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 construct 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 v1.17.0
  [f9640e96] MultiScaleArrays v1.24.0 `~/work/MultiScaleArrays.jl/MultiScaleArrays.jl`

and using this machine and Julia version.

Julia Version 1.12.6
Commit 15346901f00 (2026-04-09 19:20 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 9V74 80-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver4)
  GC: Built with stock GC
Threads: 1 default, 1 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`
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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.