diff --git a/previews/PR77/.documenter-siteinfo.json b/previews/PR77/.documenter-siteinfo.json new file mode 100644 index 00000000..9ca3f95d --- /dev/null +++ b/previews/PR77/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.10.2","generation_timestamp":"2024-03-11T20:44:34","documenter_version":"1.3.0"}} \ No newline at end of file diff --git a/previews/PR77/API/regularization/index.html b/previews/PR77/API/regularization/index.html new file mode 100644 index 00000000..8a4cff87 --- /dev/null +++ b/previews/PR77/API/regularization/index.html @@ -0,0 +1,18 @@ + +Regularization Terms · RegularizedLeastSquares.jl

API for Regularizers

This page contains documentation of the public API of the RegularizedLeastSquares. In the Julia REPL one can access this documentation by entering the help mode with ?

RegularizedLeastSquares.L21Regularization — Type
L21Regularization

Regularization term implementing the proximal map for group-soft-thresholding.

Arguments

  • λ - regularization paramter

Keywords

  • slices=1 - number of elements per group
source
RegularizedLeastSquares.LLRRegularization — Type
LLRRegularization

Regularization term implementing the proximal map for locally low rank (LLR) regularization using singular-value-thresholding.

Arguments

  • λ - regularization paramter

Keywords

  • shape::Tuple{Int}=[] - dimensions of the image
  • blockSize::Tuple{Int}=[2;2] - size of patches to perform singular value thresholding on
  • randshift::Bool=true - randomly shifts the patches to ensure translation invariance
source
RegularizedLeastSquares.NuclearRegularization — Type
NuclearRegularization

Regularization term implementing the proximal map for singular value soft-thresholding.

Arguments:

  • λ - regularization paramter

Keywords

  • svtShape::NTuple - size of the underlying matrix
source
RegularizedLeastSquares.TVRegularization — Type
TVRegularization

Regularization term implementing the proximal map for TV regularization. Calculated with the Condat algorithm if the TV is calculated only along one real-valued dimension and with the Fast Gradient Projection algorithm otherwise.

Reference for the Condat algorithm: https://lcondat.github.io/publis/Condat-fast_TV-SPL-2013.pdf

Reference for the FGP algorithm: A. Beck and T. Teboulle, "Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems", IEEE Trans. Image Process. 18(11), 2009

Arguments

  • λ::T - regularization parameter

Keywords

  • shape::NTuple - size of the underlying image
  • dims - Dimension to perform the TV along. If Integer, the Condat algorithm is called, and the FDG algorithm otherwise.
  • iterationsTV=20 - number of FGP iterations
source

Projection Regularization

Nested Regularization

RegularizedLeastSquares.innerreg — Method
innerreg(reg::AbstractNestedRegularization)

return the inner regularization term of reg. Nested regularization terms also implement the iteration interface.

source

Scaled Regularization

Misc. Nested Regularization

RegularizedLeastSquares.MaskedRegularization — Type
MaskedRegularization

Nested regularization term that only applies prox! and norm to elements of x for which the mask is true.

Examples

julia> positive = PositiveRegularization();
+
+julia> masked = MaskedRegularization(reg, [true, false, true, false]);
+
+julia> prox!(masked, fill(-1, 4))
+4-element Vector{Float64}:
+  0.0
+ -1.0
+  0.0
+ -1.0
source
RegularizedLeastSquares.TransformedRegularization — Type
TransformedRegularization(reg, trafo)

Nested regularization term that applies prox! or norm on z = trafo * x and returns (inplace) x = adjoint(trafo) * z.

Example

julia> core = L1Regularization(0.8)
+L1Regularization{Float64}(0.8)
+
+julia> wop = WaveletOp(Float32, shape = (32,32));
+
+julia> reg = TransformedRegularization(core, wop);
+
+julia> prox!(reg, randn(32*32)); # Apply soft-thresholding in Wavelet domain
source
RegularizedLeastSquares.PlugAndPlayRegularization — Type
    PlugAndPlayRegularization

Regularization term implementing a given plug-and-play proximal mapping. The actual regularization term is indirectly defined by the learned proximal mapping and as such there is no norm implemented.

Arguments

  • λ - regularization paramter

Keywords

  • model - model applied to the image
  • shape - dimensions of the image
  • input_transform - transform of image before model
source

Miscellaneous Functions

RegularizedLeastSquares.prox! — Method
prox!(reg::AbstractParameterizedRegularization, x)

perform the proximal mapping defined by reg on x. Uses the regularization parameter defined for reg.

source
RegularizedLeastSquares.prox! — Method
prox!(regType::Type{<:AbstractParameterizedRegularization}, x, λ; kwargs...)

construct a regularization term of type regType with given λ and kwargs and apply its prox! on x

source
LinearAlgebra.norm — Method
norm(reg::AbstractParameterizedRegularization, x)

returns the value of the reg regularization term on x. Uses the regularization parameter defined for reg.

source
LinearAlgebra.norm — Method
norm(regType::Type{<:AbstractParameterizedRegularization}, x, λ; kwargs...)

construct a regularization term of type regType with given λ and kwargs and apply its norm on x

source
diff --git a/previews/PR77/API/solvers/index.html b/previews/PR77/API/solvers/index.html new file mode 100644 index 00000000..934fd3b9 --- /dev/null +++ b/previews/PR77/API/solvers/index.html @@ -0,0 +1,49 @@ + +Solvers · RegularizedLeastSquares.jl

API for Solvers

This page contains documentation of the public API of the RegularizedLeastSquares. In the Julia REPL one can access this documentation by entering the help mode with ?

solve!

RegularizedLeastSquares.solve! — Method
solve!(solver::AbstractLinearSolver, b; x0 = 0, callbacks = (_, _) -> nothing)

Solves an inverse problem for the data vector b using solver.

Required Arguments

  • solver::AbstractLinearSolver - linear solver (e.g., ADMM or FISTA), containing forward/normal operator and regularizer
  • b::AbstractVector - data vector if A was supplied to the solver, back-projection of the data otherwise

Optional Keyword Arguments

  • x0::AbstractVector - initial guess for the solution; default is zero
  • callbacks - (optionally a vector of) function or callable struct that takes the two arguments callback(solver, iteration) and, e.g., stores, prints, or plots the intermediate solutions or convergence parameters. Be sure not to modify solver or iteration in the callback function as this would japaridze convergence. The default does nothing.

Examples

The optimization problem

\[ argmin_x ||Ax - b||_2^2 + λ ||x||_1\]

can be solved with the following lines of code:

julia> using RegularizedLeastSquares
+
+julia> A = [0.831658  0.96717
+            0.383056  0.39043
+            0.820692  0.08118];
+
+julia> x = [0.5932234523399985; 0.2697534345340015];
+
+julia> b = A * x;
+
+julia> S = ADMM(A);
+
+julia> x_approx = solve!(S, b)
+2-element Vector{Float64}:
+ 0.5932234523399984
+ 0.26975343453400163

Here, we use L1Regularization, which is default for ADMM. All regularization options can be found in API for Regularizers.

