:*(F, m)

Constructs F*m where F is a Jets linear (e.g. JopLn) or nonlinear (JopNl) operator.

:+(A₂, A₁)

Construct and return the linear combination of the two Jets operators A₁::Jop and A₂::Jop. Note that A₁ and A₂ must have consistent (same size and type) domains and ranges.

Example

A = 1.0*A₁ - 2.0*A₂ + 3.0*A₃

:-(A₂, A₁)

Construct and return the linear combination of the two Jets operators A₁::Jop and A₂::Jop. Note that A₁ and A₂ must have consistent (same size and type) domains and ranges.

Example

A = 1.0*A₁ - 2.0*A₂ + 3.0*A₃

:∘(A₂, A₁)

Construct the composition of the two Jets operators. Note that when applying the composition operator, operators are applied in order from right to left: first A₁ and then A₂.

Example

using Jets
dg!(d,m;mₒ) = @. d = 2*m
A₁ = JopLn(Jet(;dom=JetSpace(Float64,2), rng=JetSpace(Float64,2), df! = dg!))
A₂ = JopLn(Jet(;dom=JetSpace(Float64,2), rng=JetSpace(Float64,2), df! = dg!))
C = A₂ ∘ A₁
m = rand(domain(C))
d = C * m

adjoint(A::Union{JopLn, JopAdjoint})

Return the adjoint of A.

convert(Array, A::JopLn)

Convert a linear Jets operator into its equivalent matrix.

eltype(A::Union{Jet,Jop,JopAdjoint})

Return the element type of A.

eltype(R)

Return the element type of the space R::JetAbstractSpace.

iszero(A::Union{Jet, Jop})

Return true if A was constructed via JopZeroBlock.

length(R)

Return the dimension the space R::JetAbstractSpace

ones(R)

Construct an array of the type and size defined by R::JetAbstractSpace{T} and filled with one(T).

rand(R)

Construct an array of the type and size defined by the R::JetAbstractSpace, and filled with random values from 0 - 1.

randn(R)

Construct an array of the type and size defined by the R::JetAbstractSpace, and filled with random values from Gaussian distribution.

R = range(A)

Return R::JetAbstractSpace, which is the range of A::Union{Jet, Jop, AbstractMatrix}.

reshape(x, R)

Returns an array that is consistent with the shape of the space R::JetAbstractSpace, and shares memory with x.

size(R[,i])

Return the shape of the array associated to the Jet space R::JetAbstractSpace. If i is specifid, then returns the length along the ith array dimension.

size(A[, i])

Return the size of the range and domain of A::Union{Jet,Jop}. With no arguments, return (length(range(A)), length(domain(A))). With i specified, return length(range(A)) for i = 1 and return length(domain(A)) otherwise.

B = vec(A)

B is equivelent to A except that its domain and range are "vectorized". This is useful when calling algorithms that expect vectors in the domain and range of the operator. One example of this is the lsqr method in the IterativeSolvers package.

Example

using Jets, JetPack, IterativeSolvers

A = JopDiagonal(rand(10,11))
d = rand(range(A))
m = lsqr(vec(A), vec(d))

zeros(R)

Construct an array of the type and size defined by R::JetAbstractSpace{T} and filled with zero(T).

JopZeroBlock(dom, rng)

Construct a Jets operator that is equivalent to a matrix of zeros, and that maps from dom::JetAbstractSpace to rng::JetAbstractSpace. This can be useful when forming block operators that contain zero blocks.

R = domain(A)

Return R::JetAbstractSpace, which is the domain of A::Union{Jet, Jop, AbstractMatrix}.

lhs,rhs = dot_product_test(A, m, d; mmask, dmask)

Compute and return the left and right hand sides of the dot product test:

<d,Am> ≈ <Aᴴd,m>

Here Aᴴ is the conjugate transpose or adjoint of A, and <x, y> denotes the inner product of vectors x and y. The left and right hand sides of the dot product test are expected to be equivalent close to machine precision for operator A. If the equality does not hold this can indicate a problem with the implementation of the operator A.

This function provides the optional named arguments mmask and dmask which are vectors in the domain and range of A that are applied via elementwise multiplication to mask the vectors m and d before applying of the operator, as shown below. Here we use ∘ to represent the Hadamard product (elementwise multiplication) of two vectors.

<dmask ∘ d, A (mmask ∘ m)> ≈ <Aᵀ (dmask ∘ d), mmask ∘ m>

You can test the relative accuracy of the operator with this relation for the left hand side lhs and right hand side rhs returned by this function:

|lhs - rhs| / |lhs + rhs| < ϵ

getblock(A, i, j)

Return the block of the Jets block operator A that corresponds to row block i and column block j.

indices(R, iblock)

Return the linear indices associated with block iblock in the Jets block space R::JetBSpace.

