmd")

Computer Information:

Julia Version 1.9.4
Commit 8e5136fa297 (2023-11-14 08:46 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-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/NonStiffSDE/Project.toml`
  [f3b72e0c] DiffEqDevTools v2.42.0
  [65888b18] ParameterizedFunctions v5.16.0
  [91a5bcdd] Plots v1.39.0
  [c72e72a9] SDEProblemLibrary v0.1.6
  [31c91b34] SciMLBenchmarks v0.1.3
  [789caeaf] StochasticDiffEq v6.63.2

And the full manifest:

Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/NonStiffSDE/Manifest.toml`
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  [410a4b4d] Tricks v0.1.8
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.5.1
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.19.0
  [45397f5d] UnitfulLatexify v1.6.3
  [a7c27f48] Unityper v0.1.5
  [41fe7b60] Unzip v0.2.0
  [3d5dd08c] VectorizationBase v0.21.65
  [81def892] VersionParsing v1.3.0
  [19fa3120] VertexSafeGraphs v0.2.0
  [44d3d7a6] Weave v0.10.12
  [ddb6d928] YAML v0.4.9
  [c2297ded] ZMQ v1.2.2
  [6e34b625] Bzip2_jll v1.0.8+0
  [83423d85] Cairo_jll v1.16.1+1
  [2702e6a9] EpollShim_jll v0.0.20230411+0
  [2e619515] Expat_jll v2.5.0+0
  [b22a6f82] FFMPEG_jll v4.4.4+1
  [a3f928ae] Fontconfig_jll v2.13.93+0
  [d7e528f0] FreeType2_jll v2.13.1+0
  [559328eb] FriBidi_jll v1.0.10+0
  [0656b61e] GLFW_jll v3.3.8+0
  [d2c73de3] GR_jll v0.72.10+0
  [78b55507] Gettext_jll v0.21.0+0
  [f8c6e375] Git_jll v2.42.0+0
  [7746bdde] Glib_jll v2.76.5+0
  [3b182d85] Graphite2_jll v1.3.14+0
  [2e76f6c2] HarfBuzz_jll v2.8.1+1
  [1d5cc7b8] IntelOpenMP_jll v2024.0.0+0
  [aacddb02] JpegTurbo_jll v3.0.1+0
  [c1c5ebd0] LAME_jll v3.100.1+0
  [88015f11] LERC_jll v3.0.0+1
  [1d63c593] LLVMOpenMP_jll v15.0.4+0
  [dd4b983a] LZO_jll v2.10.1+0
⌅ [e9f186c6] Libffi_jll v3.2.2+1
  [d4300ac3] Libgcrypt_jll v1.8.7+0
  [7e76a0d4] Libglvnd_jll v1.6.0+0
  [7add5ba3] Libgpg_error_jll v1.42.0+0
  [94ce4f54] Libiconv_jll v1.17.0+0
  [4b2f31a3] Libmount_jll v2.35.0+0
  [89763e89] Libtiff_jll v4.5.1+1
  [38a345b3] Libuuid_jll v2.36.0+0
  [856f044c] MKL_jll v2024.0.0+0
  [e7412a2a] Ogg_jll v1.3.5+1
  [458c3c95] OpenSSL_jll v3.0.12+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 v6.5.3+1
  [f50d1b31] Rmath_jll v0.4.0+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.21.0+1
  [2381bf8a] Wayland_protocols_jll v1.25.0+0
  [02c8fc9c] XML2_jll v2.12.2+0
  [aed1982a] XSLT_jll v1.1.34+0
  [ffd25f8a] XZ_jll v5.4.5+0
  [f67eecfb] Xorg_libICE_jll v1.0.10+1
  [c834827a] Xorg_libSM_jll v1.2.3+0
  [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
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.4+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
  [35ca27e7] eudev_jll v3.2.9+0
  [214eeab7] fzf_jll v0.43.0+0
  [1a1c6b14] gperf_jll v3.1.1+0
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [2db6ffa8] libevdev_jll v1.11.0+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [36db933b] libinput_jll v1.18.0+0
  [b53b4c65] libpng_jll v1.6.40+0
  [a9144af2] libsodium_jll v1.0.20+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [009596ad] mtdev_jll v1.1.6+0
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+1
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.9.2
  [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 v1.0.5+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [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 v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+0
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`