The following example solves the same problem, but stores the solution x of each interation in tr:

julia> tr = Dict[]
+Dict[]
+
+julia> store_trace!(tr, solver, iteration) = push!(tr, Dict("iteration" => iteration, "x" => solver.x, "beta" => solver.β))
+store_trace! (generic function with 1 method)
+
+julia> x_approx = solve!(S, b; callbacks=(solver, iteration) -> store_trace!(tr, solver, iteration))
+2-element Vector{Float64}:
+ 0.5932234523399984
+ 0.26975343453400163
+
+julia> tr[3]
+Dict{String, Any} with 3 entries:
+  "iteration" => 2
+  "x"         => [0.593223, 0.269753]
+  "beta"      => [1.23152, 0.927611]

The last example show demonstrates how to plot the solution at every 10th iteration and store the solvers convergence metrics:

julia> using Plots
+
+julia> conv = StoreConvergenceCallback()
+
+julia> function plot_trace(solver, iteration)
+         if iteration % 10 == 0
+           display(scatter(solver.x))
+         end
+       end
+plot_trace (generic function with 1 method)
+
+julia> x_approx = solve!(S, b; callbacks = [conv, plot_trace]);

The keyword callbacks allows you to pass a (vector of) callable objects that takes the arguments solver and iteration and prints, stores, or plots intermediate result.

See also StoreSolutionCallback, StoreConvergenceCallback, CompareSolutionCallback for a number of provided callback options.

source

ADMM

RegularizedLeastSquares.ADMM — Type
ADMM(A; AHA = A'*A, precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, vary_rho = :none, iterations = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)
+ADMM( ; AHA = ,     precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, vary_rho = :none, iterations = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)

Creates an ADMM object for the forward operator A or normal operator AHA.

Required Arguments

  • A - forward operator

OR

  • AHA - normal operator (as a keyword argument)

Optional Keyword Arguments

  • AHA - normal operator is optional if A is supplied
  • precon - preconditionner for the internal CG algorithm
  • reg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms
  • regTrafo - transformation to a space in which reg is applied; if reg is a vector, regTrafo has to be a vector of the same length. Use opEye(eltype(AHA), size(AHA,1)) if no transformation is desired.
  • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
  • rho::Real - penalty of the augmented Lagrangian
  • vary_rho::Symbol - vary rho to balance primal and dual feasibility; options :none, :balance, :PnP
  • iterations::Int - maximum number of (outer) ADMM iterations
  • iterationsCG::Int - maximum number of (inner) CG iterations
  • absTol::Real - absolute tolerance for stopping criterion
  • relTol::Real - relative tolerance for stopping criterion
  • tolInner::Real - relative tolerance for CG stopping criterion
  • verbose::Bool - print residual in each iteration

ADMM differs from ISTA-type algorithms in the sense that the proximal operation is applied separately from the transformation to the space in which the penalty is applied. This is reflected by the interface which has reg and regTrafo as separate arguments. E.g., for a TV penalty, you should NOT set reg=TVRegularization, but instead use reg=L1Regularization(λ), regTrafo=RegularizedLeastSquares.GradientOp(Float64; shape=(Nx,Ny,Nz)).

See also createLinearSolver, solve!.

source

CGNR

RegularizedLeastSquares.CGNR — Type
CGNR(A; AHA = A' * A, reg = L2Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), weights = similar(AHA, 0), iterations = 10, relTol = eps(real(eltype(AHA))))
+CGNR( ; AHA = ,       reg = L2Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), weights = similar(AHA, 0), iterations = 10, relTol = eps(real(eltype(AHA))))

creates an CGNR object for the forward operator A or normal operator AHA.

Required Arguments

  • A - forward operator

OR

  • AHA - normal operator (as a keyword argument)

Optional Keyword Arguments

  • AHA - normal operator is optional if A is supplied
  • reg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms
  • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
  • weights::AbstactVector - weights for the data term; must be of same length and type as the data term
  • iterations::Int - maximum number of iterations
  • relTol::Real - tolerance for stopping criterion

See also createLinearSolver, solve!.

source

Kaczmarz

RegularizedLeastSquares.Kaczmarz — Type
Kaczmarz(A; reg = L2Regularization(0), normalizeReg = NoNormalization(), weights=nothing, randomized=false, subMatrixFraction=0.15, shuffleRows=false, seed=1234, iterations=10, regMatrix=nothing)

Creates a Kaczmarz object for the forward operator A.

Required Arguments

  • A - forward operator

Optional Keyword Arguments

  • reg::AbstractParameterizedRegularization - regularization term
  • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
  • randomized::Bool - randomize Kacmarz algorithm
  • subMatrixFraction::Real - fraction of rows used in randomized Kaczmarz algorithm
  • shuffleRows::Bool - randomize Kacmarz algorithm
  • seed::Int - seed for randomized algorithm
  • iterations::Int - number of iterations

See also createLinearSolver, solve!.

source

FISTA

RegularizedLeastSquares.FISTA — Type
FISTA(A; AHA=A'*A, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), rho=0.95, normalize_rho=true, theta=1, relTol=eps(real(eltype(AHA))), iterations=50, restart = :none, verbose = false)
+FISTA( ; AHA=,     reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), rho=0.95, normalize_rho=true, theta=1, relTol=eps(real(eltype(AHA))), iterations=50, restart = :none, verbose = false)

creates a FISTA object for the forward operator A or normal operator AHA.

Required Arguments

  • A - forward operator

OR

  • AHA - normal operator (as a keyword argument)

Optional Keyword Arguments

  • AHA - normal operator is optional if A is supplied
  • precon - preconditionner for the internal CG algorithm
  • reg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms
  • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
  • rho::Real - step size for gradient step
  • normalize_rho::Bool - normalize step size by the largest eigenvalue of AHA
  • theta::Real - parameter for predictor-corrector step
  • relTol::Real - tolerance for stopping criterion
  • iterations::Int - maximum number of iterations
  • restart::Symbol - :none, :gradient options for restarting
  • verbose::Bool - print residual in each iteration

See also createLinearSolver, solve!.

source

OptISTA

RegularizedLeastSquares.OptISTA — Type
OptISTA(A; AHA=A'*A, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), rho=0.95, normalize_rho=true, theta=1, relTol=eps(real(eltype(AHA))), iterations=50, verbose = false)
+OptISTA( ; AHA=,     reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), rho=0.95, normalize_rho=true, theta=1, relTol=eps(real(eltype(AHA))), iterations=50, verbose = false)

creates a OptISTA object for the forward operator A or normal operator AHA. OptISTA has a 2x better worst-case bound than FISTA, but actual performance varies by application. It stores 2 extra intermediate variables the size of the image compared to FISTA.

Reference:

  • Uijeong Jang, Shuvomoy Das Gupta, Ernest K. Ryu, "Computer-Assisted Design of Accelerated Composite Optimization Methods: OptISTA," arXiv:2305.15704, 2023, [https://arxiv.org/abs/2305.15704]

Required Arguments

  • A - forward operator

OR

  • AHA - normal operator (as a keyword argument)

Optional Keyword Arguments

  • AHA - normal operator is optional if A is supplied
  • reg::AbstractParameterizedRegularization - regularization term
  • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
  • rho::Real - step size for gradient step
  • normalize_rho::Bool - normalize step size by the largest eigenvalue of AHA
  • theta::Real - parameter for predictor-corrector step
  • relTol::Real - tolerance for stopping criterion
  • iterations::Int - maximum number of iterations
  • verbose::Bool - print residual in each iteration

See also createLinearSolver, solve!.

source

POGM

RegularizedLeastSquares.POGM — Type
POGM(A; AHA = A'*A, reg = L1Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), rho = 0.95, normalize_rho = true, theta = 1, sigma_fac = 1, relTol = eps(real(eltype(AHA))), iterations = 50, restart = :none, verbose = false)
+POGM( ; AHA = ,     reg = L1Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), rho = 0.95, normalize_rho = true, theta = 1, sigma_fac = 1, relTol = eps(real(eltype(AHA))), iterations = 50, restart = :none, verbose = false)

Creates a POGM object for the forward operator A or normal operator AHA. POGM has a 2x better worst-case bound than FISTA, but actual performance varies by application. It stores 3 extra intermediate variables the size of the image compared to FISTA. Only gradient restart scheme is implemented for now.

