From b10d02aa99d3db99f8e4d24437cfd4c2afb6df86 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Fri, 23 Aug 2024 09:29:32 +0000 Subject: [PATCH] build based on 20b6710 --- dev/.documenter-siteinfo.json | 2 +- dev/API/regularization/index.html | 6 +- dev/API/solvers/index.html | 14 ++-- .../examples/compressed_sensing/1e81714b.png | Bin 0 -> 33540 bytes .../examples/compressed_sensing/214b8b4f.png | Bin 33794 -> 0 bytes .../examples/compressed_sensing/9b2b83d3.png | Bin 0 -> 63662 bytes .../examples/compressed_sensing/cef6fff0.png | Bin 63039 -> 0 bytes .../examples/compressed_sensing/index.html | 68 ++++++++-------- .../examples/computed_tomography/index.html | 2 +- .../examples/getting_started/index.html | 2 +- .../explanations/regularization/index.html | 2 +- dev/generated/howto/callbacks/68bbda06.png | Bin 181124 -> 0 bytes dev/generated/howto/callbacks/e343747c.png | Bin 0 -> 179529 bytes dev/generated/howto/callbacks/index.html | 40 +++++----- .../howto/efficient_kaczmarz/index.html | 66 ++++++++-------- .../howto/gpu_acceleration/61d55e6d.png | Bin 15668 -> 0 bytes .../howto/gpu_acceleration/f94934a8.png | Bin 0 -> 15828 bytes .../howto/gpu_acceleration/index.html | 74 +++++++++--------- .../howto/multi_threading/index.html | 4 +- .../howto/normal_operator/index.html | 64 +++++++-------- dev/generated/howto/plug-and-play/index.html | 32 ++++---- dev/generated/howto/weighting/index.html | 2 +- dev/index.html | 2 +- dev/objects.inv | Bin 1608 -> 1608 bytes dev/solvers/index.html | 2 +- 25 files changed, 191 insertions(+), 191 deletions(-) create mode 100644 dev/generated/examples/compressed_sensing/1e81714b.png delete mode 100644 dev/generated/examples/compressed_sensing/214b8b4f.png create mode 100644 dev/generated/examples/compressed_sensing/9b2b83d3.png delete mode 100644 dev/generated/examples/compressed_sensing/cef6fff0.png delete mode 100644 dev/generated/howto/callbacks/68bbda06.png create mode 100644 dev/generated/howto/callbacks/e343747c.png delete mode 100644 dev/generated/howto/gpu_acceleration/61d55e6d.png create mode 100644 dev/generated/howto/gpu_acceleration/f94934a8.png diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index a0fb91a..c709031 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-08-22T15:34:42","documenter_version":"1.6.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-08-23T09:29:28","documenter_version":"1.6.0"}} \ No newline at end of file diff --git a/dev/API/regularization/index.html b/dev/API/regularization/index.html index cdf7bf4..f9510c4 100644 --- a/dev/API/regularization/index.html +++ b/dev/API/regularization/index.html @@ -1,5 +1,5 @@ -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.L21RegularizationType
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.LLRRegularizationType
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
  • fullyOverlapping::Bool=false - choose between fully overlapping block or non-overlapping blocks
source
RegularizedLeastSquares.NuclearRegularizationType
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.TVRegularizationType
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.innerregMethod
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.MaskedRegularizationType
MaskedRegularization

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

Examples

julia> positive = PositiveRegularization();
+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.L21RegularizationType
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.LLRRegularizationType
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
  • fullyOverlapping::Bool=false - choose between fully overlapping block or non-overlapping blocks
source
RegularizedLeastSquares.NuclearRegularizationType
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.TVRegularizationType
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.innerregMethod
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.MaskedRegularizationType
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]);
 
@@ -8,11 +8,11 @@
   0.0
  -1.0
   0.0
- -1.0
source
RegularizedLeastSquares.TransformedRegularizationType
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.PlugAndPlayRegularizationType
    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

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LinearAlgebra.normMethod
norm(reg::AbstractParameterizedRegularization, x)

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

source
LinearAlgebra.normMethod
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
+julia> prox!(reg, randn(32*32)); # Apply soft-thresholding in Wavelet domain
source
RegularizedLeastSquares.PlugAndPlayRegularizationType
    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.normMethod
norm(reg::AbstractParameterizedRegularization, x)

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

source
LinearAlgebra.normMethod
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/dev/API/solvers/index.html b/dev/API/solvers/index.html index fd92bdb..6fa68cb 100644 --- a/dev/API/solvers/index.html +++ b/dev/API/solvers/index.html @@ -40,10 +40,10 @@ 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.

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RegularizedLeastSquares.init!Method
init!(solver::AbstractLinearSolver, b; kwargs...)

Prepare the solver for iteration based on the given data vector b and kwargs.

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RegularizedLeastSquares.init!Method
init!(solver::AbstractLinearSolver, state::AbstractSolverState, b::AbstractMatrix; scheduler = SequentialState, kwargs...)

Initialize the solver with each column of b and pass the corresponding states to the scheduler.

source

ADMM

RegularizedLeastSquares.ADMMType
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.CGNRType
CGNR(A; AHA = A' * A, reg = L2Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), iterations = 10, relTol = eps(real(eltype(AHA))))
-CGNR( ; AHA = ,       reg = L2Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), 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()
  • iterations::Int - maximum number of iterations
  • relTol::Real - tolerance for stopping criterion

See also createLinearSolver, solve!.

source

Kaczmarz

RegularizedLeastSquares.KaczmarzType
Kaczmarz(A; reg = L2Regularization(0), normalizeReg = NoNormalization(), randomized=false, subMatrixFraction=0.15, shuffleRows=false, seed=1234, iterations=10)

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.FISTAType
FISTA(A; AHA=A'*A, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), iterations=50, verbose = false, rho = 0.95 / power_iterations(AHA), theta=1, relTol=eps(real(eltype(AHA))), restart = :none)
-FISTA( ; AHA=,     reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), iterations=50, verbose = false, rho = 0.95 / power_iterations(AHA), theta=1, relTol=eps(real(eltype(AHA))), restart = :none)

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; the default is 0.95 / max_eigenvalue as determined with power iterations.
  • 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.OptISTAType
OptISTA(A; AHA=A'*A, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), iterations=50, verbose = false, rho=0.95 / power_iterations(AHA), theta=1, relTol=eps(real(eltype(AHA))))
-OptISTA( ; AHA=,     reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), iterations=50, verbose = false, rho=0.95 / power_iterations(AHA), theta=1, relTol=eps(real(eltype(AHA))))

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; the default is 0.95 / max_eigenvalue as determined with power iterations.
  • 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.POGMType
POGM(A; AHA = A'*A, reg = L1Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), iterations = 50, verbose = false, rho = 0.95 / power_iterations(AHA), theta = 1, sigma_fac = 1, relTol = eps(real(eltype(AHA))), restart = :none)
-POGM( ; AHA = ,     reg = L1Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), iterations = 50, verbose = false, rho = 0.95 / power_iterations(AHA), theta = 1, sigma_fac = 1, relTol = eps(real(eltype(AHA))), restart = :none)

