# Contribution Guide for Abstract Iterative Solvers

An abstract iterative solver requires one struct, one constructor, and two functions in order to interface with the rest of ClimateMachine. In what follows we will describe in detail the function signatures, return values, and struct properties necessary to build with ClimateMachine.

We have the following concrete implementations:

## Basic Template for an Iterative Solver

A basic template of an iterative solver could be as follows:


export MyIterativeMethod

# struct
struct MyIterativeMethod{FT} <: AbstractIterativeSystemSolver
# minimum
rtol::FT
atol::FT
# Add more structure if necessary
end

# constructor
function MyIterativeMethod(args...)
# body of constructor
return MyIterativeMethod(contructor_args...)
end

# initialize function (1)
function initialize!(linearoperator!, Q, Qrhs, solver::MyIterativeMethod, args...)
# body of initialize function in abstract iterative solver
return Bool, Int
end

# iteration function (2)
function doiteration!(linearoperator!, Q, Qrhs, solver::MyIterativeMethod, threshold, args...)
# body of iteration
return Bool, Int, Float
end


MyIterativeMethod and function bodies would need to be replaced appropriately for a particular implementation. We will describe each component in detail in subsequent sections.

### Struct

A subset of AbstractIterativeSystemSolver needs at least two members: atol and rtol. The former represents an absolute tolerance and the latter is a relative tolerance. Both can be used to terminate the iteration to determine the convergence criteria. An example struct could be

struct MyIterativeMethod{FT} <: AbstractIterativeSystemSolver
atol::FT
rtol::FT
end

but often has more depending on the kind of iterative solver being used. For example, in a Krylov subspace method one would need to store a number of vectors which constitute the Krylov subspace.

### Constructor

The constructor for the struct can be defined any number of ways depending on the needs of the struct itself. Often times this is just used to allocate memory or convergence thresholds. This can also be a good place to define structures that make the iterative solver easier to work with. For example, for a columnwise solver one would want an easy array structure to work with vectors in a columnwise fashion.

In Basic Template for an Iterative Solver we used an outer constructor, e.g.,

# constructor
function MyIterativeMethod(args...)
# body of constructor
return MyIterativeMethod(contructor_args...)
end

but we could have also used an inner constructor if desired.

### Initialize Function

The initialize function needs the following signature

function initialize!(linearoperator!, Q, Qrhs, solver::MyIterativeMethod, args...)
# body of initialize function in abstract iterative solver
return Bool, Int
end

The intialize function has the following arguments:

1. linearoperator! (function)
2. Q (array) [OVERWRITTEN]
3. Qrhs (array)
4. solver (struct) used for dispatch
5. args... passed to linearoperator! function

The linearoperator! function is assumed to have the following signature:

linearoperator!(y, x, args...)
# body of linear operator
return nothing
end

It represents action of a linear operator $L$ on a vector $x$, that stores the value in the vector $y$, i.e. $Lx = y$. The last argument (the args...) is necessary due to how linear operators are defined within ClimateMachine.

The Q and Qrhs function arguments are supposed to represent the solution of the linear system LQ = Qrhs where L is the linear operator implicitly defined by linearoperator!.

The initialize function must have 2 return values:

1. convergence (bool)
2. iterations (int)

The return values keep track of whether or not the iterative algorithm has converged as well as how many times the linear operator was applied.

### Iteration Function

The iteration function needs the following signature

function doiteration!(linearoperator!, Q, Qrhs, solver::MyIterativeMethod, threshold, args...)
# body of iteration
return Bool, Int, Float
end

The iteration function has the following arguments:

1. linearoperator! (function)
2. Q (array) [OVERWRITTEN]
3. Qrhs (array)
4. solver (struct). used for dispatch
5. threshold (float). for the convergence criteria
6. args... passed into the linearoperator! function

The linearoperator! function is assumed to have the following signature:

linearoperator!(y, x, args...)
# body of linear operator
return nothing
end

It represents action of a linear operator $L$ on a vector $x$, that stores the value in the vector $y$, i.e. $Lx = y$. The last argument (the args...) is necessary due to how linear operators are defined within ClimateMachine.

The Q and Qrhs function arguments are supposed to represent the solution of the linear system LQ = Qrhs where L is the linear operator implicitly defined by linearoperator!.

The iteration function must have 3 return values:

1. converged (bool)
2. iterations (int)
3. residual_norm (float64)

The return values keep track of whether or not the iterative algorithm has converged as well as how many times the linear operator was applied. The residual norm is useful since it is often used to determine a stopping criteria.

## ClimateMachine Specific Considerations

An MPIStateArray Q in 3D, has the following structure by default:

size(Q) = (n_ijk, n_s, n_e)

where

1. n_ijk is the number of Gauss-Lobatto points per element
2. n_s is the number of states
3. n_e is the number of elements

In three dimensions, if one wants to operate in a column-wise fashion (with a stacked-brick topology) it is easiest to reshape the array into the following form

alias_Q = reshape(Q, (n_i, n_j, n_k, n_s, n_ev, n_eh))

where

1. n_i is the number of Gauss-Lobatto points per element within

element that are aligned with one of the horizontal directions.

1. n_j is the number of Gauss-Lobatto points per element within

element that are aligned with another one of the horizontal directions.

1. n_k is the number of Gauss-Lobatto points within element that

are aligned with the vertical direction.

1. n_s is the number of states
2. n_ev is the number of elements in the vertical direction
3. n_eh is the number of elements in the horizontal direction

Note: n_i x n_j x n_k = n_ijk and n_ev x n_eh = n_e.

Thus if one wants to operate on a column for a fixed state index (let's say the int s) and a fixed horizontal coordinate (let's say fixed ints i, j, eh), then one could operator on the state:

one_column = alias_Q[i, j, :, s, :, eh]

which are the third and fifth argument in the MPIStateArray.

Some extra tips are:

• Since GPUs have limited memory, don't take up too much memory.
• If possible define a preconditioner. Iterative methods are typically

very slow otherwise.

## Preconditioners

The code needs to be slightly restructured to allow for preconditioners.

## Writing Tests

Test on small systems where answers can be checked analytically. Check with matrices with easily computable inverses, i.e., the identity matrix or a diagonal matrix. Test with diverse matrix structures. Test with different array types: Arrays, CuArrays, MPIStateArrays, etc. Also test with balance laws to make sure that it can actually be run with IMEX solvers on the CPU/GPU and their distributed analogues.

## Performance Checks

Timing performance can be done with general CPU/GPU guidelines

## Conventions

• Q refers to the initial guess for the iterative solver that gets

overwritten with the final solution

• Qrhs refers to the right hand side of the linear system