References:

  • A.B. Taylor, J.M. Hendrickx, F. Glineur, "Exact worst-case performance of first-order algorithms for composite convex optimization," Arxiv:1512.07516, 2015, SIAM J. Opt. 2017 [http://doi.org/10.1137/16m108104x]

  • Kim, D., & Fessler, J. A. (2018). Adaptive Restart of the Optimized Gradient Method for Convex Optimization. Journal of Optimization Theory and Applications, 178(1), 240–263. [https://doi.org/10.1007/s10957-018-1287-4]

    Required Arguments

    • A - forward operator

    OR

    • AHA - normal operator (as a keyword argument)

    Optional Keyword Arguments

    • AHA - normal operator is optional if A is supplied
    • reg::AbstractParameterizedRegularization - regularization term
    • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
    • rho::Real - step size for gradient step
    • normalize_rho::Bool - normalize step size by the largest eigenvalue of AHA
    • theta::Real - parameter for predictor-corrector step
    • sigma_fac::Real - parameter for decreasing γ-momentum ∈ [0,1]
    • relTol::Real - tolerance for stopping criterion
    • iterations::Int - maximum number of iterations
    • restart::Symbol - :none, :gradient options for restarting
    • verbose::Bool - print residual in each iteration

See also createLinearSolver, solve!.

source

SplitBregman

RegularizedLeastSquares.SplitBregman — Type
SplitBregman(A; AHA = A'*A, precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, iterationsOuter = 10, iterationsInner = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)
+SplitBregman( ; AHA = ,     precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, iterationsOuter = 10, iterationsInner = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)

Creates a SplitBregman object for the forward operator A or normal operator AHA.

Required Arguments

  • A - forward operator

OR

  • AHA - normal operator (as a keyword argument)

Optional Keyword Arguments

  • AHA - normal operator is optional if A is supplied
  • precon - preconditionner for the internal CG algorithm
  • reg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms
  • regTrafo - transformation to a space in which reg is applied; if reg is a vector, regTrafo has to be a vector of the same length. Use opEye(eltype(AHA), size(AHA,1)) if no transformation is desired.
  • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
  • rho::Real - weights for condition on regularized variables; can also be a vector for multiple regularization terms
  • iterationsOuter::Int - maximum number of outer iterations. Set to 1 for unconstraint split Bregman (equivalent to ADMM)
  • iterationsInner::Int - maximum number of inner iterations
  • iterationsCG::Int - maximum number of (inner) CG iterations
  • absTol::Real - absolute tolerance for stopping criterion
  • relTol::Real - relative tolerance for stopping criterion
  • tolInner::Real - relative tolerance for CG stopping criterion
  • verbose::Bool - print residual in each iteration

This algorithm solves the constraint problem (Eq. (4.7) in Tom Goldstein and Stanley Osher), i.e. ||R(x)||₁ such that ||Ax -b||₂² < σ². In order to solve the unconstraint problem (Eq. (4.8) in Tom Goldstein and Stanley Osher), i.e. ||Ax -b||₂² + λ ||R(x)||₁, you can either set iterationsOuter=1 or use ADMM instead, which is equivalent (iterationsOuter=1 in SplitBregman in implied in ADMM and the SplitBregman variable iterationsInner is simply called iterations in ADMM)

Like ADMM, SplitBregman differs from ISTA-type algorithms in the sense that the proximal operation is applied separately from the transformation to the space in which the penalty is applied. This is reflected by the interface which has reg and regTrafo as separate arguments. E.g., for a TV penalty, you should NOT set reg=TVRegularization, but instead use reg=L1Regularization(λ), regTrafo=RegularizedLeastSquares.GradientOp(Float64; shape=(Nx,Ny,Nz)).

See also createLinearSolver, solve!.

source

Miscellaneous Functions

RegularizedLeastSquares.createLinearSolver — Function
createLinearSolver(solver::AbstractLinearSolver, A; kargs...)

This method creates a solver. The supported solvers are methods typically used for solving regularized linear systems. All solvers return an approximate solution to Ax = b.

TODO: give a hint what solvers are available

source
RegularizedLeastSquares.isapplicable — Function
isapplicable(solverType::Type{<:AbstractLinearSolver}, A, x, reg)

return true if a solver of type solverType is applicable to system matrix A, data x and regularization terms reg.

source
diff --git a/previews/PR77/assets/documenter.js b/previews/PR77/assets/documenter.js new file mode 100644 index 00000000..c6562b55 --- /dev/null +++ b/previews/PR77/assets/documenter.js @@ -0,0 +1,1050 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/previews/PR77/assets/sh.png b/previews/PR77/assets/sh.png new file mode 100644 index 00000000..9429533c Binary files /dev/null and b/previews/PR77/assets/sh.png differ diff --git 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html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content 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h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content 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html.theme--documenter-dark .navbar-dropdown,html.theme--documenter-dark .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--documenter-dark .navbar-dropdown.is-right{left:auto;right:0}html.theme--documenter-dark .navbar-divider{display:block}html.theme--documenter-dark .navbar>.container .navbar-brand,html.theme--documenter-dark .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--documenter-dark .navbar>.container .navbar-menu,html.theme--documenter-dark .container>.navbar .navbar-menu{margin-right:-.75rem}html.theme--documenter-dark .navbar.is-fixed-bottom-desktop,html.theme--documenter-dark .navbar.is-fixed-top-desktop{left:0;position:fixed;right:0;z-index:30}html.theme--documenter-dark 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a.navbar-item.is-active:not(:focus):not(:hover),html.theme--documenter-dark .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--documenter-dark .navbar-item.has-dropdown:focus .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown:hover .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--documenter-dark .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--documenter-dark .pagination{font-size:1rem;margin:-.25rem}html.theme--documenter-dark .pagination.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--documenter-dark .pagination.is-medium{font-size:1.25rem}html.theme--documenter-dark .pagination.is-large{font-size:1.5rem}html.theme--documenter-dark .pagination.is-rounded .pagination-previous,html.theme--documenter-dark #documenter .docs-sidebar 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.hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/previews/PR77/assets/themeswap.js b/previews/PR77/assets/themeswap.js new file mode 100644 index 00000000..9f5eebe6 --- /dev/null +++ b/previews/PR77/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/previews/PR77/assets/warner.js b/previews/PR77/assets/warner.js new file mode 100644 index 00000000..3f6f5d00 --- /dev/null +++ b/previews/PR77/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; + + document.getElementsByTagName("head")[0].appendChild(meta); + } + + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); +} else { + maybeAddWarning(); +} diff --git a/previews/PR77/gettingStarted/index.html b/previews/PR77/gettingStarted/index.html new file mode 100644 index 00000000..696baf90 --- /dev/null +++ b/previews/PR77/gettingStarted/index.html @@ -0,0 +1,11 @@ + +Getting Started · RegularizedLeastSquares.jl

Getting Started

To get familiar with the different aspects of RegularizedLeastSquares.jl, we will go through a simple example from the field of Compressed Sensing.

