Filtration.jl private functions

Documentation

Base.show — Method

show(io::IO, z::Control)

Print a Control

Base.show — Method

show(io::IO, z::Costate)

Print a Costate

Base.show — Method

show(io::IO, z::Extremal)

Print an Extremal

Base.show — Method

show(io::IO, model::Model)

Print a Model. It provide the print given by ModelingToolkit.jl for the associated ODE system.

Base.show — Method

show(io::IO, z::MultiModel)

Print a MultiModel

Base.show — Method

show(io::IO, z::MultiSynthesis)

Print a MultiSynthesis

Base.show — Method

show(io::IO, odelike::ODELike)

Print an ODELike

Base.show — Method

show(io::IO, z::Pareto)

Print a Pareto

Base.show — Method

show(io::IO, z::SimulationModel)

Print a SimulationModel

Base.show — Method

show(io::IO, z::Solution)

Print a Solution

Base.show — Method

show(io::IO, z::State)

Print a State

Base.show — Method

show(io::IO, z::Synthesis)

Print a Synthesis

Filtration.__AichoucheModel — Method

__AichoucheModel(p::Vector{<:Real}; initial_guess) -> Model

Create the Aichouche model.

Arguments

p : Parameter vector for the Aichouche model.

Keyword Arguments (optional)

initial_guess : initial guess forwarded to the constructed Model.

Return

model : the constructed Model.

Filtration.__BenyahiaModel — Method

__BenyahiaModel(p::Vector{<:Real}; initial_guess) -> Model

Create the Benyahia model.

Arguments

p : Parameter vector for the Benyahia model.

Keyword Arguments (optional)

initial_guess : initial guess forwarded to the constructed Model.

Return

model : the constructed Model.

Filtration.__CoganModel — Method

__CoganModel(p::Vector{<:Real}; initial_guess) -> Model

Create the Cogan model.

Arguments

p : Parameter vector for the Cogan model.

Keyword Arguments (optional)

initial_guess : initial guess forwarded to the constructed Model.

Return

model : the constructed Model.

Filtration.__bound_value — Method

__bound_value(bound::Real, t::Real) -> Real

Evaluate a bound (a constant, a ODESolution cut, or another ODELike) at a given $t$.

Filtration.__compute_phi — Method

__compute_phi(
    t::Real,
    extremal::Extremal,
    model::Model;
    p
) -> Real

Compute the value of the function $t \mapsto \phi(x(t), p(t))$ on a given time $t$ of an Extremal computed on a Model.

Arguments

t : a real time $t$
extremal : the Extremal
model : the Model

Keyword arguments (optional)

p : a vector of parameter for the model (default is the parametes of the model).

Return

phi : the value of $\phi(t)$.

Filtration.__get_m — Method

__get_m(
    synthesis1::Synthesis,
    synthesis2::Synthesis
) -> Real

Get the end of the DL2 curve of synthesis1, ie. the value of $x_1$ at which it meets synthesis2's cut₊ curve.

Arguments

synthesis1 : the below Synthesis
synthesis2 : the above Synthesis

Return

m : the value of $x_1$ at the junction

Filtration.classification_plot — Method

classification_plot(
    s::Synthesis;
    SL_nb_points,
    kwargs...
) -> Plots.Plot

Plot a Synthesis, in a classification of optimal control form

Arguments

s : a Synthesis

Keyword arguments (optional)

SLnbpoints : Number of points for the switching locus curve. Initialized to nothing (points of the ODESolution)
kwargs... : keyword arguments for plot

Returns

plt : the plot

Filtration.dispersal_locus_2_continuation_method — Method

dispersal_locus_2_continuation_method(
    model::Model,
    xlim::Tuple{Real, Real},
    ylim::Tuple{Real, Real},
    DL::Union{ODELike, SciMLBase.ODESolution}
) -> SciMLBase.ODESolution

Compute the second dispersal locus by continuation method.

Warning

This function must be used only when a second dispersal locus happened, ie. when the end of the first dispersal locus is below ylim[2].

Arguments

model : a Model
xlim : the limits of state $x_2$
ylim : the limits of state $x_1$
DL : the first dispersal locus curve

Returns

DL2 : the second dispersal locus curve

Filtration.dispersal_locus_2_direct — Method

dispersal_locus_2_direct(
    model::Model,
    xlim::Tuple{Real, Real},
    ylim::Tuple{Real, Real},
    DL::Union{ODELike, SciMLBase.ODESolution},
    cutSL::SciMLBase.ODESolution;
    n
) -> ODELike

Compute the second dispersal locus by direct method.

Warning

This function must be used only when a second dispersal locus happened, ie. when the end of the first dispersal locus is below ylim[2].

