# SparseDiffTools.jl

This package is for exploiting sparsity in Jacobians and Hessians to accelerate computations. Matrix-free Jacobian-vector product and Hessian-vector product operators are provided that are compatible with AbstractMatrix-based libraries like IterativeSolvers.jl for easy and efficient Newton-Krylov implementation. It is possible to perform matrix coloring, and utilize coloring in Jacobian and Hessian construction.

Optionally, automatic and numerical differentiation are utilized.

## Example

Suppose we had the function

```
fcalls = 0
function f(y,x) # in-place
global fcalls += 1
for i in 2:length(x)-1
y[i] = x[i-1] - 2x[i] + x[i+1]
end
y[1] = -2x[1] + x[2]
y[end] = x[end-1] - 2x[end]
nothing
end
function g(x) # out-of-place
global fcalls += 1
y = zero(x)
for i in 2:length(x)-1
y[i] = x[i-1] - 2x[i] + x[i+1]
end
y[1] = -2x[1] + x[2]
y[end] = x[end-1] - 2x[end]
y
end
```

For this function, we know that the sparsity pattern of the Jacobian is a
`Tridiagonal`

matrix. However, if we didn't know the sparsity pattern for
the Jacobian, we could use the `Symbolics.jacobian_sparsity`

function to automatically
detect the sparsity pattern. We declare that the function `f`

outputs a
vector of length 30 and takes in a vector of length 30, and `jacobian_sparsity`

returns
a `SparseMatrixCSC`

:

```
using Symbolics
input = rand(30)
output = similar(input)
sparsity_pattern = Symbolics.jacobian_sparsity(f,output,input)
jac = Float64.(sparsity_pattern)
```

Now we call `matrix_colors`

to get the colorvec vector for that matrix:

```
using SparseDiffTools
colors = matrix_colors(jac)
```

Since `maximum(colors)`

is 3, this means that finite differencing can now
compute the Jacobian in just 4 `f`

-evaluations. Generating the sparsity
pattern used 1 (pseudo) `f`

-evaluation, so the total number of times that
`f`

is called to compute the sparsity pattern plus the entire 30x30 Jacobian
is 5 times:

```
using FiniteDiff
FiniteDiff.finite_difference_jacobian!(jac, f, rand(30), colorvec=colors)
@show fcalls # 5
```

In addition, a faster forward-mode autodiff call can be utilized as well:

`forwarddiff_color_jacobian!(jac, f, x, colorvec = colors)`

If one only needs to compute products, one can use the operators. For example,

```
x = rand(30)
J = JacVec(f,x)
```

makes `J`

into a matrix-free operator which calculates `J*v`

products. For
example:

```
v = rand(30)
res = similar(v)
mul!(res,J,v) # Does 1 f evaluation
```

makes `res = J*v`

. Additional operators for `HesVec`

exists, including
`HesVecGrad`

which allows one to utilize a gradient function. These operators
are compatible with iterative solver libraries like IterativeSolvers.jl, meaning
the following performs the Newton-Krylov update iteration:

```
using IterativeSolvers
gmres!(res,J,v)
```

## Documentation

### Matrix Coloring

This library extends the common `ArrayInterfaceCore.matrix_colors`

function to allow
for coloring sparse matrices using graphical techniques.

Matrix coloring allows you to reduce the number of times finite differencing
requires an `f`

call to `maximum(colors)+1`

, or reduces automatic differentiation
to using `maximum(colors)`

partials. Since normally these values are `length(x)`

,
this can be significant savings.

The API for computing the colorvec vector is:

```
matrix_colors(A::AbstractMatrix,alg::SparseDiffToolsColoringAlgorithm = GreedyD1Color();
partition_by_rows::Bool = false)
```

The first argument is the abstract matrix which represents the sparsity pattern
of the Jacobian. The second argument is the optional choice of coloring algorithm.
It will default to a greedy distance 1 coloring, though if your special matrix
type has more information, like is a `Tridiagonal`

or `BlockBandedMatrix`

, the
colorvec vector will be analytically calculated instead. The keyword argument
`partition_by_rows`

allows you to partition the Jacobian on the basis of rows instead
of columns and generate a corresponding coloring vector which can be used for
reverse-mode AD. Default value is false.

The result is a vector which assigns a colorvec to each column (or row) of the matrix.

### Colorvec-Assisted Differentiation

Colorvec-assisted differentiation for numerical differentiation is provided by FiniteDiff.jl and for automatic differentiation is provided by ForwardDiff.jl.

For FiniteDiff.jl, one simply has to use the provided `colorvec`

keyword
argument. See
the FiniteDiff Jacobian documentation
for more details.

For forward-mode automatic differentiation, use of a colorvec vector is provided by the following function:

```
forwarddiff_color_jacobian!(J::AbstractMatrix{<:Number},
f,
x::AbstractArray{<:Number};
dx = nothing,
colorvec = eachindex(x),
sparsity = nothing)
```

Notice that if a sparsity pattern is not supplied then the built Jacobian will
be the compressed Jacobian: `sparsity`

must be a sparse matrix or a structured matrix
(`Tridiagonal`

, `Banded`

, etc. conforming to the ArrayInterfaceCore.jl specs) with the
appropriate sparsity pattern to allow for decompression.

This call will allocate the cache variables each time. To avoid allocating the cache, construct the cache in advance:

```
ForwardColorJacCache(f,x,_chunksize = nothing;
dx = nothing,
colorvec=1:length(x),
sparsity = nothing)
```

and utilize the following signature:

```
forwarddiff_color_jacobian!(J::AbstractMatrix{<:Number},
f,
x::AbstractArray{<:Number},
jac_cache::ForwardColorJacCache)
```

`dx`

is a pre-allocated output vector which is used to declare the output size,
and thus allows for specifying a non-square Jacobian.

