Pseudo-arclength continuation (PALC)

PALC

Data type for storing all information for PALC algorithm.

PALC(; kwargs...)

Contructor for PALC algorithm.

Kwargs

• predicter::AbstractPredictor: Prediction method. Default (recommended): Bordered()
• inner_product::AbstractInnerProduct: Inner product to use throughout. Default: StandardDotProduct()
• linesearch: Line search method provided by LineSearch.jl. Default: LiFukushimaLineSearch()
• linsolve: Linear solve algorithm provided by LinearSolve.jl. Default SVDFactorization()
• termcond: Termination condition for solvers. Default: termcond=NonlinearSolve.AbsTerminationMode()

Examples

pred = Secant() # secant predictor
alg = PALC(; predicter=pred) # all defaults except prediction method

PALCCache{MT<:Union{Matrix{Float64},SparseMatrixCSC{Float64,Int}}}

A cache for the PALC algorithm.

This cache includes all preallocated storage required for PALC. It also contains the zero curve (solution), return code, and any special detected points.

Fields

• ds::Float64: PALC step size
• br::Vector{Tuple{Vector{Float64},Float64}}: Zero curve storage, in format [(u0,λ0); (u1,λ1); ...; (un,λn)] for n iterations.
• detected_points::Vector{Tuple{Vector{Float64},Float64, Symbol}}: Storage for any points of interest detected by FoldBifurcationDetectionCallback() or any other detection callbacks (once implemented).
• uλ0::Vector{Float64}: Full [u;λ] current iterate
• u0::Vector{Float64}: Unknowns u current iterate.
• λ0::Float64: Continuation parameter λ current iterate.
• λn::Float64: Natural continuation parameter. Only used in select instances when required.
• bordered_mat::MT: Bordered matrix storage
• bordered_b::Vector{Float64}: Storage for RHS of bordered system.
• δuλ0::Vector{Float64}: Storage for full tangent [δu; δλ]
• δu0::Vector{Float64}: Storage for tangent components δu
• δλ::Float64: Storage for tangent component δλ
• δuλ0_initial::Vector{Float64}: Storage for initalized tangent computed before first iteration.
• uλpred::Vector{Float64}: Storage for predicted point
• Ffun::Vector{Float64}: Storage for continuation function residuals (not including hyperplane constraint)
• Jfun::MT: Storage for non-square jacobian of system ∂F(u;λ)/∂[u;λ]. (not including hyperplane constraint)
• u_0::Vector{Float64}: Storage for regula-falsi solver
• u_1::Vector{Float64}: Storage for regula-falsi solver
• u_t::Vector{Float64}: Storage for regula-falsi solver
• ret::Symbol: Exit condition return code.

struct CorrectionStepLimiter

A type to limit the step size in the correction step.

Restricts the Euclidean distance between the prediction and correction to be less than some frac of Δs at the current iteration. If the distance is larger, the correction step is rejected despite success of the Newton-Raphson solver.

Fields

• frac::Float64: The fraction of the current PALC step size Δs that the correction step length should be less than.

function CorrectionStepLimiter(; frac=0.1)

Constructor for the correction step limiter.

Kwargs

• frac::Float64: The fraction of the current PALC step size Δs that the correction step length should be less than. Defaults to 0.1.