MultiScaleArrays.jl: High-Performance Matrix Exponentiation and Products

MultiScaleArrays.jl allows you to easily build multiple-scale models that are fully compatible with native Julia scientific computing packages like DifferentialEquations.jl or Optim.jl. These models utilize a tree structure to describe phenomena on multiple scales, but the interface allows you to describe equations on different levels, using aggregations from lower levels to describe complex systems. Their structure allows for complex and dynamic models to be developed with only a small performance difference. In the end, they present themselves as an AbstractArray to standard solvers, allowing them to be used in place of a Vector in any appropriately made Julia package.

Installation

To install MultiScaleArrays.jl, use the Julia package manager:

using Pkg
Pkg.add("MultiScaleArrays")

Example

The usage is best described by an example. Here we build a hierarchy where Embryos contain Tissues which contain Populations which contain Cells, and the cells contain proteins whose concentrations are modeled as simply a vector of numbers (it can be anything linearly indexable).

using MultiScaleArrays
struct Cell{B} <: AbstractMultiScaleArrayLeaf{B}
    values::Vector{B}
end
struct Population{T <: AbstractMultiScaleArray, B <: Number} <: AbstractMultiScaleArray{B}
    nodes::Vector{T}
    values::Vector{B}
    end_idxs::Vector{Int}
end
struct Tissue{T <: AbstractMultiScaleArray, B <: Number} <: AbstractMultiScaleArray{B}
    nodes::Vector{T}
    values::Vector{B}
    end_idxs::Vector{Int}
end
struct Embryo{T <: AbstractMultiScaleArray, B <: Number} <: AbstractMultiScaleArrayHead{B}
    nodes::Vector{T}
    values::Vector{B}
    end_idxs::Vector{Int}
end

This setup defines a type structure that is both a tree and an array. A picture of a possible version is the following:

Let's build a version of this. Using the constructors, we can directly construct leaf types:

cell1 = Cell([1.0; 2.0; 3.0])
cell2 = Cell([4.0; 5.0])

and build types higher up in the hierarchy by using the constuct method. The method is construct(T::AbstractMultiScaleArray, nodes, values), though, if values is not given, it's taken to be empty.

cell3 = Cell([3.0; 2.0; 5.0])
cell4 = Cell([4.0; 6.0])
population = construct(Population, deepcopy([cell1, cell3, cell4]))
population2 = construct(Population, deepcopy([cell1, cell3, cell4]))
population3 = construct(Population, deepcopy([cell1, cell3, cell4]))
tissue1 = construct(Tissue, deepcopy([population, population2, population3])) # Make a Tissue from Populations
tissue2 = construct(Tissue, deepcopy([population2, population, population3]))
embryo = construct(Embryo, deepcopy([tissue1, tissue2])) # Make an embryo from Tissues

Idea

The idea behind MultiScaleArrays is simple. The *DiffEq solvers (OrdinaryDiffEq.jl, StochasticDiffEq.jl, DelayDiffEq.jl, etc.) and native optimization packages like Optim.jl in their efficient in-place form all work with any Julia-defined AbstractArray which has a linear index. Thus, to define our multiscale model, we develop a type which has an efficient linear index. One can think of representing cells with proteins as each being an array with values for each protein. The linear index of the multiscale model would be indexing through each protein of each cell. With proper index overloads, one can define a type such that a[i] does just that, and thus it will work in the differential equation solvers. MultiScaleArrays.jl takes that further by allowing one to recursively define an arbitrary n-level hierarchical model which has efficient indexing structures. The result is a type which models complex behavior, but the standard differential equation solvers will work directly and efficiently on this type, making it easy to develop novel models without having to re-develop advanced adaptive/stiff/stochastic/etc. solving techniques for each new model.