In Addtion to RegularizedLeastSquares.jl, we will need the packages LinearOperatorCollection.jl, Images.jl and Random.jl, as well as PyPlot for visualization.

using RegularizedLeastSquares, LinearOperatorCollection, Images, PyPlot, Random

To get started, let us generate a simple phantom

N = 256
+I = shepp_logan(N)

In this example, we consider an operator which randomly samples half of the pixels in the image. Such an operator and the corresponding measurement can be generated by calling

# sampling operator
+idx = sort( shuffle( collect(1:N^2) )[1:div(N^2,2)] )
+A = SamplingOp(eltype(I), pattern = idx , shape = (N,N))
+
+# generate undersampled data
+y = A*vec(I)

To recover the image, we solve the TV-regularized least squares problem

\[\begin{equation} + \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{y} \vert\vert_2^2 + λTV(\mathbf{x}) . +\end{equation}\]

For this purpose we build a TV regularizer with regularization parameter $λ=0.01$

reg = TVRegularization(0.01; shape=(N,N))

To solve the CS problem, the Alternating Direction Method of Multipliers can be used. Thus, we build the corresponding solver

solver = createLinearSolver(ADMM, A; reg=reg, ρ=0.1, iterations=20)

and apply it to our measurement

Ireco = solve!(solver,y)
+Ireco = reshape(Ireco,N,N)

The original phantom and the reconstructed image are shown below

Phantom Reconstruction

diff --git a/previews/PR77/index.html b/previews/PR77/index.html new file mode 100644 index 00000000..be7d3360 --- /dev/null +++ b/previews/PR77/index.html @@ -0,0 +1,3 @@ + +Home · RegularizedLeastSquares.jl

RegularizedLeastSquares.jl

Solvers for Linear Inverse Problems using Regularization Techniques

Introduction

RegularizedLeastSquares.jl is a Julia package for solving large scale linear systems using different types of algorithms. Ill-conditioned problems arise in many areas of practical interest. To solve these problems, one often resorts to regularization techniques and non-linear problem formulations. This packages provides implementations for a variety of solvers, which are used in fields such as MPI and MRI.

The implemented methods range from the $l_2$-regularized CGNR method to more general optimizers such as the Alternating Direction of Multipliers Method (ADMM) or the Split-Bregman method.

For convenience, implementations of popular regularizers, such as $l_1$-regularization and TV regularization, are provided. On the other hand, hand-crafted regularizers can be used quite easily. For this purpose, a Regularization object needs to be build. The latter mainly contains the regularization parameter and a function to calculate the proximal map of a given input.

Depending on the problem, it becomes unfeasible to store the full system matrix at hand. For this purpose, RegularizedLeastSquares.jl allows for the use of matrix-free operators. Such operators can be realized using the interface provided by the package LinearOperators.jl. Other interfaces can be used as well, as long as the product *(A,x) and the adjoint adjoint(A) are provided. A number of common matrix-free operators are provided by the package LinearOperatorColection.jl.

Installation

Within Julia, use the package manager:

using Pkg
+Pkg.add("RegularizedLeastSquares")

This adds the latest release of the package is added. To install a different version, please consult the Pkg documentation.

Usage

diff --git a/previews/PR77/objects.inv b/previews/PR77/objects.inv new file mode 100644 index 00000000..8bca9162 --- /dev/null +++ b/previews/PR77/objects.inv @@ -0,0 +1,7 @@ +# Sphinx inventory version 2 +# Project: RegularizedLeastSquares.jl +# Version: 0.13.1-DEV +# The remainder of this file is compressed using zlib. +xXr0ʴա!dz1N25b+T,SIΐdd}>S21gOJk + ]{ >,-^{>舀>c...&!N^AeM7]4;x-y =,6 0c!:69%P>FOCY 4S"g |,qzfP\d +@Cf1VYL9hRIIhfΉW9"F|a.TLm!&Re۵%Tם~|wVល#,dQ_Bze@iAqAK).bz!CX?҂eKYmo_\mnzOb/wHٯCWVm7QxϸKFՍѳgꦢ&籢"_RB]kdHK,ΡE0_͗sMh]_?;80 ܢ ,0rȵIizLG!5 A<󌂑R≩ASB*VQ<6CtRmm+7p\XOWE996Ō?ɪ{m 'm6FZk54\XSxi3t h!B6gk$A j|f6s\S|< 56ئ4 91t|:\1;bTN~ՓKF E~D7U\$0*Vsp<,l|N~ou;Τ\pԀ6;pj" \ No newline at end of file diff --git a/previews/PR77/regularization/index.html b/previews/PR77/regularization/index.html new file mode 100644 index 00000000..a9f9ebd5 --- /dev/null +++ b/previews/PR77/regularization/index.html @@ -0,0 +1,39 @@ + +Regularization · RegularizedLeastSquares.jl

Regularization

When formulating inverse problems, a Regularizer is formulated as an additional term in a cost function, which has to be minimized. Popular optimizers often deal with a regularizers $g$, by computing the proximal map

\[\begin{equation} + prox_g (\mathbf{x}) = \underset{\mathbf{u}}{argmin} \frac{1}{2}\vert\vert \mathbf{u}-\mathbf{x} \vert {\vert}^2 + g(\mathbf{x}). +\end{equation}\]

In order to implement those kinds of algorithms,RegularizedLeastSquares defines the following type hierarchy:

abstract type AbstractRegularization
+prox!(reg::AbstractRegularization, x)
+norm(reg::AbstractRegularization, x)

Here prox!(reg, x) is an in-place function which computes the proximal map on the input-vector x. The function norm computes the value of the corresponding term in the inverse problem. RegularizedLeastSquares.jl provides AbstractParameterizedRegularization and AbstractProjectionRegularization as core regularization types.

Parameterized Regularization Terms

This group of regularization terms features a regularization parameter λ that is used during the prox! and normcomputations. Examples of this regulariztion group are L1, L2 or LLR (locally low rank) regularization terms.

These terms are constructed by supplying a λ and optionally term specific keyword arguments:

julia> l2 = L2Regularization(0.3)
+L2Regularization{Float64}(0.3)

Parameterized regularization terms implement:

prox!(reg::AbstractParameterizedRegularization, x, λ)
+norm(reg::AbstractParameterizedRegularization, x, λ)

where λ by default is filled with the value used during construction.

Invoking λ on a parameterized term retrieves its regularization parameter. This can be used in a solver to scale and overwrite the parameter as follows:

julia> prox!(l2, [1.0])
+1-element Vector{Float64}:
+ 0.625
+
+julia> param = λ(l2)
+0.3
+
+julia> prox!(l2, [1.0], param*0.2)
+1-element Vector{Float64}:
+ 0.8928571428571428
+

Projection Regularization Terms

This group of regularization terms implement projections, such as a positivity constraint or a projection with a given convex projection function.

julia> positive = PositiveRegularization()
+PositiveRegularization()
+
+julia> prox!(positive, [2.0, -0.2])
+2-element Vector{Float64}:
+ 2.0
+ 0.0

Nested Regularization Terms

Nested regularization terms are terms that act as decorators to the core regularization terms. These terms can be nested around other terms and add functionality to a regularization term, such as scaling λ based on the provided system matrix or applying a transform, such as the Wavelet, to x:

julia> core = L1Regularization(0.8)
+L1Regularization{Float64}(0.8)
+
+julia> wop = WaveletOp(Float32, shape = (32,32));
+
+julia> reg = TransformedRegularization(core, wop);
+
+julia> prox!(reg, randn(32*32)); # Apply soft-thresholding in Wavelet domain

