NSDETimeParallel.jl

An abstract type for parameters of Parareals.

An abstract type for caching intermediate computations in AbstractTimeParallelSolvers.

An abstract type for iterates in AbstractTimeParallelSolutions.

An abstract type for parameters in AbstractTimeParallelSolvers.

An abstract type for time-parallel solutions of NSDEBase.AbstractInitialValueProblems.

An abstract type for time-parallel solvers of NSDEBase.AbstractInitialValueProblems.

An abstract type for tolerance parameters in AbstractTimeParallelSolvers.

An abstract type for proximity-function weights used by AbstractTimeParallelSolvers.

Parareal <: AbstractTimeParallelSolver

A composite type for the Parareal algorithm.

Constructors

Parareal(finesolver, coarsesolver, parameters, tolerance)
Parareal(finesolver, coarsesolver; parameters=PararealParameters(), tolerance=Tolerance())

Arguments

• finesolver :: AbstractInitialValueSolver : fine solver (accurate but expensive).
• coarsesolver :: AbstractInitialValueSolver : coarse solver (rough but quick).
• parameters :: AbstractPararealParameters : parameters for the correction step.
• tolerance :: AbstractTolerance : tolerance and error mechanism.

Methods

(parareal::Parareal)(solution::PararealSolution, problem::AbstractInitialValueProblem)
(parareal::Parareal)(problem::AbstractInitialValueProblem)

returns the solution of a problem using parareal.

PararealIterate <: AbstractTimeParallelIterate

A composite type for a single iterate in a PararealSolution.

Constructors

PararealIterate(chunks::AbstractVector{𝕊}) where 𝕊<:AbstractInitialValueSolution
PararealIterate(problem::AbstractInitialValueProblem, parareal::Parareal)

Functions

• firstindex : first index.
• getindex : get chunk.
• lastindex : last index.
• length : number of chunks.
• numchunks : number of chunks.
• setindex! : set chunk.

Methods

(iterate::PararealIterate)(t::Real)

returns the value of iterate at t via interpolation.

PararealParameters <: AbstractPararealParameters

A composite type for the basic parameters of Parareal.

Constructors

PararealParameters(N, K)
PararealParameters(; N=10, K=N)

Arguments

• N :: Integer : number of time chunks/processors.
• K :: Integer : maximum number of iterations.

PararealSolution <: AbstractTimeParallelSolution

A composite type for an AbstractTimeParallelSolution obtained using Parareal.

Constructors

PararealSolution(lastiterate, errors)
PararealSolution(problem::AbstractInitialValueProblem, parareal::Parareal)

Arguments

• lastiterate :: PararealIterate
• errors :: AbstractVector{ℝ} where ℝ<:Real : iteration errors.

Functions

• firstindex : first index.
• getindex : get iterate.
• lastindex : last index.
• numiterates : number of iterates.
• numchunks : number of chunks of last iterate.
• setindex! : set iterate.

Methods

(solution::PararealSolution)(t::Real)

returns the value of solution at t via interpolation.

Tolerance <: AbstractTolerance

A composite type for the tolerance mechanism of an time-parallel solver.

Constructors

Tolerance(ϵ, ψ, weights)
Tolerance(; ϵ=1e-12, ψ=ψ₁, weights=Weights())

Arguments

• ϵ :: Real : tolerance.
• ψ :: Function : error function.
• weights :: Weights : weights for ψ.

Weights <: AbstractWeights

A composite type for the weights of Tolerance.

Constructors

Weights(; w=1.0, updatew=false)

Arguments

• w :: Union{AbstractVector{ℝ}, ℝ} where ℝ<:Real : weighting factor for ψ.
• updatew :: Bool : flags when to update! w using (an approximation of) the Lipschitz function of the fine solver.

Functions

• update! : updates w using (an approximation of) the Lipschitz function of the fine solver.

firstindex(iterate::PararealIterate)

returns the first index of iterate.

firstindex(solution::PararealSolution)

returns the first index of solution.

getindex(iterate::PararealIterate, n::Integer)

returns the n-th chunk of iterate.

getindex(solution::PararealSolution, n::Integer)

returns the n-th chunk of the last iteration of a PararealSolution.

lastindex(iterate::PararealIterate)

returns the last index of iterate.

lastindex(solution::PararealSolution)

returns the last index of solution.

length(iterate::PararealIterate)

returns the number of chunks of iterate.

length(solution::PararealSolution)

returns the number of chunks of solution.

setindex!(iterate::PararealIterate, value::AbstractInitialValueSolution, n::Integer)

stores value into the n-th chunk of iterate.

setindex!(solution::PararealSolution, chunk::AbstractInitialValueSolution, n::Integer)

stores an AbstractInitialValueSolution as the n-th chunk of the last iteration of a PararealSolution.

solve!(solution::AbstractTimeParallelSolution, problem, solver::AbstractTimeParallelSolver; kwargs...) :: AbstractTimeParallelSolution

computes the solution of problem using solver.

solve(problem, solver::AbstractTimeParallelSolver; kwargs...) :: AbstractTimeParallelSolution

computes the solution of problem using solver.

TimeParallelSolution(problem::AbstractInitialValueProblem, parareal::Parareal)

returns a PararealSolution constructor for the solution of problem with parareal.

numchunks(iterate::PararealIterate)

returns the number of chunks of iterate.

numchunks(solution::PararealSolution)

returns the number of chunks of solution.

numiterates(solution::PararealSolution)

returns the number of iterates of solution.

Distributed implementation of Parareal.

MPI implementation of Parareal.

update!(weights::Weights, U<:AbstractVector{𝕍}, F<:AbstractVector{𝕍}) where 𝕍<:AbstractVector{ℂ} where ℂ<:Number

updates weights.w based on U and F.

standard error function.

weighted error function.