Contributing

Please refer to the SciML ColPrac: Contributor's Guide on Collaborative Practices for Community Packages for guidance on PRs, issues, and other matters relating to contributing to SciML.
See the SciML Style Guide for common coding practices and other style decisions.
There are a few community forums:
- The #diffeq-bridged and #sciml-bridged channels in the Julia Slack
- The #diffeq-bridged and #sciml-bridged channels in the Julia Zulip
- On the Julia Discourse forums
- See also SciML Community page

Reproducibility

The documentation of this SciML package was built using these direct dependencies,

Status `~/work/MultiScaleArrays.jl/MultiScaleArrays.jl/docs/Project.toml`
  [e30172f5] Documenter v1.13.0
  [f9640e96] MultiScaleArrays v1.14.0 `~/work/MultiScaleArrays.jl/MultiScaleArrays.jl`

and using this machine and Julia version.

Julia Version 1.11.5
Commit 760b2e5b739 (2025-04-14 06:53 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)

A more complete overview of all dependencies and their versions is also provided.

Status `~/work/MultiScaleArrays.jl/MultiScaleArrays.jl/docs/Manifest.toml`
  [47edcb42] ADTypes v1.15.0
  [a4c015fc] ANSIColoredPrinters v0.0.1
  [1520ce14] AbstractTrees v0.4.5
  [7d9f7c33] Accessors v0.1.42
  [79e6a3ab] Adapt v4.3.0
  [66dad0bd] AliasTables v1.1.3
  [ec485272] ArnoldiMethod v0.4.0
  [4fba245c] ArrayInterface v7.19.0
  [4c555306] ArrayLayouts v1.11.1
  [62783981] BitTwiddlingConvenienceFunctions v0.1.6
  [70df07ce] BracketingNonlinearSolve v1.3.0
  [2a0fbf3d] CPUSummary v0.2.6
  [d360d2e6] ChainRulesCore v1.25.2
  [fb6a15b2] CloseOpenIntervals v0.1.13
  [944b1d66] CodecZlib v0.7.8
  [38540f10] CommonSolve v0.2.4
  [bbf7d656] CommonSubexpressions v0.3.1
  [f70d9fcc] CommonWorldInvalidations v1.0.0
  [34da2185] Compat v4.16.0
  [a33af91c] CompositionsBase v0.1.2
  [2569d6c7] ConcreteStructs v0.2.3
  [187b0558] ConstructionBase v1.6.0
  [adafc99b] CpuId v0.3.1
  [a8cc5b0e] Crayons v4.1.1
  [9a962f9c] DataAPI v1.16.0
  [864edb3b] DataStructures v0.18.22
  [e2d170a0] DataValueInterfaces v1.0.0
  [2b5f629d] DiffEqBase v6.176.0
  [459566f4] DiffEqCallbacks v4.8.0
  [77a26b50] DiffEqNoiseProcess v5.24.1
  [163ba53b] DiffResults v1.1.0
  [b552c78f] DiffRules v1.15.1
  [a0c0ee7d] DifferentiationInterface v0.7.1
  [b4f34e82] Distances v0.10.12
  [31c24e10] Distributions v0.25.120
  [ffbed154] DocStringExtensions v0.9.5
  [e30172f5] Documenter v1.13.0
  [4e289a0a] EnumX v1.0.5
  [f151be2c] EnzymeCore v0.8.12
  [d4d017d3] ExponentialUtilities v1.27.0
  [e2ba6199] ExprTools v0.1.10
  [55351af7] ExproniconLite v0.10.14
  [7034ab61] FastBroadcast v0.3.5
  [9aa1b823] FastClosures v0.3.2
  [442a2c76] FastGaussQuadrature v1.0.2
  [a4df4552] FastPower v1.1.3
  [1a297f60] FillArrays v1.13.0