The type of regularization term a nested term can be wrapped around depends on the concrete type of the nested term. However generally, they can be nested arbitrarly deep, adding new functionality with each layer. Each nested regularization term can return its inner regularization. Furthermore, all regularization terms implement the iteration interface to iterate over the nesting. The innermost regularization term of a nested term must be a core regularization term and it can be returned by the sink function:

julia> RegularizedLeastSquares.innerreg(reg) == core
+true
+
+julia> sink(reg) == core
+true
+
+julia> foreach(r -> println(nameof(typeof(r))), reg)
+TransformedRegularization
+L1Regularization
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Uses the regularization parameter defined for reg.\n\n\n\n\n\n","category":"method"},{"location":"API/regularization/#RegularizedLeastSquares.prox!-Tuple{Type{<:AbstractParameterizedRegularization}, Any, Any}","page":"Regularization Terms","title":"RegularizedLeastSquares.prox!","text":"prox!(regType::Type{<:AbstractParameterizedRegularization}, x, λ; kwargs...)\n\nconstruct a regularization term of type regType with given λ and kwargs and apply its prox! on x\n\n\n\n\n\n","category":"method"},{"location":"API/regularization/#LinearAlgebra.norm-Tuple{AbstractParameterizedRegularization, AbstractArray}","page":"Regularization Terms","title":"LinearAlgebra.norm","text":"norm(reg::AbstractParameterizedRegularization, x)\n\nreturns the value of the reg regularization term on x. Uses the regularization parameter defined for reg.\n\n\n\n\n\n","category":"method"},{"location":"API/regularization/#RegularizedLeastSquares.λ-Tuple{AbstractParameterizedRegularization}","page":"Regularization Terms","title":"RegularizedLeastSquares.λ","text":"λ(reg::AbstractParameterizedRegularization)\n\nreturn the regularization parameter λ of reg\n\n\n\n\n\n","category":"method"},{"location":"API/regularization/#LinearAlgebra.norm-Tuple{Type{<:AbstractParameterizedRegularization}, Any, Any}","page":"Regularization Terms","title":"LinearAlgebra.norm","text":"norm(regType::Type{<:AbstractParameterizedRegularization}, x, λ; kwargs...)\n\nconstruct a regularization term of type regType with given λ and kwargs and apply its norm on x\n\n\n\n\n\n","category":"method"},{"location":"solvers/#Solvers","page":"Solvers","title":"Solvers","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.jl provides a variety of solvers, which are used in fields such as MPI and MRI. The following is a non-exhaustive list of the implemented solvers:","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"Kaczmarz algorithm (Kaczmarz)\nConjugate Gradients Normal Residual method (CGNR)\nFast Iterative Shrinkage Thresholding Algorithm (FISTA)\nAlternating Direction of Multipliers Method (ADMM)","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"The solvers are organized in a type-hierarchy and inherit from:","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"abstract type AbstractLinearSolver","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"The type hierarchy is further differentiated into solver categories such as AbstractRowAtionSolver, AbstractPrimalDualSolver or AbstractProximalGradientSolver. A list of all available solvers can be returned by the linearSolverList function.","category":"page"},{"location":"solvers/#Creating-a-Solver","page":"Solvers","title":"Creating a Solver","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"To create a solver, one can invoke the method createLinearSolver as in","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"solver = createLinearSolver(ADMM, A; reg=reg, kwargs...)","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"Here A denotes the system matrix and reg are the Regularization terms to be used by the solver. All further solver parameters can be passed as keyword arguments and are solver specific. To make things more compact, it can be usefull to collect all parameters in a Dict{Symbol,Any}. In this way, the code snippet above can be written as","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"params=Dict{Symbol,Any}()\nparams[:reg] = ...\n...\n\nsolver = createLinearSolver(ADMM, A; params...)","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"This notation can be convenient when a large number of parameters are set manually.","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"It is possible to check if a given solver is applicable to the wanted arguments, as not all solvers are applicable to all system matrix and data (element) types or regularization terms combinations. This is achieved with the isapplicable function:","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"isapplicable(Kaczmarz, A, x, [L21Regularization(0.4f0)])\nfalse","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"For a given set of arguments the list of applicable solvers can be retrieved with applicableSolverList.","category":"page"},{"location":"API/solvers/#API-for-Solvers","page":"Solvers","title":"API for Solvers","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"This page contains documentation of the public API of the RegularizedLeastSquares. In the Julia REPL one can access this documentation by entering the help mode with ?","category":"page"},{"location":"API/solvers/#solve!","page":"Solvers","title":"solve!","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.solve!(::AbstractLinearSolver, ::Any)","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.solve!-Tuple{AbstractLinearSolver, Any}","page":"Solvers","title":"RegularizedLeastSquares.solve!","text":"solve!(solver::AbstractLinearSolver, b; x0 = 0, callbacks = (_, _) -> nothing)\n\nSolves an inverse problem for the data vector b using solver.\n\nRequired Arguments\n\nsolver::AbstractLinearSolver - linear solver (e.g., ADMM or FISTA), containing forward/normal operator and regularizer\nb::AbstractVector - data vector if A was supplied to the solver, back-projection of the data otherwise\n\nOptional Keyword Arguments\n\nx0::AbstractVector - initial guess for the solution; default is zero\ncallbacks - (optionally a vector of) function or callable struct that takes the two arguments callback(solver, iteration) and, e.g., stores, prints, or plots the intermediate solutions or convergence parameters. Be sure not to modify solver or iteration in the callback function as this would japaridze convergence. The default does nothing.\n\nExamples\n\nThe optimization problem\n\n\targmin_x Ax - b_2^2 + λ x_1\n\ncan be solved with the following lines of code:\n\njulia> using RegularizedLeastSquares\n\njulia> A = [0.831658 0.96717\n 0.383056 0.39043\n 0.820692 0.08118];\n\njulia> x = [0.5932234523399985; 0.2697534345340015];\n\njulia> b = A * x;\n\njulia> S = ADMM(A);\n\njulia> x_approx = solve!(S, b)\n2-element Vector{Float64}:\n 0.5932234523399984\n 0.26975343453400163\n\nHere, we use L1Regularization, which is default for ADMM. All regularization options can be found in API for Regularizers.\n\nThe following example solves the same problem, but stores the solution x of each interation in tr:\n\njulia> tr = Dict[]\nDict[]\n\njulia> store_trace!(tr, solver, iteration) = push!(tr, Dict(\"iteration\" => iteration, \"x\" => solver.x, \"beta\" => solver.β))\nstore_trace! (generic function with 1 method)\n\njulia> x_approx = solve!(S, b; callbacks=(solver, iteration) -> store_trace!(tr, solver, iteration))\n2-element Vector{Float64}:\n 0.5932234523399984\n 0.26975343453400163\n\njulia> tr[3]\nDict{String, Any} with 3 entries:\n \"iteration\" => 2\n \"x\" => [0.593223, 0.269753]\n \"beta\" => [1.23152, 0.927611]\n\nThe last example show demonstrates how to plot the solution at every 10th iteration and store the solvers convergence metrics:\n\njulia> using Plots\n\njulia> conv = StoreConvergenceCallback()\n\njulia> function plot_trace(solver, iteration)\n if iteration % 10 == 0\n display(scatter(solver.x))\n end\n end\nplot_trace (generic function with 1 method)\n\njulia> x_approx = solve!(S, b; callbacks = [conv, plot_trace]);\n\nThe keyword callbacks allows you to pass a (vector of) callable objects that takes the arguments solver and iteration and prints, stores, or plots intermediate result.\n\nSee also StoreSolutionCallback, StoreConvergenceCallback, CompareSolutionCallback for a number of provided callback options.\n\n\n\n\n\n","category":"method"},{"location":"API/solvers/#ADMM","page":"Solvers","title":"ADMM","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.ADMM","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.ADMM","page":"Solvers","title":"RegularizedLeastSquares.ADMM","text":"ADMM(A; AHA = A'*A, precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, vary_rho = :none, iterations = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)\nADMM( ; AHA = , precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, vary_rho = :none, iterations = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)\n\nCreates an ADMM object for the forward operator A or normal operator AHA.\n\nRequired Arguments\n\nA - forward operator\n\nOR\n\nAHA - normal operator (as a keyword argument)\n\nOptional Keyword Arguments\n\nAHA - normal operator is optional if A is supplied\nprecon - preconditionner for the internal CG algorithm\nreg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms\nregTrafo - transformation to a space in which reg is applied; if reg is a vector, regTrafo has to be a vector of the same length. Use opEye(eltype(AHA), size(AHA,1)) if no transformation is desired.\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nrho::Real - penalty of the augmented Lagrangian\nvary_rho::Symbol - vary rho to balance primal and dual feasibility; options :none, :balance, :PnP\niterations::Int - maximum number of (outer) ADMM iterations\niterationsCG::Int - maximum number of (inner) CG iterations\nabsTol::Real - absolute tolerance for stopping criterion\nrelTol::Real - relative tolerance for stopping criterion\ntolInner::Real - relative tolerance for CG stopping criterion\nverbose::Bool - print residual in each iteration\n\nADMM differs from ISTA-type algorithms in the sense that the proximal operation is applied separately from the transformation to the space in which the penalty is applied. This is reflected by the interface which has reg and regTrafo as separate arguments. E.g., for a TV penalty, you should NOT set reg=TVRegularization, but instead use reg=L1Regularization(λ), regTrafo=RegularizedLeastSquares.GradientOp(Float64; shape=(Nx,Ny,Nz)).\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#CGNR","page":"Solvers","title":"CGNR","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.CGNR","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.CGNR","page":"Solvers","title":"RegularizedLeastSquares.CGNR","text":"CGNR(A; AHA = A' * A, reg = L2Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), weights = similar(AHA, 0), iterations = 10, relTol = eps(real(eltype(AHA))))\nCGNR( ; AHA = , reg = L2Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), weights = similar(AHA, 0), iterations = 10, relTol = eps(real(eltype(AHA))))\n\ncreates an CGNR object for the forward operator A or normal operator AHA.