Example

Consider a block operator with 2 row-blocks and 3 column-blocks. We can use indices to determine the elements of the vector that are associatd with the first block of its domain:

using Pkg
pkg"add Jets JetPack"
using Jets, JetPack
A = @blockop [JopDiagonal(rand(10)) for irow=1:2, icol=1:3]
indices(domain(A), 1) # returns indices 1:10

isblockop(A)

Return true if A is a Jets block operator.

jacobian!(F, mₒ)

Return the jacobian of F::Union{Jet, Jop, AbstractMatrix} at the linearization point m₀. The jacobian shares the underlying Jet with F. This means that if the jacobian may mutate F.

jacobian(F, mₒ)

Return the jacobian of F::Union{Jet, Jop, AbstractMatrix} at the point m₀. The linearization constructs a new underlying Jet.

jet(A)

Return the Jet associated with A::Union{Jop, JopAdjoint}.

lhs,rhs = linearity_test(A::Jop)

test the the linear Jet operator A satisfies the following test for linearity:

A(m_1+m_2)=Am_1 + A_m2

μobs, μexp = linearization_test(F, mₒ; μ)

Thest that the jacobian, J, of F satisfies the Taylor expansion:

F(m) = F(m_o) + F'(m_o)δm + O(δm^2)

nblocks(R)

Return the number of blocks in the Jets block space R::JetBSpace.

nblocks(A[, i])

Return the number of blocks in the range and domain of the Jets block operator A::Union{Jet, Jop}. With i specified, return nblocks(range(jet)) for i = 1 and return nblocks(domain(jet)) otherwise.

perfstat(A)

Return a Dictionary with performance information for A::Union{Jet,Jop,JopAdjoint}. the perfstat(jet::Jet) method that can be implemented by the author of an operator to track performance metrics.

point!(F, mₒ)

Update the linearization point (model vector) for F::Union{Jet, JopLn, JopAdjoint} to model vector mₒ.

point(F)

Return the linearization point (model vector) mₒ associated with F::Union{Jet, JopLn, JopAdjoint}.

shape(A[, i])

Return the shape of the range and domain of A::Union{Jet, Jop, AbstractMatrix}. With no arguments, return (size(range(A)), size(domain(A))). With i specified, return size(range(A)) for i = 1 and return size(domain(A)) otherwise.

space(R, iblock)

Return the Jets sub-space associated with block iblock in the Jets block space R::JetBSpace.

Example

Consider a block operator with 2 row-blocks and 3 column-blocks. We can use space to determine the sub-space associated with the first block of its domain:

using Pkg
pkg"add Jets JetPack"
using Jets, JetPack
A = @blockop [JopDiagonal(rand(10)) for irow=1:2, icol=1:3]
space(domain(A), 1) # JetSpace(Float64,10)

state!(A::Union{Jet,Jop,JopAdjoint}, s)

Updates and merges the state of the A with s.

state(A::Union{Jet,Jop,JopAdjoint}[, key])

If key::Symbol is specified, then return the state corresponding to key. Otherwise, return the state of A as a NamedTuple.

mul!(d, F, m)

In place version of d=F*m where F is a Jets linear (e.g. JopLn) or nonlinear (JopNl) operator.

randperm(R)

Construct a list of random linear indices over the dimensions of R::JetAbstractSpace. The list is useful for selecting a random subset of a multi-dimensional image.

Example

using Jets
R = JetSpace(Float64, 10, 2)
x = rand(R)
y = x[randperm(R)[1:10]] # get 10 elements at random from x

Array(R)

Construct an uninitialized array of the type and size defined by R::JetsAbstractSpace.

Jet(;dom, rng, f!, df!, df′!, upstate!, s)

Return a Jet with domain dom::JetAbstractSpace, range rng::JetSAbstractpace, with forward mapping f!::Function, linearized forward mapping df!::Function, linearized adjoint mapping df′!::Function, Jacobian state modification function upstate!::Function, and state s::NamedTuple.

A jet describes a function f! and its linearization (forward df!, and adjointdf′!``) about a point.

If one of f! or df! is specified, and df′! is not, then we assume that f!=df!=df′!. This means that the operator is linear and self-adjoint.

If f! and df! are sepecified, bug df′! is not, then we assume that df′!=df!. This means that the operator is nonlinear and self-adjoint.

Example

Consider a nonlinear mapping with a self-adjoint linearization $f(x)=x^2$

using Jets
g!(m) = m.^2
dg!(δm; mₒ) = @. 2*mₒ*δm
jet = Jet(;dom=JetSpace(Float64,2), rng=JetSpace(Float64,2), f! = g!, df! = dg!)

JetSSpace(_T, n, M, map::F)

Construct and return a symmetric space JetSSpace.

parameters

• _T is the type, usually Complex{Float32} or Complex{Float64}.
• n is a tuple that defines the dimensionality of the space.
• M is a tuple that defines which dimensions are symmetric. Note that currently only a single symmetric dimension is supported by the API.
• F is a function that maps indices for the symmetric dimension, described below.

An example requiring a JetSSpace is the Fourier transform: the Fourier transform of a real vector is in a complex space with Hermittian symmetry. Only the positive frequencies are needed, and the spectrum at negative frequencies is the Hermittian conjugate of the spectrum at the corresponding positive frequencies: S(-f) = conj(S(f). For this example the map F is a function that returns the multi-dimensional index of f when given the multi-dimensional index of -f.

See also: JopFft in the JetPackTransforms package.

JetSpace(T, n)

Construct and return a JetSpace of type T and size n

Examples

Create a 100 dimension space with array elelment type Float64 and array size (100,)

R1 = JetSpace(Float64, 100)

Create a 100 dimension space with array element type Float32 and array size (5, 20)

R2 = JetSpace(Float32, 5, 20)

JopLn(; kwargs ...)

Construct a JopLn with Jet constructed from keyword arguments kwargs. This is equivalent to JopLn(Jet(;kwargs...)). Please see Jet for more information.

JopNl(; kwargs ...)

Construct a JopNl with Jet constructed from keyword arguments kwargs. This is equivalent to JopNl(Jet(;kwargs...)). Please see Jet for more information.