  [6a86dc24] FiniteDiff v2.27.0
⌅ [f6369f11] ForwardDiff v0.10.38
  [069b7b12] FunctionWrappers v1.1.3
  [77dc65aa] FunctionWrappersWrappers v0.1.3
  [d9f16b24] Functors v0.5.2
  [46192b85] GPUArraysCore v0.2.0
  [c145ed77] GenericSchur v0.5.5
  [d7ba0133] Git v1.4.0
  [86223c79] Graphs v1.13.0
  [34004b35] HypergeometricFunctions v0.3.28
  [b5f81e59] IOCapture v0.2.5
  [615f187c] IfElse v0.1.1
  [d25df0c9] Inflate v0.1.5
  [3587e190] InverseFunctions v0.1.17
  [92d709cd] IrrationalConstants v0.2.4
  [82899510] IteratorInterfaceExtensions v1.0.0
  [692b3bcd] JLLWrappers v1.7.0
  [682c06a0] JSON v0.21.4
  [ae98c720] Jieko v0.2.1
  [ccbc3e58] JumpProcesses v9.16.0
  [ba0b0d4f] Krylov v0.10.1
  [b964fa9f] LaTeXStrings v1.4.0
  [10f19ff3] LayoutPointers v0.1.17
  [0e77f7df] LazilyInitializedFields v1.3.0
  [5078a376] LazyArrays v2.6.1
  [2d8b4e74] LevyArea v1.0.0
  [87fe0de2] LineSearch v0.1.4
  [d3d80556] LineSearches v7.4.0
  [7ed4a6bd] LinearSolve v3.18.1
  [2ab3a3ac] LogExpFunctions v0.3.29
  [1914dd2f] MacroTools v0.5.16
  [d125e4d3] ManualMemory v0.1.8
  [d0879d2d] MarkdownAST v0.1.2
  [bb5d69b7] MaybeInplace v0.1.4
  [e1d29d7a] Missings v1.2.0
  [2e0e35c7] Moshi v0.3.5
  [46d2c3a1] MuladdMacro v0.2.4
  [f9640e96] MultiScaleArrays v1.14.0 `~/work/MultiScaleArrays.jl/MultiScaleArrays.jl`
  [d41bc354] NLSolversBase v7.10.0
  [2774e3e8] NLsolve v4.5.1
  [77ba4419] NaNMath v1.1.3
  [8913a72c] NonlinearSolve v4.9.0
  [be0214bd] NonlinearSolveBase v1.12.0
  [5959db7a] NonlinearSolveFirstOrder v1.5.0
  [9a2c21bd] NonlinearSolveQuasiNewton v1.6.0
  [26075421] NonlinearSolveSpectralMethods v1.2.0
  [429524aa] Optim v1.13.2
  [bac558e1] OrderedCollections v1.8.1
  [1dea7af3] OrdinaryDiffEq v6.98.0
  [89bda076] OrdinaryDiffEqAdamsBashforthMoulton v1.2.0
  [6ad6398a] OrdinaryDiffEqBDF v1.6.0
  [bbf590c4] OrdinaryDiffEqCore v1.26.1
  [50262376] OrdinaryDiffEqDefault v1.4.0
  [4302a76b] OrdinaryDiffEqDifferentiation v1.10.0
  [9286f039] OrdinaryDiffEqExplicitRK v1.1.0
  [e0540318] OrdinaryDiffEqExponentialRK v1.4.0
  [becaefa8] OrdinaryDiffEqExtrapolation v1.5.0
  [5960d6e9] OrdinaryDiffEqFIRK v1.12.0
  [101fe9f7] OrdinaryDiffEqFeagin v1.1.0
  [d3585ca7] OrdinaryDiffEqFunctionMap v1.1.1
  [d28bc4f8] OrdinaryDiffEqHighOrderRK v1.1.0
  [9f002381] OrdinaryDiffEqIMEXMultistep v1.3.0
  [521117fe] OrdinaryDiffEqLinear v1.3.0
  [1344f307] OrdinaryDiffEqLowOrderRK v1.2.0
  [b0944070] OrdinaryDiffEqLowStorageRK v1.3.0
  [127b3ac7] OrdinaryDiffEqNonlinearSolve v1.10.0
  [c9986a66] OrdinaryDiffEqNordsieck v1.1.0
  [5dd0a6cf] OrdinaryDiffEqPDIRK v1.3.1
  [5b33eab2] OrdinaryDiffEqPRK v1.1.0
  [04162be5] OrdinaryDiffEqQPRK v1.1.0
  [af6ede74] OrdinaryDiffEqRKN v1.1.0
  [43230ef6] OrdinaryDiffEqRosenbrock v1.11.0