\n\nRequired Arguments\n\nA - forward operator\n\nOR\n\nAHA - normal operator (as a keyword argument)\n\nOptional Keyword Arguments\n\nAHA - normal operator is optional if A is supplied\nreg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nweights::AbstactVector - weights for the data term; must be of same length and type as the data term\niterations::Int - maximum number of iterations\nrelTol::Real - tolerance for stopping criterion\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#Kaczmarz","page":"Solvers","title":"Kaczmarz","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.Kaczmarz","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.Kaczmarz","page":"Solvers","title":"RegularizedLeastSquares.Kaczmarz","text":"Kaczmarz(A; reg = L2Regularization(0), normalizeReg = NoNormalization(), weights=nothing, randomized=false, subMatrixFraction=0.15, shuffleRows=false, seed=1234, iterations=10, regMatrix=nothing)\n\nCreates a Kaczmarz object for the forward operator A.\n\nRequired Arguments\n\nA - forward operator\n\nOptional Keyword Arguments\n\nreg::AbstractParameterizedRegularization - regularization term\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nrandomized::Bool - randomize Kacmarz algorithm\nsubMatrixFraction::Real - fraction of rows used in randomized Kaczmarz algorithm\nshuffleRows::Bool - randomize Kacmarz algorithm\nseed::Int - seed for randomized algorithm\niterations::Int - number of iterations\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#FISTA","page":"Solvers","title":"FISTA","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.FISTA","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.FISTA","page":"Solvers","title":"RegularizedLeastSquares.FISTA","text":"FISTA(A; AHA=A'*A, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), rho=0.95, normalize_rho=true, theta=1, relTol=eps(real(eltype(AHA))), iterations=50, restart = :none, verbose = false)\nFISTA( ; AHA=, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), rho=0.95, normalize_rho=true, theta=1, relTol=eps(real(eltype(AHA))), iterations=50, restart = :none, verbose = false)\n\ncreates a FISTA object for the forward operator A or normal operator AHA.\n\nRequired Arguments\n\nA - forward operator\n\nOR\n\nAHA - normal operator (as a keyword argument)\n\nOptional Keyword Arguments\n\nAHA - normal operator is optional if A is supplied\nprecon - preconditionner for the internal CG algorithm\nreg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nrho::Real - step size for gradient step\nnormalize_rho::Bool - normalize step size by the largest eigenvalue of AHA\ntheta::Real - parameter for predictor-corrector step\nrelTol::Real - tolerance for stopping criterion\niterations::Int - maximum number of iterations\nrestart::Symbol - :none, :gradient options for restarting\nverbose::Bool - print residual in each iteration\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#OptISTA","page":"Solvers","title":"OptISTA","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.OptISTA","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.OptISTA","page":"Solvers","title":"RegularizedLeastSquares.OptISTA","text":"OptISTA(A; AHA=A'*A, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), rho=0.95, normalize_rho=true, theta=1, relTol=eps(real(eltype(AHA))), iterations=50, verbose = false)\nOptISTA( ; AHA=, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), rho=0.95, normalize_rho=true, theta=1, relTol=eps(real(eltype(AHA))), iterations=50, verbose = false)\n\ncreates a OptISTA object for the forward operator A or normal operator AHA. OptISTA has a 2x better worst-case bound than FISTA, but actual performance varies by application. It stores 2 extra intermediate variables the size of the image compared to FISTA.\n\nReference:\n\nUijeong Jang, Shuvomoy Das Gupta, Ernest K. Ryu, \"Computer-Assisted Design of Accelerated Composite Optimization Methods: OptISTA,\" arXiv:2305.15704, 2023, [https://arxiv.org/abs/2305.15704]\n\nRequired Arguments\n\nA - forward operator\n\nOR\n\nAHA - normal operator (as a keyword argument)\n\nOptional Keyword Arguments\n\nAHA - normal operator is optional if A is supplied\nreg::AbstractParameterizedRegularization - regularization term\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nrho::Real - step size for gradient step\nnormalize_rho::Bool - normalize step size by the largest eigenvalue of AHA\ntheta::Real - parameter for predictor-corrector step\nrelTol::Real - tolerance for stopping criterion\niterations::Int - maximum number of iterations\nverbose::Bool - print residual in each iteration\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#POGM","page":"Solvers","title":"POGM","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.POGM","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.POGM","page":"Solvers","title":"RegularizedLeastSquares.POGM","text":"POGM(A; AHA = A'*A, reg = L1Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), rho = 0.95, normalize_rho = true, theta = 1, sigma_fac = 1, relTol = eps(real(eltype(AHA))), iterations = 50, restart = :none, verbose = false)\nPOGM( ; AHA = , reg = L1Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), rho = 0.95, normalize_rho = true, theta = 1, sigma_fac = 1, relTol = eps(real(eltype(AHA))), iterations = 50, restart = :none, verbose = false)\n\nCreates a POGM object for the forward operator A or normal operator AHA. POGM has a 2x better worst-case bound than FISTA, but actual performance varies by application. It stores 3 extra intermediate variables the size of the image compared to FISTA. Only gradient restart scheme is implemented for now.\n\nReferences:\n\nA.B. Taylor, J.M. Hendrickx, F. Glineur, \"Exact worst-case performance of first-order algorithms for composite convex optimization,\" Arxiv:1512.07516, 2015, SIAM J. Opt. 2017 [http://doi.org/10.1137/16m108104x]\nKim, D., & Fessler, J. A. (2018). Adaptive Restart of the Optimized Gradient Method for Convex Optimization. Journal of Optimization Theory and Applications, 178(1), 240–263. [https://doi.org/10.1007/s10957-018-1287-4]\nRequired Arguments\nA - forward operator\nOR\nAHA - normal operator (as a keyword argument)\nOptional Keyword Arguments\nAHA - normal operator is optional if A is supplied\nreg::AbstractParameterizedRegularization - regularization term\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nrho::Real - step size for gradient step\nnormalize_rho::Bool - normalize step size by the largest eigenvalue of AHA\ntheta::Real - parameter for predictor-corrector step\nsigma_fac::Real - parameter for decreasing γ-momentum ∈ [0,1]\nrelTol::Real - tolerance for stopping criterion\niterations::Int - maximum number of iterations\nrestart::Symbol - :none, :gradient options for restarting\nverbose::Bool - print residual in each iteration\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#SplitBregman","page":"Solvers","title":"SplitBregman","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.SplitBregman","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.SplitBregman","page":"Solvers","title":"RegularizedLeastSquares.SplitBregman","text":"SplitBregman(A; AHA = A'*A, precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, iterationsOuter = 10, iterationsInner = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)\nSplitBregman( ; AHA = , precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, iterationsOuter = 10, iterationsInner = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)\n\nCreates a SplitBregman object for the forward operator A or normal operator AHA.\n\nRequired Arguments\n\nA - forward operator\n\nOR\n\nAHA - normal operator (as a keyword argument)\n\nOptional Keyword Arguments\n\nAHA - normal operator is optional if A is supplied\nprecon - preconditionner for the internal CG algorithm\nreg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms\nregTrafo - transformation to a space in which reg is applied; if reg is a vector, regTrafo has to be a vector of the same length. Use opEye(eltype(AHA), size(AHA,1)) if no transformation is desired.\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nrho::Real - weights for condition on regularized variables; can also be a vector for multiple regularization terms\niterationsOuter::Int - maximum number of outer iterations. Set to 1 for unconstraint split Bregman (equivalent to ADMM)\niterationsInner::Int - maximum number of inner iterations\niterationsCG::Int - maximum number of (inner) CG iterations\nabsTol::Real - absolute tolerance for stopping criterion\nrelTol::Real - relative tolerance for stopping criterion\ntolInner::Real - relative tolerance for CG stopping criterion\nverbose::Bool - print residual in each iteration\n\nThis algorithm solves the constraint problem (Eq. (4.7) in Tom Goldstein and Stanley Osher), i.e. ||R(x)||₁ such that ||Ax -b||₂² < σ². In order to solve the unconstraint problem (Eq. (4.8) in Tom Goldstein and Stanley Osher), i.e. ||Ax -b||₂² + λ ||R(x)||₁, you can either set iterationsOuter=1 or use ADMM instead, which is equivalent (iterationsOuter=1 in SplitBregman in implied in ADMM and the SplitBregman variable iterationsInner is simply called iterations in ADMM)\n\nLike ADMM, SplitBregman differs from ISTA-type algorithms in the sense that the proximal operation is applied separately from the transformation to the space in which the penalty is applied. This is reflected by the interface which has reg and regTrafo as separate arguments. E.g., for a TV penalty, you should NOT set reg=TVRegularization, but instead use reg=L1Regularization(λ), regTrafo=RegularizedLeastSquares.GradientOp(Float64; shape=(Nx,Ny,Nz)).\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#Miscellaneous-Functions","page":"Solvers","title":"Miscellaneous Functions","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.StoreSolutionCallback\nRegularizedLeastSquares.StoreConvergenceCallback\nRegularizedLeastSquares.CompareSolutionCallback\nRegularizedLeastSquares.linearSolverList\nRegularizedLeastSquares.createLinearSolver\nRegularizedLeastSquares.applicableSolverList\nRegularizedLeastSquares.isapplicable","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.StoreSolutionCallback","page":"Solvers","title":"RegularizedLeastSquares.StoreSolutionCallback","text":"StoreSolutionCallback(T)\n\nCallback that accumlates the solvers solution per iteration. Results are stored in the solutions field.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#RegularizedLeastSquares.StoreConvergenceCallback","page":"Solvers","title":"RegularizedLeastSquares.StoreConvergenceCallback","text":"StoreConvergenceCallback()\n\nCallback that accumlates the solvers convergence metrics per iteration. Results are stored in the convMeas field.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#RegularizedLeastSquares.CompareSolutionCallback","page":"Solvers","title":"RegularizedLeastSquares.CompareSolutionCallback","text":"CompareSolutionCallback(ref, cmp)\n\nCallback that compares the solvers current solution with the given reference via cmp(ref, solution) per iteration. Results are stored in the results field.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#RegularizedLeastSquares.linearSolverList","page":"Solvers","title":"RegularizedLeastSquares.linearSolverList","text":"Return a list of all available linear solvers\n\n\n\n\n\n","category":"function"},{"location":"API/solvers/#RegularizedLeastSquares.createLinearSolver","page":"Solvers","title":"RegularizedLeastSquares.createLinearSolver","text":"createLinearSolver(solver::AbstractLinearSolver, A; kargs...)\n\nThis method creates a solver. The supported solvers are methods typically used for solving regularized linear systems. All solvers return an approximate solution to Ax = b.\n\nTODO: give a hint what solvers are available\n\n\n\n\n\n","category":"function"},{"location":"API/solvers/#RegularizedLeastSquares.applicableSolverList","page":"Solvers","title":"RegularizedLeastSquares.applicableSolverList","text":"applicable(args...)