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; the default is 0.95 / max_eigenvalue as determined with power iterations.
    • 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.SplitBregmanType
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, iterations = 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, iterations = 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
  • iterations::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 iterations=1 or use ADMM instead, which is equivalent (iterations=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

RegularizedLeastSquares.solverstateFunction
solverstate(solver::AbstractLinearSolver)

Return the current state of the solver

source
RegularizedLeastSquares.solversolutionFunction
solversolution(solver::AbstractLinearSolver)

Return the current solution of the solver

source
solversolution(state::AbstractSolverState)

Return the current solution of the solver's state

source
RegularizedLeastSquares.solverconvergenceFunction
solverconvergence(solver::AbstractLinearSolver)

Return a named tuple of the solvers current convergence metrics

source
RegularizedLeastSquares.StoreSolutionCallbackType
StoreSolutionCallback(T)

Callback that accumlates the solvers solution per iteration. Results are stored in the solutions field.

source
RegularizedLeastSquares.StoreConvergenceCallbackType
StoreConvergenceCallback()

Callback that accumlates the solvers convergence metrics per iteration. Results are stored in the convMeas field.

source
RegularizedLeastSquares.CompareSolutionCallbackType
CompareSolutionCallback(ref, cmp)

Callback that compares the solvers current solution with the given reference via cmp(ref, solution) per iteration. Results are stored in the results field.

source
RegularizedLeastSquares.linearSolverListFunction

Return a list of all available linear solvers

source
RegularizedLeastSquares.createLinearSolverFunction
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.applicableSolverListFunction
applicable(args...)

list all solvers that are applicable to the given arguments. Arguments are the same as for isapplicable without the solver type.

See also isapplicable, linearSolverList.

source
RegularizedLeastSquares.isapplicableFunction
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
+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
RegularizedLeastSquares.init!Method
init!(solver::AbstractLinearSolver, b; kwargs...)

Prepare the solver for iteration based on the given data vector b and kwargs.

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RegularizedLeastSquares.init!Method
init!(solver::AbstractLinearSolver, state::AbstractSolverState, b::AbstractMatrix; scheduler = SequentialState, kwargs...)

Initialize the solver with each column of b and pass the corresponding states to the scheduler.

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ADMM

RegularizedLeastSquares.ADMMType
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!.

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CGNR

RegularizedLeastSquares.CGNRType
CGNR(A; AHA = A' * A, reg = L2Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), iterations = 10, relTol = eps(real(eltype(AHA))))
+CGNR( ; AHA = ,       reg = L2Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), 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()
  • iterations::Int - maximum number of iterations
  • relTol::Real - tolerance for stopping criterion

See also createLinearSolver, solve!.

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Kaczmarz

RegularizedLeastSquares.KaczmarzType
Kaczmarz(A; reg = L2Regularization(0), normalizeReg = NoNormalization(), randomized=false, subMatrixFraction=0.15, shuffleRows=false, seed=1234, iterations=10)

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!.

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FISTA

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

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; the default is 0.95 / max_eigenvalue as determined with power iterations.
  • 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!.

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OptISTA

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

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; the default is 0.95 / max_eigenvalue as determined with power iterations.
  • 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!.

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POGM

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

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; the default is 0.95 / max_eigenvalue as determined with power iterations.
    • 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!.

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SplitBregman

RegularizedLeastSquares.SplitBregmanType
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, iterations = 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, iterations = 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
  • iterations::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 iterations=1 or use ADMM instead, which is equivalent (iterations=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!.

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Miscellaneous

RegularizedLeastSquares.solverstateFunction
solverstate(solver::AbstractLinearSolver)

Return the current state of the solver

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RegularizedLeastSquares.solversolutionFunction
solversolution(solver::AbstractLinearSolver)

Return the current solution of the solver

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solversolution(state::AbstractSolverState)

Return the current solution of the solver's state

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RegularizedLeastSquares.solverconvergenceFunction
solverconvergence(solver::AbstractLinearSolver)

Return a named tuple of the solvers current convergence metrics

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RegularizedLeastSquares.StoreSolutionCallbackType
StoreSolutionCallback(T)

Callback that accumlates the solvers solution per iteration. Results are stored in the solutions field.

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RegularizedLeastSquares.StoreConvergenceCallbackType
StoreConvergenceCallback()

Callback that accumlates the solvers convergence metrics per iteration. Results are stored in the convMeas field.

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RegularizedLeastSquares.CompareSolutionCallbackType
CompareSolutionCallback(ref, cmp)

Callback that compares the solvers current solution with the given reference via cmp(ref, solution) per iteration. Results are stored in the results field.

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RegularizedLeastSquares.linearSolverListFunction

Return a list of all available linear solvers

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RegularizedLeastSquares.createLinearSolverFunction
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

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RegularizedLeastSquares.applicableSolverListFunction
applicable(args...)

list all solvers that are applicable to the given arguments. Arguments are the same as for isapplicable without the solver type.

See also isapplicable, linearSolverList.

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RegularizedLeastSquares.isapplicableFunction
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.

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z$b}W6g9+Goc9@WQKpL35F^%A1u)-f4n}!xi1;Zia#p2a5PdOw-U&!|2~fRe0dQT zHto4tS42SPs}q@o9P-;R&*(z#{}`$hP%U5VqnI8Lk?My<=6YC1Y1EbV{~IB@lmGg# zH51a;%d8-gYf*Clf5D#I2t-(t>umMYNx~?(7~(?K|DH@he^|BOctmElD`oj7b`8>Z zVXA*rd za4LQ>UUbF>bpAi05@xQe0OCKNMbwD4W&Rh1h5hdelK(_c3V^f!|MkE4cG3U0K6&y@ Xkm#!syQISf0WTQ|Me$Nmqk#Vb&Xv=v diff --git a/dev/generated/examples/compressed_sensing/index.html b/dev/generated/examples/compressed_sensing/index.html index 0bdf0b2..8d2e653 100644 --- a/dev/generated/examples/compressed_sensing/index.html +++ b/dev/generated/examples/compressed_sensing/index.html @@ -8,22 +8,22 @@ 1 4 5 - 10 - 17 - 18 - 19 - 20 - 21 - 26 + 6 + 7 + 9 + 11 + 12 + 22 + 25 ⋮ - 65516 - 65517 - 65518 + 65501 + 65502 + 65506 + 65514 65521 65522 - 65528 - 65530 - 65532 + 65526 + 65529 65534

Afterwards we build a sampling operator which samples the image at the selected indices.

