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https://github.com/kul-optec/abstractoperators.jl

Abstract operators for large scale optimization in Julia

https://github.com/kul-optec/abstractoperators.jl

Science Score: 13.0%

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Keywords

automatic-differentiation back-propagation derivatives julia-language large-scale optimization

Keywords from Contributors

nonlinear-programming optimization-algorithms
Last synced: 4 months ago · JSON representation

Repository

Abstract operators for large scale optimization in Julia

Basic Info
  • Host: GitHub
  • Owner: kul-optec
  • License: other
  • Language: Julia
  • Default Branch: master
  • Homepage:
  • Size: 950 KB
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  • Stars: 30
  • Watchers: 3
  • Forks: 9
  • Open Issues: 2
  • Releases: 9
Topics
automatic-differentiation back-propagation derivatives julia-language large-scale optimization
Created over 8 years ago · Last pushed almost 2 years ago
Metadata Files
Readme License

README.md

AbstractOperators.jl

Build status codecov

Description

Abstract operators extend the syntax typically used for matrices to linear mappings of arbitrary dimensions and nonlinear functions. Unlike matrices however, abstract operators apply the mappings with specific efficient algorithms that minimize memory requirements. This is particularly useful in iterative algorithms and in first order large-scale optimization algorithms.

Installation

To install the package, hit ] from the Julia command line to enter the package manager, then

julia pkg> add AbstractOperators

Usage

With using AbstractOperators the package imports several methods like multiplication * and adjoint transposition ' (and their in-place methods mul!).

For example, one can create a 2-D Discrete Fourier Transform as follows:

julia julia> A = DFT(3,4) ℱ ℝ^(3, 4) -> ℂ^(3, 4) Here, it can be seen that A has a domain of dimensions size(A,2) = (3,4) and of type domainType(A) = Float64 and a codomain of dimensions size(A,1) = (3,4) and type codomainType(A) = Complex{Float64}.

This linear transformation can be evaluated as follows:

```julia julia> x = randn(3,4); #input matrix

julia> y = A*x 3×4 Array{Complex{Float64},2}: -1.11412+0.0im 3.58654-0.724452im -9.10125+0.0im 3.58654+0.724452im -0.905575+1.98446im 0.441199-0.913338im 0.315788+3.29666im 0.174273+0.318065im -0.905575-1.98446im 0.174273-0.318065im 0.315788-3.29666im 0.441199+0.913338im

julia> mul!(y, A, x) == A*x #in-place evaluation true

julia> all(A'y - *(size(x)...)x .< 1e-12) true

julia> mul!(x, A',y) #in-place evaluation 3×4 Array{Float64,2}: -2.99091 9.45611 -19.799 1.6327 -11.1841 11.2365 -26.3614 11.7261 5.04815 7.61552 -6.00498 6.25586

```

Notice that inputs and outputs are not necessarily Vectors.

It is also possible to combine multiple AbstractOperators using different calculus rules.

For example AbstractOperators can be concatenated horizontally:

```julia julia> B = Eye(Complex{Float64},(3,4)) I ℂ^(3, 4) -> ℂ^(3, 4)

julia> H = [A B] [ℱ,I] ℝ^(3, 4) ℂ^(3, 4) -> ℂ^(3, 4) ```

In this case H has a domain of dimensions size(H,2) = ((3, 4), (3, 4)) and type domainType(H) = (Float64, Complex{Float64}).

When an AbstractOperators have multiple domains, this must be multiplied using an ArrayPartition (using RecursiveArrayTools with corresponding size and domain, for example:

```julia julia> using RecursiveArrayTools

julia> H*ArrayPartition(x, complex(x)) 3×4 Array{Complex{Float64},2}: -16.3603+0.0im 52.4946-8.69342im -129.014+0.0im 44.6712+8.69342im -22.051+23.8135im 16.5309-10.9601im -22.5719+39.5599im 13.8174+3.81678im -5.81874-23.8135im 9.70679-3.81678im -2.21552-39.5599im 11.5502+10.9601im ```

Similarly, when an AbstractOperators have multiple codomains, this will return an ArrayPartition, for example: ```julia julia> V = VCAT(Eye(3,3),FiniteDiff((3,3))) [I;δx] ℝ^(3, 3) -> ℝ^(3, 3) ℝ^(2, 3)

julia> V*ones(3,3) ([1.0 1.0 1.0; 1.0 1.0 1.0; 1.0 1.0 1.0], [0.0 0.0 0.0; 0.0 0.0 0.0])

```

A list of the available AbstractOperators and calculus rules can be found in the documentation.

Related packages

Credits

AbstractOperators.jl is developed by Niccolò Antonello and Lorenzo Stella at KU Leuven, ESAT/Stadius,

Owner

  • Name: OPTEC
  • Login: kul-optec
  • Kind: organization

KU Leuven Center of Excellence: Optimization in Engineering

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juliahub.com: AbstractOperators

Abstract operators for large scale optimization in Julia

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