\n\nlist all solvers that are applicable to the given arguments. Arguments are the same as for isapplicable without the solver type.\n\nSee also isapplicable, linearSolverList.\n\n\n\n\n\n","category":"function"},{"location":"API/solvers/#RegularizedLeastSquares.isapplicable","page":"Solvers","title":"RegularizedLeastSquares.isapplicable","text":"isapplicable(solverType::Type{<:AbstractLinearSolver}, A, x, reg)\n\nreturn true if a solver of type solverType is applicable to system matrix A, data x and regularization terms reg.\n\n\n\n\n\n","category":"function"},{"location":"#RegularizedLeastSquares.jl","page":"Home","title":"RegularizedLeastSquares.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Solvers for Linear Inverse Problems using Regularization Techniques","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"RegularizedLeastSquares.jl is a Julia package for solving large scale linear systems using different types of algorithms. Ill-conditioned problems arise in many areas of practical interest. To solve these problems, one often resorts to regularization techniques and non-linear problem formulations. This packages provides implementations for a variety of solvers, which are used in fields such as MPI and MRI.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The implemented methods range from the l_2-regularized CGNR method to more general optimizers such as the Alternating Direction of Multipliers Method (ADMM) or the Split-Bregman method.","category":"page"},{"location":"","page":"Home","title":"Home","text":"For convenience, implementations of popular regularizers, such as l_1-regularization and TV regularization, are provided. On the other hand, hand-crafted regularizers can be used quite easily. For this purpose, a Regularization object needs to be build. The latter mainly contains the regularization parameter and a function to calculate the proximal map of a given input.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Depending on the problem, it becomes unfeasible to store the full system matrix at hand. For this purpose, RegularizedLeastSquares.jl allows for the use of matrix-free operators. Such operators can be realized using the interface provided by the package LinearOperators.jl. Other interfaces can be used as well, as long as the product *(A,x) and the adjoint adjoint(A) are provided. A number of common matrix-free operators are provided by the package LinearOperatorColection.jl.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Within Julia, use the package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg\nPkg.add(\"RegularizedLeastSquares\")","category":"page"},{"location":"","page":"Home","title":"Home","text":"This adds the latest release of the package is added. To install a different version, please consult the Pkg documentation.","category":"page"},{"location":"#Usage","page":"Home","title":"Usage","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"See Getting Started for an introduction to using the package","category":"page"},{"location":"gettingStarted/#Getting-Started","page":"Getting Started","title":"Getting Started","text":"","category":"section"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"To get familiar with the different aspects of RegularizedLeastSquares.jl, we will go through a simple example from the field of Compressed Sensing.","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"In Addtion to RegularizedLeastSquares.jl, we will need the packages LinearOperatorCollection.jl, Images.jl and Random.jl, as well as PyPlot for visualization.","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"using RegularizedLeastSquares, LinearOperatorCollection, Images, PyPlot, Random","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"To get started, let us generate a simple phantom","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"N = 256\nI = shepp_logan(N)","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"In this example, we consider an operator which randomly samples half of the pixels in the image. Such an operator and the corresponding measurement can be generated by calling","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"# sampling operator\nidx = sort( shuffle( collect(1:N^2) )[1:div(N^2,2)] )\nA = SamplingOp(eltype(I), pattern = idx , shape = (N,N))\n\n# generate undersampled data\ny = A*vec(I)","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"To recover the image, we solve the TV-regularized least squares problem","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"beginequation\n undersetmathbfxargmin frac12vertvert mathbfAmathbfx-mathbfy vertvert_2^2 + λTV(mathbfx) \nendequation","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"For this purpose we build a TV regularizer with regularization parameter λ=001","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"reg = TVRegularization(0.01; shape=(N,N))","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"To solve the CS problem, the Alternating Direction Method of Multipliers can be used. Thus, we build the corresponding solver","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"solver = createLinearSolver(ADMM, A; reg=reg, ρ=0.1, iterations=20)","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"and apply it to our measurement","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"Ireco = solve!(solver,y)\nIreco = reshape(Ireco,N,N)","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"The original phantom and the reconstructed image are shown below","category":"page"},{"location":"gettingStarted/","page":"Getting Started","title":"Getting Started","text":"(Image: Phantom) (Image: Reconstruction)","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"DocTestSetup = quote\n using RegularizedLeastSquares, Wavelets, LinearOperatorCollection\nend","category":"page"},{"location":"regularization/#Regularization","page":"Regularization","title":"Regularization","text":"","category":"section"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"When formulating inverse problems, a Regularizer is formulated as an additional term in a cost function, which has to be minimized. Popular optimizers often deal with a regularizers g, by computing the proximal map","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"beginequation\n prox_g (mathbfx) = undersetmathbfuargmin frac12vertvert mathbfu-mathbfx vert vert^2 + g(mathbfx)\nendequation","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"In order to implement those kinds of algorithms,RegularizedLeastSquares defines the following type hierarchy:","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"abstract type AbstractRegularization\nprox!(reg::AbstractRegularization, x)\nnorm(reg::AbstractRegularization, x)","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"Here prox!(reg, x) is an in-place function which computes the proximal map on the input-vector x. The function norm computes the value of the corresponding term in the inverse problem. RegularizedLeastSquares.jl provides AbstractParameterizedRegularization and AbstractProjectionRegularization as core regularization types.","category":"page"},{"location":"regularization/#Parameterized-Regularization-Terms","page":"Regularization","title":"Parameterized Regularization Terms","text":"","category":"section"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"This group of regularization terms features a regularization parameter λ that is used during the prox! and normcomputations. Examples of this regulariztion group are L1, L2 or LLR (locally low rank) regularization terms.","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"These terms are constructed by supplying a λ and optionally term specific keyword arguments:","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"julia> l2 = L2Regularization(0.3)\nL2Regularization{Float64}(0.3)","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"Parameterized regularization terms implement:","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"prox!(reg::AbstractParameterizedRegularization, x, λ)\nnorm(reg::AbstractParameterizedRegularization, x, λ)","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"where λ by default is filled with the value used during construction.","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"Invoking λ on a parameterized term retrieves its regularization parameter. This can be used in a solver to scale and overwrite the parameter as follows:","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"julia> prox!(l2, [1.0])\n1-element Vector{Float64}:\n 0.625\n\njulia> param = λ(l2)\n0.3\n\njulia> prox!(l2, [1.0], param*0.2)\n1-element Vector{Float64}:\n 0.8928571428571428\n","category":"page"},{"location":"regularization/#Projection-Regularization-Terms","page":"Regularization","title":"Projection Regularization Terms","text":"","category":"section"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"This group of regularization terms implement projections, such as a positivity constraint or a projection with a given convex projection function.","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"julia> positive = PositiveRegularization()\nPositiveRegularization()\n\njulia> prox!(positive, [2.0, -0.2])\n2-element Vector{Float64}:\n 2.0\n 0.0","category":"page"},{"location":"regularization/#Nested-Regularization-Terms","page":"Regularization","title":"Nested Regularization Terms","text":"","category":"section"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"Nested regularization terms are terms that act as decorators to the core regularization terms. These terms can be nested around other terms and add functionality to a regularization term, such as scaling λ based on the provided system matrix or applying a transform, such as the Wavelet, to x:","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"julia> core = L1Regularization(0.8)\nL1Regularization{Float64}(0.8)\n\njulia> wop = WaveletOp(Float32, shape = (32,32));\n\njulia> reg = TransformedRegularization(core, wop);\n\njulia> prox!(reg, randn(32*32)); # Apply soft-thresholding in Wavelet domain","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"The type of regularization term a nested term can be wrapped around depends on the concrete type of the nested term. However generally, they can be nested arbitrarly deep, adding new functionality with each layer. Each nested regularization term can return its inner regularization. Furthermore, all regularization terms implement the iteration interface to iterate over the nesting. The innermost regularization term of a nested term must be a core regularization term and it can be returned by the sink function:","category":"page"},{"location":"regularization/","page":"Regularization","title":"Regularization","text":"julia> RegularizedLeastSquares.innerreg(reg) == core\ntrue\n\njulia> sink(reg) == core\ntrue\n\njulia> foreach(r -> println(nameof(typeof(r))), reg)\nTransformedRegularization\nL1Regularization","category":"page"}] +} diff --git a/previews/PR77/siteinfo.js b/previews/PR77/siteinfo.js new file mode 100644 index 00000000..5a540c5a --- /dev/null +++ b/previews/PR77/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR77"; diff --git a/previews/PR77/solvers/index.html b/previews/PR77/solvers/index.html new file mode 100644 index 00000000..cdb12683 --- /dev/null +++ b/previews/PR77/solvers/index.html @@ -0,0 +1,7 @@ + +Solvers · RegularizedLeastSquares.jl