A = SamplingOp(eltype(image), pattern = sampledIndices , shape = size(image));

Then we apply the sampling operator to the vectorized image to obtain the sampled measurement vector

b = A*vec(image);

To visualize our image we can use CairoMakie:

using CairoMakie
 function plot_image(figPos, img; title = "", width = 150, height = 150)
   ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height)
@@ -36,30 +36,30 @@
 samplingMask[sampledIndices] .= true
 plot_image(fig[1,2], image .* samplingMask, title = "Sampled Image")
 resize_to_layout!(fig)
-fig
Example block output

As we can see in the right image, only a third of the pixels are sampled. The goal of the inverse problem is to recover the original image from this measurement vector.

Solving the Inverse Problem

To recover the image from the measurement vector, we solve the TV-regularized least squares problem:

\[\begin{equation} +figExample block output

As we can see in the right image, only a third of the pixels are sampled. The goal of the inverse problem is to recover the original image from this measurement vector.

Solving the Inverse Problem

To recover the image from the measurement vector, we solve the TV-regularized least squares problem:

\[\begin{equation} \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_2^2 + \vert\vert\mathbf{x}\vert\vert_{\lambda\text{TV}} . \end{equation}\]

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

using RegularizedLeastSquares
 reg = TVRegularization(0.01; shape=size(image));

We will use the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) to solve our inverse problem. Thus, we build the corresponding solver

solver = createLinearSolver(FISTA, A; reg=reg, iterations=20);

and apply it to our measurement vector

img_approx = solve!(solver,b)
65536-element Vector{Float32}:
- 1.1679568f-18
- 2.1116658f-18
- 4.5817737f-18
- 1.0172477f-17
- 2.268474f-17
- 5.0405303f-17
- 1.1135638f-16
- 2.4575f-16
- 5.3614856f-16
- 1.1372157f-15
+ 4.408601f-19
+ 8.141939f-19
+ 1.8912686f-18
+ 4.657992f-18
+ 1.13304495f-17
+ 2.6190603f-17
+ 5.750876f-17
+ 1.2296451f-16
+ 2.6434892f-16
+ 5.744004f-16
  ⋮
- 3.745113f-16
- 1.8518066f-16
- 8.505684f-17
- 3.59355f-17
- 1.43278f-17
- 5.6754496f-18
- 2.348305f-18
- 1.0605327f-18
- 6.0551384f-19

To visualize the reconstructed image, we need to reshape the result vector to the correct shape. Afterwards we can use CairoMakie again:

img_approx = reshape(img_approx,size(image));
+ 5.001874f-16
+ 2.3916176f-16
+ 1.1642602f-16
+ 5.6809295f-17
+ 2.6450612f-17
+ 1.1868527f-17
+ 5.449228f-18
+ 2.69539f-18
+ 1.6241735f-18

To visualize the reconstructed image, we need to reshape the result vector to the correct shape. Afterwards we can use CairoMakie again:

img_approx = reshape(img_approx,size(image));
 plot_image(fig[1,3], img_approx, title = "Reconstructed Image")
 resize_to_layout!(fig)
-fig
Example block output

This page was generated using Literate.jl.

+figExample block output

This page was generated using Literate.jl.

diff --git a/dev/generated/examples/computed_tomography/index.html b/dev/generated/examples/computed_tomography/index.html index e1354e0..14ca478 100644 --- a/dev/generated/examples/computed_tomography/index.html +++ b/dev/generated/examples/computed_tomography/index.html @@ -62,4 +62,4 @@ 0.0

To visualize the reconstructed image, we need to reshape the result vector to the correct shape. Afterwards we can use CairoMakie again:

img_approx = reshape(img_approx,size(image));
 plot_image(fig[1,4], img_approx, title = "Reconstructed Image")
 resize_to_layout!(fig)
-fig
Example block output

This page was generated using Literate.jl.

+figExample block output

This page was generated using Literate.jl.

diff --git a/dev/generated/examples/getting_started/index.html b/dev/generated/examples/getting_started/index.html index 813e69b..d723c90 100644 --- a/dev/generated/examples/getting_started/index.html +++ b/dev/generated/examples/getting_started/index.html @@ -9,4 +9,4 @@ \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_2^2 + \vert\vert\mathbf{x}\vert\vert^2_2 . \end{equation}\]

The corresponding solver can be built with the L2 regularization term:

solver = createLinearSolver(CGNR, A; reg = L2Regularization(0.0001), iterations=32);
 x_approx = solve!(solver, b)
-isapprox(x, x_approx, rtol = 0.001)
true

This page was generated using Literate.jl.

+isapprox(x, x_approx, rtol = 0.001)
true

This page was generated using Literate.jl.

diff --git a/dev/generated/explanations/regularization/index.html b/dev/generated/explanations/regularization/index.html index 11affb5..2339c5a 100644 --- a/dev/generated/explanations/regularization/index.html +++ b/dev/generated/explanations/regularization/index.html @@ -55,4 +55,4 @@ plot_image(fig[1,4], img_prox_wavelet, title = "Reg. Wavelet Domain") resize_to_layout!(fig) figExample block output

Generally, regularization terms can be nested arbitrarly deep, adding new functionality with each layer. Each nested regularization term can return its inner regularization term. 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:

RegularizedLeastSquares.innerreg(reg) == core
true
sink(reg) == core
true
foreach(r -> println(nameof(typeof(r))), reg)
TransformedRegularization
-L1Regularization

This page was generated using Literate.jl.

+L1Regularization

This page was generated using Literate.jl.

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65536-element Vector{Float32}:
- 5.547791f-19
- 9.346682f-19
- 1.8597913f-18
- 3.7886526f-18
- 8.0094f-18
- 1.7636859f-17
- 3.9217387f-17
- 8.7732907f-17
- 1.9868275f-16
- 4.3741385f-16
+ 8.419296f-19
+ 1.4342351f-18
+ 3.1738705f-18
+ 7.515546f-18
+ 1.7523383f-17
+ 3.9682643f-17
+ 8.913537f-17
+ 1.9674716f-16
+ 4.1574712f-16
+ 8.3140503f-16
  ⋮
- 5.8828746f-16
- 2.8533803f-16
- 1.371053f-16
- 6.317723f-17
- 2.817526f-17
- 1.2458892f-17
- 5.4933825f-18
- 2.4608964f-18
- 1.3337157f-18

In the callback we have to copy the solution, as the solver will update it in place. As is explained in the solver section, each features fields that are intended to be immutable during a solve! call and a state that is modified in each iteration. Depending on the solvers parameters and the measurement input, the state can differ in its fields and their type. Ideally, one tries to avoid accessing the state directly and uses the provided functions to access the state.

The resulting figure shows the reconstructed image after 0, 4, 8, 12, 16 and 20 iterations:

resize_to_layout!(fig)
-fig
Example block output
+ 3.993051f-16 + 1.8685763f-16 + 9.106389f-17 + 4.4373068f-17 + 2.0190658f-17 + 8.47266f-18 + 3.4212165f-18 + 1.3973583f-18 + 6.7898433f-19

In the callback we have to copy the solution, as the solver will update it in place. As is explained in the solver section, each features fields that are intended to be immutable during a solve! call and a state that is modified in each iteration. Depending on the solvers parameters and the measurement input, the state can differ in its fields and their type. Ideally, one tries to avoid accessing the state directly and uses the provided functions to access the state.