If one is using an out-of-place function `f(x)`

, then the alternative form
ca be used:

```
jacout = forwarddiff_color_jacobian(g, x,
dx = similar(x),
colorvec = 1:length(x),
sparsity = nothing,
jac_prototype = nothing)
```

Note that the out-of-place form is efficient and compatible with StaticArrays.
One can specify the type and shape of the output Jacobian by giving an
additional `jac_prototype`

to the out-of place `forwarddiff_color_jacobian`

function, otherwise it will become a dense matrix. If `jac_prototype`

and
`sparsity`

are not specified, then the oop Jacobian will assume that the
function has a *square* Jacobian matrix. If it is not the case, please specify
the shape of output by giving `dx`

.

Similar functionality is available for Hessians, using finite differences of forward derivatives. Given a scalar function `f(x)`

, a vector value for `x`

,
and a color vector and sparsity pattern, this can be accomplished using
`numauto_color_hessian`

or its in-place form `numauto_color_hessian!`

.

```
H = numauto_color_hessian(f, x, colorvec, sparsity)
numauto_color_hessian!(H, f, x, colorvec, sparsity)
```

To avoid unnecessary allocations every time the Hessian is computed,
construct a `ForwardColorHesCache`

beforehand:

```
hescache = ForwardColorHesCache(f, x, colorvec, sparsity)
numauto_color_hessian!(H, f, x, hescache)
```

By default, these methods use a mix of numerical and automatic differentiation,
namely by taking finite differences of gradients calculated via ForwardDiff.jl.
Alternatively, if you have your own custom gradient function `g!`

, you can specify
it as an argument to `ForwardColorHesCache`

:

`hescache = ForwardColorHesCache(f, x, colorvec, sparsity, g!)`

Note that any user-defined gradient needs to have the signature `g!(G, x)`

,
i.e. updating the gradient `G`

in place.

### Jacobian-Vector and Hessian-Vector Products

Matrix-free implementations of Jacobian-Vector and Hessian-Vector products is
provided in both an operator and function form. For the functions, each choice
has the choice of being in-place and out-of-place, and the in-place versions
have the ability to pass in cache vectors to be non-allocating. When in-place
the function signature for Jacobians is `f!(du,u)`

, while out-of-place has
`du=f(u)`

. For Hessians, all functions must be `f(u)`

which returns a scalar

The functions for Jacobians are:

```
auto_jacvec!(dy, f, x, v,
cache1 = ForwardDiff.Dual{DeivVecTag}.(x, v),
cache2 = ForwardDiff.Dual{DeivVecTag}.(x, v))
auto_jacvec(f, x, v)
# If compute_f0 is false, then `f(cache1,x)` will be computed
num_jacvec!(dy,f,x,v,cache1 = similar(v),
cache2 = similar(v);
compute_f0 = true)
num_jacvec(f,x,v,f0=nothing)
```

For Hessians, the following are provided:

```
num_hesvec!(dy,f,x,v,
cache1 = similar(v),
cache2 = similar(v),
cache3 = similar(v))
num_hesvec(f,x,v)
numauto_hesvec!(dy,f,x,v,
cache = ForwardDiff.GradientConfig(f,v),
cache1 = similar(v),
cache2 = similar(v))
numauto_hesvec(f,x,v)
autonum_hesvec!(dy,f,x,v,
cache1 = similar(v),
cache2 = ForwardDiff.Dual{DeivVecTag}.(x, v),
cache3 = ForwardDiff.Dual{DeivVecTag}.(x, v))
autonum_hesvec(f,x,v)
```

In addition,
the following forms allow you to provide a gradient function `g(dy,x)`

or `dy=g(x)`

respectively:

```
num_hesvecgrad!(dy,g,x,v,
cache2 = similar(v),
cache3 = similar(v))
num_hesvecgrad(g,x,v)
auto_hesvecgrad!(dy,g,x,v,
cache2 = ForwardDiff.Dual{DeivVecTag}.(x, v),
cache3 = ForwardDiff.Dual{DeivVecTag}.(x, v))
auto_hesvecgrad(g,x,v)
```

The `numauto`

and `autonum`

methods both mix numerical and automatic differentiation, with
the former almost always being more efficient and thus being recommended.

Optionally, if you load Zygote.jl, the following `numback`

and `autoback`

methods are available and allow numerical/ForwardDiff over reverse mode
automatic differentiation respectively, where the reverse-mode AD is provided by Zygote.jl.
Currently these methods are not competitive against `numauto`

, but as Zygote.jl gets
optimized these will likely be the fastest.

```
using Zygote # Required
numback_hesvec!(dy,f,x,v,
cache1 = similar(v),
cache2 = similar(v))
numback_hesvec(f,x,v)
# Currently errors! See https://github.com/FluxML/Zygote.jl/issues/241
autoback_hesvec!(dy,f,x,v,
cache2 = ForwardDiff.Dual{DeivVecTag}.(x, v),
cache3 = ForwardDiff.Dual{DeivVecTag}.(x, v))
autoback_hesvec(f,x,v)
```

*v and H*v Operators

JThe following produce matrix-free operators which are used for calculating
Jacobian-vector and Hessian-vector products where the differentiation takes
place at the vector `u`

:

```
JacVec(f,x::AbstractArray;autodiff=true)
HesVec(f,x::AbstractArray;autodiff=true)
HesVecGrad(g,x::AbstractArray;autodiff=false)
```

These all have the same interface, where `J*v`

utilizes the out-of-place
Jacobian-vector or Hessian-vector function, whereas `mul!(res,J,v)`

utilizes
the appropriate in-place versions. To update the location of differentiation
in the operator, simply mutate the vector `u`

: `J.u .= ...`

.