Solvers

RegularizedLeastSquares.jl provides a variety of solvers, which are used in fields such as MPI and MRI. The following is a non-exhaustive list of the implemented solvers:

  • Kaczmarz algorithm (Kaczmarz)
  • Conjugate Gradients Normal Residual method (CGNR)
  • Fast Iterative Shrinkage Thresholding Algorithm (FISTA)
  • Alternating Direction of Multipliers Method (ADMM)

The solvers are organized in a type-hierarchy and inherit from:

abstract type AbstractLinearSolver

The type hierarchy is further differentiated into solver categories such as AbstractRowAtionSolver, AbstractPrimalDualSolver or AbstractProximalGradientSolver. A list of all available solvers can be returned by the linearSolverList function.

Creating a Solver

To create a solver, one can invoke the method createLinearSolver as in

solver = createLinearSolver(ADMM, A; reg=reg, kwargs...)

Here A denotes the system matrix and reg are the Regularization terms to be used by the solver. All further solver parameters can be passed as keyword arguments and are solver specific. To make things more compact, it can be usefull to collect all parameters in a Dict{Symbol,Any}. In this way, the code snippet above can be written as

params=Dict{Symbol,Any}()
+params[:reg] = ...
+...
+
+solver = createLinearSolver(ADMM, A; params...)

This notation can be convenient when a large number of parameters are set manually.

It is possible to check if a given solver is applicable to the wanted arguments, as not all solvers are applicable to all system matrix and data (element) types or regularization terms combinations. This is achieved with the isapplicable function:

isapplicable(Kaczmarz, A, x, [L21Regularization(0.4f0)])
+false

For a given set of arguments the list of applicable solvers can be retrieved with applicableSolverList.