The resulting figure shows the reconstructed image after 0, 4, 8, 12, 16 and 20 iterations:

resize_to_layout!(fig)
+fig
Example block output

This page was generated using Literate.jl.

diff --git a/dev/generated/howto/efficient_kaczmarz/index.html b/dev/generated/howto/efficient_kaczmarz/index.html index 2894003..a08dcda 100644 --- a/dev/generated/howto/efficient_kaczmarz/index.html +++ b/dev/generated/howto/efficient_kaczmarz/index.html @@ -5,44 +5,44 @@ b = A*x;

The dot_with_matrix_row function calculates the dot product between a row of A and the current approximate solution of x:

row = 1
 isapprox(RegularizedLeastSquares.dot_with_matrix_row(A, x, row), sum(A[row, :] .* x))
true

Since in Julia, dense arrays are stored in column-major order, such a row-based operation is quite inefficient. A workaround is to transpose the matrix then pass it to a Kaczmarz solver.

At = collect(transpose(A))
 A_eff = transpose(At)
256×256 transpose(::Matrix{Float64}) with eltype Float64:
- -0.189399     0.601485  -1.95593    …  -0.64707     -0.0105908   1.0391
- -0.706348     0.624658   0.613219      -0.884453     0.0349683  -0.243366
-  1.4048       1.71171   -0.0580304     -0.0822498    0.655972    0.931729
-  1.04614     -1.15006    0.916264       0.00714401   1.63534    -0.484932
- -0.623837    -0.548661  -0.863925       1.12634     -1.98437    -0.675121
-  0.886654     0.396911   0.0913785  …   1.19359      0.664691   -0.200409
-  0.545648     0.802177  -0.320662      -1.33509     -1.7281     -0.064476
-  1.74603      1.61793   -0.590559      -0.98909      1.34828    -2.01368
- -1.85396      0.874325  -1.43171       -0.496901     0.434827    1.42408
- -0.181865     0.767489   1.10993       -0.863647    -1.17524    -0.3089
-  ⋮                                  ⋱                            ⋮
- -0.299713     0.128175   0.139161      -1.42344     -0.775597    1.85231
-  0.184957     1.11683    0.390196      -0.791326     0.915318   -0.545659
- -1.19324      0.174939   0.0729529      1.16968      0.642389    0.187389
- -0.00103597   0.280875  -0.549137   …   0.603096     2.10453     0.127906
-  0.233185    -2.60052   -1.83491        2.74383      0.540197    0.165311
-  2.46164      0.777437   0.967356       0.387718    -0.458552    0.509111
- -0.625099     0.228578   0.0813318      0.455351    -0.328492   -1.67416
-  0.0983612    2.2171     1.81932        0.0991705   -0.769969   -1.31368
- -0.993177    -0.50764    0.0906102  …   1.39881      1.15723    -1.32507

Note that the transpose function can return a lazy transpose object, so we first collect the transpose into a dense matrix. Then we transpose it again to get the efficient representation of the matrix.

We can compare the performance using the BenchmarkTools.jl package. First for the original matrix:

using BenchmarkTools
+  2.47154    -1.69437    -0.416084   …   0.367591    1.10383     1.77787
+  0.599265    0.463391    1.2313         0.0313625   0.351771   -1.01844
+ -0.447882   -0.380744    0.0878082      0.601217   -0.753224    0.898901
+  0.910174   -0.0798059  -0.422644       0.0915945  -0.537778   -2.04373
+  0.713189    1.72562     0.840623       0.221132   -0.122292    0.430983
+ -0.146743    0.535785   -0.83828    …  -0.764071    1.67458    -1.00984
+  1.3032      1.58242     0.772758      -1.09469    -1.63271    -0.819563
+  0.334873   -0.149519    0.624688       1.10821     0.491696    0.128356
+  1.92588     0.980394   -0.569869      -1.5654     -0.688463   -0.311905
+ -0.335922    0.0428925  -0.393854      -1.80722    -1.94559    -0.919267
+  ⋮                                  ⋱                           ⋮
+  0.610916    0.0116187   0.0444647      0.910798   -0.478441   -1.07877
+ -0.158691   -0.224015    1.68594       -0.358469    1.34047     0.30642
+ -0.437568    0.599318    0.417718       1.89221     0.378605    0.372428
+ -0.823511    1.10764    -1.21642    …   0.579675   -0.0919037   0.519975
+ -2.02905    -0.365303   -0.133571       0.779717    0.668506    0.570041
+ -0.756667    1.82197     1.13116       -1.12231     0.99631    -0.855226
+ -0.0424238   1.1162     -1.93996       -0.607697   -0.55596    -0.469157
+  0.230504    1.28832     0.226337       0.219362    1.34469     0.487648
+ -2.16356    -0.259803    0.0375218  …   1.87657     0.0422485  -0.185053

Note that the transpose function can return a lazy transpose object, so we first collect the transpose into a dense matrix. Then we transpose it again to get the efficient representation of the matrix.

We can compare the performance using the BenchmarkTools.jl package. First for the original matrix:

using BenchmarkTools
 solver = createLinearSolver(Kaczmarz, A; reg = L2Regularization(0.0001), iterations=100)
 @benchmark solve!(solver, b) samples = 100
BenchmarkTools.Trial: 100 samples with 1 evaluation.
- Range (minmax):  25.174 ms 27.299 ms   GC (min … max): 0.00% … 0.00%
- Time  (median):     27.136 ms                GC (median):    0.00%
- Time  (mean ± σ):   27.104 ms ± 216.639 μs   GC (mean ± σ):  0.00% ± 0.00%
+ Range (minmax):  27.875 ms28.118 ms   GC (min … max): 0.00% … 0.00%
+ Time  (median):     28.045 ms               GC (median):    0.00%
+ Time  (mean ± σ):   28.045 ms ± 29.736 μs   GC (mean ± σ):  0.00% ± 0.00%
 
-                                                         ▁    
-  ▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃▁▁▁▅█▄▂ ▂
-  25.2 ms         Histogram: frequency by time         27.2 ms <
+                                        ▁▅█▁▄▁                
+  ▃▁▁▁▁▁▁▁▁▁▁▁▃▁▁▁▁▃▁▁▁▁▁▁▁▁▁▁▃▁▁▁▁▁▁▄▄▆██████▅▅▇▅▁▃▁▃▁▁▁▃▁▃ ▃
+  27.9 ms         Histogram: frequency by time        28.1 ms <
 
  Memory estimate: 17.31 KiB, allocs estimate: 504.

