# Specifying (Non)Linear Solvers

One of the key features of DifferentialEquations.jl is its flexibility. Keeping with this trend, many of the native Julia solvers provided by DifferentialEquations.jl allow you to choose the method for linear and nonlinear solving. This section details how to make that choice.

## Linear Solvers: linsolve Specification

For differential equation integrators which use linear solvers, an argument to the method linsolve determines the linear solver which is used. The signature is:

linsolve! = linsolve(Val{:init},f,x;kwargs...)
linsolve!(x,A,b,matrix_updated=false;kwargs...)

This is an in-place function which updates x by solving Ax=b. The user should specify the function linsolve(Val{:init},f,x) which returns a linsolve! function. The setting matrix_updated determines whether the matrix A has changed from the last call. This can be used to smartly cache factorizations.

Note that linsolve! needs to accept splatted keyword arguments. The possible arguments passed to the linear solver are as follows:

• Pl, a pre-specified left preconditioner which utilizes the internal adaptive norm estimates
• Pr, a pre-specified right preconditioner which utilizes the internal adaptive norm estimates
• tol, a linear solver tolerance specified from the ODE solver's implicit handling

### Pre-Built Linear Solver Choices

The following choices of pre-built linear solvers exist:

• DefaultLinSolve
• LinSolveFactorize
• LinSolveGPUFactorize
• LinSolveGMRES
• LinSolveCG
• LinSolveBiCGStabl
• LinSolveChebyshev
• LinSolveMINRES
• LinSolveIterativeSolvers

### DefaultLinSolve

The default linear solver is DefaultLinSolve. This method is adaptive, and automatically chooses an LU factorization choose for dense and sparse arrays, and is compatible with GPU-based arrays. When the Jacobian is an AbstractDiffEqOperator, i.e. is matrix-free, DefaultLinSolve defaults to using a gmres iterative solver.

### Basic linsolve method choice: Factorization by LinSolveFactorize

The easiest way to specify a linsolve is by a factorization function which generates a type on which \ (or A_ldiv_B!) is called. This is done through the helper function LinSolveFactorize which makes the appropriate function. For example, the Rosenbrock23 takes in a linsolve function, which we can choose to be a QR-factorization by:

Rosenbrock23(linsolve=LinSolveFactorize(qrfact!))

LinSolveFactorize takes in a function which returns an object that can \. Direct methods like qrfact! will automatically cache the factorization, making it efficient for small dense problems.

However, for large sparse problems, you can let \ be an iterative method. For example, using PETSc.jl, we can define our factorization function to be:

linsolve = LinSolveFactorize((A) -> KSP(A, ksp_type="gmres", ksp_rtol=1e-6))

This function creates a KSP type which makes \ perform the GMRES iterative method provided by PETSc.jl. Thus if we pass this function into the algorithm as the factorization method, all internal linear solves will happen by PETSc.jl.

If one has a problem with a sufficiently large Jacobian (~100x100) and a sufficiently powerful GPU, it can make sense to offload the factorization and backpropogation steps to the GPU. For this, the LinSolveGPUFactorize linear solver is provided. It works similarly to LinSolveFactorize, but the matrix is automatically sent to the GPU as a CuArray and the ldiv! is performed against a CUDA QR factorization of the matrix.

Note that this method requires that you have done using CuArrays in your script. A working installation of CuArrays.jl is required, which requires an installation of CUDA Toolkit.

### IterativeSolvers.jl-Based Methods

The signature for LinSolveIterativeSolvers is:

LinSolveIterativeSolvers(generate_iterator,args...;
Pl=IterativeSolvers.Identity(),
Pr=IterativeSolvers.Identity(),
kwargs...)

where Pl is the left preconditioner, Pr is the right preconditioner, and the other args... and kwargs... are passed into the iterative solver chosen in generate_iterator which designates the construction of an iterator from IterativeSolvers.jl. For example, using gmres_iterable! would make a version that uses IterativeSolvers.gmres. The following are aliases to common choices:

• LinSolveGMRES – GMRES
• LinSolveCG – CG (Conjugate Gradient)
• LinSolveBiCGStabl – BiCGStabl Stabilized Bi-Conjugate Gradient
• LinSolveChebyshev – Chebyshev
• LinSolveMINRES – MINRES

which all have the same arguments as LinSolveIterativeSolvers except with generate_iterator pre-specified.

### Implementing Your Own LinSolve: How LinSolveFactorize Was Created

In order to make your own linsolve functions, let's look at how the LinSolveFactorize function is created. For example, for an LU-Factorization, we would like to use lufact! to do our linear solving. We can directly write this as:

function linsolve!(::Type{Val{:init}},f,u0,kwargs...)
function _linsolve!(x,A,b,update_matrix=false,kwargs...)
_A = lufact!(A)
ldiv!(x,_A,b)
end
end

This initialization function returns a linear solving function that always computes the LU-factorization and then does the solving. This method works fine and you can pass it to the methods like

Rosenbrock23(linsolve=linsolve!)

and it will work, but this method does not cache _A, the factorization. This means that, even if A has not changed, it will re-factorize the matrix.

To change this, we can instead create a call-overloaded type. The generalized form of this is:

mutable struct LinSolveFactorize{F}
factorization::F
A
end
LinSolveFactorize(factorization) = LinSolveFactorize(factorization,nothing)
function (p::LinSolveFactorize)(x,A,b,matrix_updated=false)
if matrix_updated
p.A = p.factorization(A)
end
A_ldiv_B!(x,p.A,b)
end
function (p::LinSolveFactorize)(::Type{Val{:init}},f,u0_prototype)
LinSolveFactorize(p.factorization,nothing)
end
linsolve = LinSolveFactorize(lufact!)

LinSolveFactorize is a type which holds the factorization method and the pre-factorized matrix. When linsolve is passed to the ODE/SDE/etc. solver, it will use the function linsolve(Val{:init},f,u0_prototype) to create a LinSolveFactorize object which holds the factorization method and a cache for holding a factorized matrix. Then

function (p::LinSolveFactorize)(x,A,b,matrix_updated=false)
if matrix_updated
p.A = p.factorization(A)
end
A_ldiv_B!(x,p.A,b)
end

is what's used in the solver's internal loop. If matrix_updated is true, it will re-compute the factorization. Otherwise it just solves the linear system with the cached factorization. This general idea of using a call-overloaded type can be employed to do many other things.