  [2d112036] OrdinaryDiffEqSDIRK v1.3.0
  [669c94d9] OrdinaryDiffEqSSPRK v1.3.0
  [e3e12d00] OrdinaryDiffEqStabilizedIRK v1.3.0
  [358294b1] OrdinaryDiffEqStabilizedRK v1.1.0
  [fa646aed] OrdinaryDiffEqSymplecticRK v1.3.0
  [b1df2697] OrdinaryDiffEqTsit5 v1.1.0
  [79d7bb75] OrdinaryDiffEqVerner v1.2.0
  [90014a1f] PDMats v0.11.35
  [d96e819e] Parameters v0.12.3
  [69de0a69] Parsers v2.8.3
  [e409e4f3] PoissonRandom v0.4.4
  [f517fe37] Polyester v0.7.18
  [1d0040c9] PolyesterWeave v0.2.2
  [85a6dd25] PositiveFactorizations v0.2.4
  [d236fae5] PreallocationTools v0.4.27
⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.4.3
  [08abe8d2] PrettyTables v2.4.0
  [43287f4e] PtrArrays v1.3.0
  [1fd47b50] QuadGK v2.11.2
  [74087812] Random123 v1.7.1
  [e6cf234a] RandomNumbers v1.6.0
  [3cdcf5f2] RecipesBase v1.3.4
  [731186ca] RecursiveArrayTools v3.33.0
  [189a3867] Reexport v1.2.2
  [2792f1a3] RegistryInstances v0.1.0
  [ae029012] Requires v1.3.1
  [ae5879a3] ResettableStacks v1.1.1
  [79098fc4] Rmath v0.8.0
  [7e49a35a] RuntimeGeneratedFunctions v0.5.15
  [94e857df] SIMDTypes v0.1.0
  [0bca4576] SciMLBase v2.102.1
  [19f34311] SciMLJacobianOperators v0.1.6
  [c0aeaf25] SciMLOperators v1.3.1
  [53ae85a6] SciMLStructures v1.7.0
  [efcf1570] Setfield v1.1.2
  [727e6d20] SimpleNonlinearSolve v2.5.0
  [699a6c99] SimpleTraits v0.9.4
  [ce78b400] SimpleUnPack v1.1.0
  [a2af1166] SortingAlgorithms v1.2.1
  [0a514795] SparseMatrixColorings v0.4.21
  [276daf66] SpecialFunctions v2.5.1
  [aedffcd0] Static v1.2.0
  [0d7ed370] StaticArrayInterface v1.8.0
  [90137ffa] StaticArrays v1.9.13
  [1e83bf80] StaticArraysCore v1.4.3
  [10745b16] Statistics v1.11.1
  [82ae8749] StatsAPI v1.7.1
  [2913bbd2] StatsBase v0.34.5
  [4c63d2b9] StatsFuns v1.5.0
  [789caeaf] StochasticDiffEq v6.80.0
  [7792a7ef] StrideArraysCore v0.5.7
  [892a3eda] StringManipulation v0.4.1
  [2efcf032] SymbolicIndexingInterface v0.3.41
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.12.1
  [8290d209] ThreadingUtilities v0.5.5
  [a759f4b9] TimerOutputs v0.5.29
  [3bb67fe8] TranscodingStreams v0.11.3
  [781d530d] TruncatedStacktraces v1.4.0
  [3a884ed6] UnPack v1.0.2
  [2e619515] Expat_jll v2.6.5+0
  [f8c6e375] Git_jll v2.50.0+0
  [1d5cc7b8] IntelOpenMP_jll v2025.0.4+0
  [94ce4f54] Libiconv_jll v1.18.0+0
  [856f044c] MKL_jll v2025.0.1+1
  [9bd350c2] OpenSSH_jll v10.0.1+0
  [458c3c95] OpenSSL_jll v3.5.0+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [f50d1b31] Rmath_jll v0.5.1+0
  [1317d2d5] oneTBB_jll v2022.0.0+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
  [1a1011a3] SharedArrays 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 ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`

You can also download the manifest file and the project file.