And then for the efficient matrix:

solver_eff = createLinearSolver(Kaczmarz, A_eff; reg = L2Regularization(0.0001), iterations=100)
 @benchmark solve!(solver_eff, b) samples = 100
BenchmarkTools.Trial: 100 samples with 1 evaluation.
- Range (minmax):  1.887 ms 2.196 ms   GC (min … max): 0.00% … 0.00%
- Time  (median):     1.909 ms               GC (median):    0.00%
- Time  (mean ± σ):   1.915 ms ± 33.987 μs   GC (mean ± σ):  0.00% ± 0.00%
+ Range (minmax):  1.847 ms 1.940 ms   GC (min … max): 0.00% … 0.00%
+ Time  (median):     1.861 ms               GC (median):    0.00%
+ Time  (mean ± σ):   1.863 ms ± 13.091 μs   GC (mean ± σ):  0.00% ± 0.00%
 
-       ▁ ▁█                                                
-  ▃▃▅█▆█▇███▇▇▅▃▄▄▃▃▃▁▁▃▁▁▁▁▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃▁▁▁▁▁▁▁▁▃ ▃
-  1.89 ms        Histogram: frequency by time        2.03 ms <
+        ▄█       ▄▃ ▂                                        
+  ▃▁▃▇█▃██▇▅▆▆▇██▆█▅▃▁▆▁▁▃▅▁▁▁▃▃▁▁▁▁▃▁▁▁▁▁▁▁▁▃▁▁▁▁▁▁▁▁▁▃▃ ▃
+  1.85 ms        Histogram: frequency by time        1.91 ms <
 
- Memory estimate: 17.34 KiB, allocs estimate: 506.

We can also combine the efficient matrix with a weighting matrix, as is shown in the Weighting example.

Custom operators need to implement the dot_with_matrix_row function to be used with the Kaczmarz solver. Ideally, such an implementation is allocation free.


This page was generated using Literate.jl.

+ Memory estimate: 17.34 KiB, allocs estimate: 506.

We can also combine the efficient matrix with a weighting matrix, as is shown in the Weighting example.

Custom operators need to implement the dot_with_matrix_row function to be used with the Kaczmarz solver. Ideally, such an implementation is allocation free.


This page was generated using Literate.jl.

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zwIxzQA;)J@c)I^_dR-8WFL+WbFxFXIH!zyGtEmZT>S`01Y ziZEp^kMN!aF@ig8L!iE4de?7@!7_Ety|U(0q!s0#R=^vY#H{|A>Z?;kMTpZQ;B|J6 zetdi1At+!Rb}pxjrbr5P8*bD2_C}BJ)S|YCv-6UJ?>+@;_v9of<7&XApH?8!!NVMZ*A&%ds}OzHpeyg9APlLIj!b-)HF( zu|KnI6S7Jx3DZw-qsU~QC803-Yk3-y;VPWYHaG#dB9yQ@2A<|(t8yczpHv}_{%T1Z z+V)R^8U5!44I{JqZ5ArroBL!N*ZLj)v74XTeZ`Z0mI-cXb^FI`dr

Solvers adapt their states based on the type of the given measurement vector. This means that the solver will automatically switch to GPU acceleration if a GPU array is passed as the measurement vector.

solver = createLinearSolver(CGNR, A; reg = L2Regularization(0.0001), iterations=32);
 x_approx = solve!(solver, b)
16-element JLArrays.JLArray{Float32, 1}:
- 0.6934705
- 0.7766296
- 0.94692737
- 0.1807007
- 0.3425655
- 0.31208363
- 0.81724966
- 0.830756
- 0.8948325
- 0.16529448
- 0.21269572
- 0.5512471
- 0.31383783
- 0.13994116
- 0.16612688
- 0.6538022

This adaption does not include the operator. So if we want to compare with CPU result, we need to construct a new solver with a CPU operator.

solver = createLinearSolver(CGNR, Array(A); reg = L2Regularization(0.0001), iterations=32);
+ 0.17287469
+ 0.62701166
+ 0.14676003
+ 0.6033243
+ 0.4049614
+ 0.5988155
+ 0.73063433
+ 0.6497745
+ 0.76667565
+ 0.89490557
+ 0.4916952
+ 0.832206
+ 0.9998267
+ 0.8995493
+ 0.57370514
+ 0.9202157

This adaption does not include the operator. So if we want to compare with CPU result, we need to construct a new solver with a CPU operator.

solver = createLinearSolver(CGNR, Array(A); reg = L2Regularization(0.0001), iterations=32);
 x_cpu = solve!(solver, Array(b))
 isapprox(Array(x_approx), x_cpu)
true

Matrix-Free Operators

A special case is the usage of matrix-free operators. Since these operators do not have a concrete matrix representation, their GPU support depends on their implementation. Since not all multiplications within a solver approximation are in-place, the operator also needs to support the * operation and construct an appropriate result vector. For matrix-free operators based on LinearOperators.jl, this can be achieved by implementing the LinearOperators.storage_type method.

In the following, we will take another look at the CS example and execute it on the GPU. Note that for the JLArray example we chose a small phantom, since the JLArray implementation is not optimized for performance:

using ImagePhantoms, ImageGeoms
 N = 32
@@ -35,26 +35,26 @@
 reg = TVRegularization(0.01; shape=size(image))
 solver = createLinearSolver(FISTA, A; reg=reg, iterations=20)
 img_approx = solve!(solver,b);

To visualize the reconstructed image, we need to reshape the result vector to the correct shape and convert it to a CPU array:

img_approx = reshape(Array(img_approx),size(image))
32×32 Matrix{Float32}:
- 0.000912645  0.00102413   0.00122618   …  0.00211412  0.00203619  0.00196657
- 0.00100939   0.00111489   0.00131014      0.00234097  0.00223807  0.00215573
- 0.00119764   0.00129061   0.00148883      0.00278423  0.00263765  0.00253882
- 0.00149607   0.00158425   0.00183043      0.00346889  0.00323034  0.0031038
- 0.00195311   0.00208215   0.00247151      0.0044958   0.00402462  0.00379264
- 0.00267045   0.0029247    0.00356319   …  0.00592881  0.00504037  0.00456432
- 0.00382276   0.00426547   0.0051361       0.00780656  0.0064372   0.00564293
- 0.00558921   0.00614037   0.00696765      0.00955767  0.00835509  0.00724997
- 0.00789791   0.00818732   0.00844248      0.00955043  0.0101324   0.00937727
- 0.0102126    0.00979983   0.00927027      0.0171683   0.0107284   0.0109629
- ⋮                                      ⋱              ⋮           
- 0.00698968   0.00707629   0.00661722      0.00692743  0.00762787  0.00743909
- 0.00494124   0.00558412   0.00612437      0.00649439  0.0060451   0.00553446
- 0.00335053   0.00403693   0.00499535   …  0.00512907  0.00434914  0.00385641
- 0.00231572   0.00275555   0.00357199      0.00351391  0.00295388  0.0026981
- 0.00167929   0.00191321   0.00243001      0.00237004  0.00211695  0.00203543
- 0.00124699   0.00138703   0.00170899      0.00175277  0.00166778  0.00164885
- 0.00093072   0.00103587   0.00127261      0.0014443   0.00139817  0.00137467
- 0.000709084  0.000801504  0.00100721   …  0.00127143  0.00122053  0.00118586
- 0.000589848  0.000678071  0.000873482     0.00118211  0.00112658  0.00109054

We will again use CairoMakie for visualization:

using CairoMakie
+ 0.00151533  0.00158607  0.00168952  …  0.00158273  0.00153376  0.00149802
+ 0.00162358  0.00171204  0.00185874     0.00181336  0.0017391   0.00168684
+ 0.00186577  0.00199836  0.00225735     0.00230486  0.00220547  0.00214068
+ 0.00224282  0.00245885  0.0029228      0.00310273  0.00296836  0.00290371
+ 0.00267109  0.00302994  0.00379927     0.0043548   0.00410756  0.0040141
+ 0.0031312   0.00371525  0.00489249  …  0.00620918  0.00578974  0.00561371
+ 0.00382538  0.0046552   0.00608891     0.00836054  0.00790062  0.00778584
+ 0.00498189  0.00583883  0.00712563     0.0100626   0.00976795  0.0102514
+ 0.00648297  0.00706044  0.00794547     0.0107663   0.0107969   0.0127489
+ 0.00823372  0.00840476  0.00865798     0.0285679   0.0258083   0.0155364
+ ⋮                                   ⋱              ⋮           
+ 0.0064767   0.00727178  0.00812502     0.0141754   0.0143571   0.0141576
+ 0.00535655  0.00615326  0.0069996      0.012134    0.0117438   0.0122558
+ 0.00441364  0.00513068  0.00622997  …  0.00954313  0.00887906  0.00717651
+ 0.00359766  0.00407462  0.00499565     0.00695663  0.00637404  0.00602849
+ 0.00285175  0.00310944  0.00368035     0.00479004  0.0043906   0.00422618
+ 0.00215721  0.0022935   0.00261727     0.00329971  0.00304902  0.0029376
+ 0.00157507  0.00164998  0.00183977     0.00248079  0.00228835  0.00218101
+ 0.00121028  0.00124639  0.00135594  …  0.00207952  0.00191114  0.00180711
+ 0.0010553   0.00107158  0.00113994     0.00189981  0.00175045  0.0016557

We will again use CairoMakie for visualization:

using CairoMakie
 function plot_image(figPos, img; title = "", width = 150, height = 150)
   ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height)
   hidedecorations!(ax)
@@ -67,4 +67,4 @@
 plot_image(fig[1,2], image .* samplingMask, title = "Sampled Image")
 plot_image(fig[1,3], img_approx, title = "Reconstructed Image")
 resize_to_layout!(fig)
-fig
Example block output

This page was generated using Literate.jl.

+figExample block output

This page was generated using Literate.jl.

diff --git a/dev/generated/howto/multi_threading/index.html b/dev/generated/howto/multi_threading/index.html index 58a9949..c60c4ba 100644 --- a/dev/generated/howto/multi_threading/index.html +++ b/dev/generated/howto/multi_threading/index.html @@ -10,7 +10,7 @@ xs_approx[i] = solve!(solver, bs[i]) end

Operator Based Multi-Threading

This type of multi-threading involves linear operators or proximal maps that can be implemnted in parallel. Examples of this include the proximal map of the TV regularization term, which is based on the multi-threaded GradientOp from LinearOperatorCollection. GPU acceleration also falls under this approach, see GPU Acceleration for more information.

Measurement Based Multi-Threading

This level of multi-threading applies the same solver (and its parameters) to multiple measurement vectors or rather a measurement matrix B. This is useful in the case of multiple measurements that can be solved in parallel and can reuse the same solver. This approach is not applicable if the operator is stateful.

To use this approach we first build a measurement matrix B and a corresponding solver:

A = first(As)
 B = mapreduce(x -> A*x, hcat, xs)
-solver = createLinearSolver(CGNR, A; iterations=32)
CGNR{Matrix{Float64}, Matrix{Float64}, L2Regularization{Float64}, Vector{Any}}([0.7154886197403123 0.6892179810320691 … 0.9485452385663927 0.22706059872974949; 0.5247020200532362 0.44108857804693324 … 0.7340644121945139 0.09277225794448651; … ; 0.16710428921996134 0.08144393947608375 … 0.41187194775833236 0.5541574393627506; 0.9668381754204286 0.8669994370355492 … 0.4802453564381014 0.6885290918221507], [10.307447307522143 8.622280539511276 … 8.842164952639337 6.920548393382659; 8.622280539511276 12.42411352661935 … 9.544452373488586 8.797214325325626; … ; 8.842164952639337 9.544452373488586 … 11.893826801356274 7.731494275677011; 6.920548393382659 8.797214325325626 … 7.731494275677011 9.904338898917135], L2Regularization{Float64}(0.0), Any[], NoNormalization(), 32, RegularizedLeastSquares.CGNRState{Float64, Float64, Vector{Float64}}([4.0e-323, 5.4e-323, 1.0e-322, 1.14e-322, 1.24e-322, 1.3e-322, 1.5e-322, 1.53e-322, 1.6e-322, 1.8e-322, 1.83e-322, 1.93e-322, 2.0e-322, 2.03e-322, 2.17e-322, 2.2e-322], [1.5e-323, 1.5e-323, 6.9411664211838e-310, 6.94116642118695e-310, 2.0e-323, 2.0e-323, 6.9411664211901e-310, 6.94116642119328e-310, 2.5e-323, 3.0e-323, 6.94116642119644e-310, 6.9411664211996e-310, 3.5e-323, 4.0e-323, 6.94116642120276e-310, 6.9411664212059e-310], [1.398106786369495e-76, 1.7364951552627087e156, 1.7364950529692583e156, 4.083488118020984e233, 1.4142891847151206e-76, 4.474713132109184e40, 1.398106786369495e-76, 4.391451444348257e155, 1.21975e-319, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [6.94120106112977e-310, 6.94120106112977e-310, 6.9411251996269e-310, 6.94112519963006e-310, 6.94112519963164e-310, 6.94120106112977e-310, 6.94120106112977e-310, 6.9411251996332e-310, 6.9411251996348e-310, 6.9411925761922e-310, 6.94112519963796e-310, 6.94112519963954e-310, 6.94112519964113e-310, 6.9412010613357e-310, 6.9411251996443e-310, 6.94112519964587e-310], 0.0, 0.0, 0.0, 0, 2.220446049250313e-16, 0.0))

We can then simply pass the measurement matrix to the solver. The result will be the same as if we passed each colument of B seperately:

x_approx = solve!(solver, B)
+solver = createLinearSolver(CGNR, A; iterations=32)
CGNR{Matrix{Float64}, Matrix{Float64}, L2Regularization{Float64}, Vector{Any}}([0.6527068141140602 0.041093330727839494 … 0.3647100855894174 0.7474582012388873; 0.11001127724690773 0.7096476440856907 … 0.5016092758550019 0.5961800090879167; … ; 0.6314198674939352 0.5359764127574309 … 0.3054068564525615 0.4478050811054971; 0.22800764011737606 0.041849349616210674 … 0.948829559816521 0.7626690668119286], [12.546066420202758 9.718629092250646 … 9.726700025831162 8.03541407846781; 9.718629092250646 11.074378689407043 … 8.93865114968054 8.010013422365175; … ; 9.726700025831162 8.93865114968054 … 12.363709622818673 8.50001448956229; 8.03541407846781 8.010013422365175 … 8.50001448956229 9.272001596436729], L2Regularization{Float64}(0.0), Any[], NoNormalization(), 32, RegularizedLeastSquares.CGNRState{Float64, Float64, Vector{Float64}}([5.0e-324, NaN, 1.5e-323, 2.0e-323, NaN, 3.0e-323, 3.5e-323, NaN, NaN, 5.0e-323, 5.4e-323, 6.0e-323, 6.4e-323, 7.0e-323, 7.4e-323, NaN], [5.0e-324, 1.0e-323, 1.5e-323, 2.0e-323, 2.5e-323, 3.0e-323, 3.5e-323, 4.0e-323, 4.4e-323, 5.0e-323, 5.4e-323, 6.0e-323, 6.4e-323, 7.0e-323, 7.4e-323, 8.0e-323], [1.5e-323, 2.5e-323, 6.92240780026115e-310, 6.9224078011133e-310, 3.0e-323, 4.0e-323, 6.9224078002643e-310, 6.92240780026747e-310, 4.4e-323, 4.4e-323, 6.92240780027064e-310, 6.9224078002738e-310, 5.0e-323, 8.0e-323, 6.92240780027696e-310, 6.9224078002801e-310], [5.0e-324, 0.0, 0.0, 5.0e-324, 0.0, 5.0e-324, 5.0e-324, 0.0, 0.0, 0.0, 5.0e-324, 5.0e-324, 5.0e-324, 5.0e-324, 5.0e-324, 0.0], 0.0, 0.0, 0.0, 0, 2.220446049250313e-16, 0.0))

We can then simply pass the measurement matrix to the solver. The result will be the same as if we passed each colument of B seperately:

x_approx = solve!(solver, B)
 size(x_approx)
(16, 4)

The previous solve! call was still executed sequentially. To execute it in parallel, we have to specify a multi-threaded scheduler as a keyword-argument of the solve! call. RegularizedLeastSquares.jl provides a MultiThreadingState scheduler that can be used for this purpose. This scheduler is based on the Threads.@threads macro:

x_multi = solve!(solver, B; scheduler = MultiThreadingState)
 x_approx == x_multi
true

Custom Scheduling

It is possible to implement custom scheduling. The following code shows how to implement this for the Threads.@spawn macro. Usually one this to implement multi-threading with a package such as FLoop.jl or ThreadPools.jl for thread pinning:

Since most solver have conv. criteria, they can finish at different iteration numbers, which we track this information with flags.

 mutable struct SpawnState{S, ST <: AbstractSolverState{S}} <: RegularizedLeastSquares.AbstractMatrixSolverState{S}
    states::Vector{ST}
@@ -25,4 +25,4 @@
   end
   return state.active, state
 end

Now we can simply use the SpawnState scheduler in the solve! call:

x_custom = solve!(solver, B; scheduler = SpawnState)
-x_approx == x_multi
true

This page was generated using Literate.jl.

+x_approx == x_multi
true

This page was generated using Literate.jl.

diff --git a/dev/generated/howto/normal_operator/index.html b/dev/generated/howto/normal_operator/index.html index 2132645..77e5672 100644 --- a/dev/generated/howto/normal_operator/index.html +++ b/dev/generated/howto/normal_operator/index.html @@ -14,22 +14,22 @@ solver = createLinearSolver(CGNR, A; AHA = adjoint(A) * A, reg = L2Regularization(0.0001), iterations=32); x_approx = solve!(solver, b)
16-element Vector{Float64}:
- -1.4722416522658646
- -1.347082491121041
-  0.022170470402297634
-  0.07736300250573858
- -0.2024295635951121
-  0.17370313463167475
-  1.845707647047999
-  0.782538972200631
- -0.15770522802816953
-  1.0957228682293045
-  1.53989966090913
-  0.21029993870061472
-  0.8152953191744927
-  0.49544840980877697
- -0.8217253233668218
-  0.581125395480638

The normal operator can also be computed using the normalOperator function from LinearOperatorCollection.jl. This is useful if the normal operator is not directly available or shouldn't be stored in memory. This function is opinionated and attempts to optimize the resulting operator for iterative applications. Specifying a custom method for a custom operator allows one to control this optimization.

An example of such an optimization is a matrix-free weighting of $\mathbf{A}$ as shown in the Weighting example:

using LinearOperatorCollection
+  0.31534446277010153
+  0.5143626685048083
+ -1.8130746610813213
+  0.8170643327599431
+ -0.6078294016264768
+ -0.8370171044562775
+  0.9144783514875474
+ -1.4181712951259942
+ -2.1224277232727706
+ -1.2446555217831627
+ -0.14558674090152143
+  1.4802105846687157
+ -1.1921219710737887
+  0.182549642953892
+  0.6360943737883537
+  0.3727096693333945

The normal operator can also be computed using the normalOperator function from LinearOperatorCollection.jl. This is useful if the normal operator is not directly available or shouldn't be stored in memory. This function is opinionated and attempts to optimize the resulting operator for iterative applications. Specifying a custom method for a custom operator allows one to control this optimization.

An example of such an optimization is a matrix-free weighting of $\mathbf{A}$ as shown in the Weighting example:

using LinearOperatorCollection
 weights = rand(32)
 WA = ProdOp(WeightingOp(weights), A)
 AHA = LinearOperatorCollection.normalOperator(WA)
Linear operator
@@ -50,19 +50,19 @@
   \mathbf{A}^*\tilde{\mathbf{W}}\mathbf{A}
 \end{equation}\]

The optimized normal operator can then be passed to the solver:

solver = createLinearSolver(CGNR, WA; AHA = AHA, reg = L2Regularization(0.0001), iterations=32);
 x_approx2 = solve!(solver, weights .* b)
16-element Vector{Float64}:
- -1.4722392223282568
- -1.3470927044575378
-  0.022171942294678518
-  0.07735244949331373
- -0.20243776633090127
-  0.17367906768125957
-  1.8457002250464758
-  0.7825290615850105
- -0.15769518575899574
-  1.09572961848627
-  1.5398720644874901
-  0.21028472966029874
-  0.8152877867726963
-  0.49542635216135034
- -0.8217331287752728
-  0.5811079898228333

Of course it is also possible to optimize a normal operator with other means and pass it to the solver via the AHA keyword argument.

It is also possible to only supply the normal operator to these solvers, however on then needs to supply $\mathbf{A^*b}$ intead of $\mathbf{b}$.


This page was generated using Literate.jl.

+ 0.31533408825332054 + 0.5143594211110782 + -1.8130445546585328 + 0.8170601568386201 + -0.6078024961280396 + -0.8370016365926805 + 0.9144077453768623 + -1.4181878527663558 + -2.1223786996393796 + -1.2445972624696904 + -0.14558250213206408 + 1.4802264923614155 + -1.1920466364049744 + 0.18257887901512915 + 0.6360983962297961 + 0.37269551395181366

Of course it is also possible to optimize a normal operator with other means and pass it to the solver via the AHA keyword argument.

It is also possible to only supply the normal operator to these solvers, however on then needs to supply $\mathbf{A^*b}$ intead of $\mathbf{b}$.


This page was generated using Literate.jl.

diff --git a/dev/generated/howto/plug-and-play/index.html b/dev/generated/howto/plug-and-play/index.html index c19e93c..12a5f13 100644 --- a/dev/generated/howto/plug-and-play/index.html +++ b/dev/generated/howto/plug-and-play/index.html @@ -5,19 +5,19 @@ b = A*x;

For the documentation we will just use the identity function as a placeholder for the PnP prior.

model = identity
identity (generic function with 1 method)

In practice, you would replace this with a neural network:

using Flux
 model = Flux.loadmodel!(model, ...)

The model can then be used together with the PnPRegularization term:

reg = PnPRegularization(1.0; model = model, shape = [16]);

Since models often expect a specific input range, we can use the MinMaxTransform to normalize the input:

reg = PnPRegularization(1.0; model = model, shape = [16], input_transform = RegularizedLeastSquares.MinMaxTransform);

Custom input transforms can be implemented by passing something callable as the input_transform keyword argument. For more details see the PnPRegularization documentation.

The regularization term can then be used in the solver:

solver = createLinearSolver(Kaczmarz, A; reg = reg, iterations = 32)
 x_approx = solve!(solver, b)
16-element Vector{Float64}:
- -0.3949633544832487
-  1.7205196811540353
- -0.1685508922632013
-  0.28281630348455344
-  0.1632164644498708
- -0.7017620326139309
- -2.564067625921601
-  0.017764258364315832
- -0.44593284864452576
-  0.39522723946562843
-  0.37085146139811664
-  1.7079945071451283
-  0.3692138462057746
- -0.060338669509075515
- -0.5051250661075604
-  2.2843383570517193

This page was generated using Literate.jl.

+ -0.01252802242361617 + 1.7232060596321972 + 0.1786745699644965 + -0.5295680487236962 + -0.470419591708525 + -1.033594758849739 + -1.3815784212027002 + -1.6930830276031839 + 1.237712467692882 + 0.49215477471771507 + -1.9074198244180813 + 1.3969294818927342 + 0.5436982068756469 + 1.2384479677615357 + -0.9861053249705004 + -1.562446013442756

This page was generated using Literate.jl.

diff --git a/dev/generated/howto/weighting/index.html b/dev/generated/howto/weighting/index.html index c8b791c..0556116 100644 --- a/dev/generated/howto/weighting/index.html +++ b/dev/generated/howto/weighting/index.html @@ -12,4 +12,4 @@ P = ProdOp(W, A) solver = createLinearSolver(Kaczmarz, P; reg = L2Regularization(0.0001), iterations=10) x_approx2 = solve!(solver, W * b) -isapprox(x_approx, x_approx2)
true

This page was generated using Literate.jl.

+isapprox(x_approx, x_approx2)
true

This page was generated using Literate.jl.

diff --git a/dev/index.html b/dev/index.html index 47fe2e9..daa3906 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -Home · RegularizedLeastSquares.jl

RegularizedLeastSquares.jl

Solvers for Linear Inverse Problems using Regularization Techniques

Introduction

RegularizedLeastSquares.jl is a Julia package for solving large linear systems using various types of algorithms. Ill-conditioned problems arise in many areas of practical interest. Regularisation techniques and nonlinear problem formulations are often used to solve these problems. This package provides implementations for a variety of solvers used in areas such as MPI and MRI. In particular, this package serves as the optimization backend of the Julia packages MPIReco.jl and MRIReco.jl.

The implemented methods range from the $l^2_2$-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.

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.

Features

  • Variety of optimization algorithms optimized for least squares problems
  • Support for matrix-free operators
  • GPU support

Usage

See also

Packages:

Organizations:

+Home · RegularizedLeastSquares.jl

RegularizedLeastSquares.jl

Solvers for Linear Inverse Problems using Regularization Techniques

Introduction

RegularizedLeastSquares.jl is a Julia package for solving large linear systems using various types of algorithms. Ill-conditioned problems arise in many areas of practical interest. Regularisation techniques and nonlinear problem formulations are often used to solve these problems. This package provides implementations for a variety of solvers used in areas such as MPI and MRI. In particular, this package serves as the optimization backend of the Julia packages MPIReco.jl and MRIReco.jl.

The implemented methods range from the $l^2_2$-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.

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.

Features

  • Variety of optimization algorithms optimized for least squares problems
  • Support for matrix-free operators
  • GPU support

Usage

See also

Packages:

Organizations:

diff --git a/dev/objects.inv b/dev/objects.inv index 121d3d4fbba1d0fcc41e9b4582f71eb6aa329a07..afd86b657fb2616d1a0367596dc276b900364526 100644 GIT binary patch delta 12 TcmX@XbAo3=Fr&%FkT0wNAE*Ss delta 12 TcmX@XbAo3=Fr)FtkT0wNAEN}m diff --git a/dev/solvers/index.html b/dev/solvers/index.html index e589383..3b806a9 100644 --- a/dev/solvers/index.html +++ b/dev/solvers/index.html @@ -17,4 +17,4 @@ rho::T # ... iteration::Int64 -end

States are subtypes of the parametric AbstractSolverState{S} type. The state fields of solvers can be exchanged with different state belonging to the correct solver S. This means that the states can be used to realize custom variants of an existing solver: ```julia mutable struct VariantState{T, tempT} <: AbstractSolverState{Solver} x::tempT other::tempT

...

iteration::Int64 end

SolverVariant(A; kwargs...) = Solver(A, VariantState(kwargs...))

function iterate(solver::Solver, state::VarianteState)

Custom iteration

end

+end

States are subtypes of the parametric AbstractSolverState{S} type. The state fields of solvers can be exchanged with different state belonging to the correct solver S. This means that the states can be used to realize custom variants of an existing solver: ```julia mutable struct VariantState{T, tempT} <: AbstractSolverState{Solver} x::tempT other::tempT

...

iteration::Int64 end

SolverVariant(A; kwargs...) = Solver(A, VariantState(kwargs...))

function iterate(solver::Solver, state::VarianteState)

Custom iteration

end

This page was generated using Literate.jl.