Arguments

model : a Model
xlim : the limits of state $x_2$
ylim : the limits of state $x_1$
DL : the first dispersal locus curve
cutSL : the cutSL curve

Returns

DL2 : the second dispersal locus curve

Filtration.dispersal_locus_continuation_method — Method

dispersal_locus_continuation_method(
    model::Model,
    xlim::Tuple{Real, Real},
    ylim::Tuple{Real, Real},
    SL::SciMLBase.ODESolution,
    cut₋::SciMLBase.ODESolution
) -> SciMLBase.ODESolution

Compute the first dispersal locus by continuation method.

Warning

This function must be used only when a dispersal locus happened, ie. when the end of the switching curve is below ylim[2].

Arguments

model : a Model
xlim : the limits of state $x_2$
ylim : the limits of state $x_1$
SL : the switching locus curve
cut₋ : the cut₋ curve

Returns

DL : the dispersal locus curve

Filtration.dispersal_locus_direct — Method

Compute the dispersal locus with a direct method.

Arguments

model : the Model
xlim : the limits of state $x_2$
ylim : the limits of state $x_1$
SL : the switching locus curve
cut₋ : the cut₋ curve

Returns

DL : the dispersal locus curve, as an ODELike solution

Filtration.fit_known — Method

fit_known(
    model::Model,
    control::Control,
    time::Vector{<:Real},
    data_x0::Vector{<:Real},
    initial_p::Vector{<:Real},
    x0::Vector{<:Real};
    lb,
    ub,
    verbose
) -> Any

Fit the model parameters to the given data by minimizing the least square error between the simulated and observed data. The initial state is supposed to be known.

Arguments

model : a Model
control : a Control
time : vector of time points where data are observed
datax0 : vector of observed data for state ``x0``
initial_p : initial guess for the model parameters
x0 : initial state

Keyword Arguments

lb : lower bounds for the model parameters (default : zeros)
ub : upper bounds for the model parameters (default : Inf)
verbose : whether to print optimization details (default : false)

Filtration.fit_unknown — Method

fit_unknown(
    model::Model,
    control::Control,
    time::Vector{<:Real},
    data_x0::Vector{<:Real},
    initial_p::Vector{<:Real},
    x0::Vector{<:Real},
    get_m::Function,
    u::Real,
    slope::Real;
    lb,
    ub,
    verbose
) -> Any

Fit the model parameters to the given data by minimizing the least square error between the simulated and observed data. The initial state is supposed to be partially unknown, with one unknown state variable being fitted via a constraint on its derivative at initial time. See here for more information.

Arguments

model : a Model
control : a Control
time : vector of time points where data are observed
datax0 : vector of observed data for state ``x0``
initial_p : initial guess for the model parameters
x0 : initial state and initial guess for the unknown state variables
get_m : function to compute the unknown initial state variable from control and parameters
u : singular control applied before initial time
slope : slope of state $x_0$ before initial time

Keyword Arguments

lb : lower bounds for the model parameters (default : zeros)
ub : upper bounds for the model parameters (default : Inf)
verbose : whether to print optimization details (default : false)

Filtration.get_arrow_time — Method

get_arrow_time(
    value::Vector{<:Real},
    sol::SciMLBase.ODESolution,
    axis::Symbol
) -> Vector{<:Real}

Compute the time $t$ for which sol(t) ∈ value, along the given axis.

Arguments

value : vector of coordinate values to locate on the trajectory
sol : state trajectory as an ODESolution
axis : the axis on which value is expressed, either :x or :y

Return

times : Vector of times

Filtration.get_y — Method

Get the y-coordinate of the ode at a given x-coordinate.

Arguments

x : the x-coordinate
ode : the ode solution

Returns

y : the y-coordinate

Filtration.real_solve_control — Method

real_solve_control(
    model::Model,
    init_x₂::Real,
    end_x₂::Real,
    init_x₁::Real,
    N::Int64
) -> Solution

Provide the real solution, where the control is only composed by -1 and 1 values. In this method, the control is approximated by -1 and 1 with respect to the value of uₛ.

Arguments

model : a Model
init_x₂ : the initial state $x_2(t_0)$
end_x₂ : the final state $x_2(t_f)$
init_x₁ : the initial state $x_1(t_0)$
N : the number of subdivision of the singular arc

Return

real_sol : the Solution

Filtration.real_solve_optimal — Method

real_solve_optimal(
    model::Model,
    init_x₂::Real,
    end_x₂::Real,
    init_x₁::Real,
    N::Int64;
    verbose
) -> Solution

Provide the optimal real solution, where the control is only composed by -1 and 1 values. In this method, the global constraints non linear optimization problem is constructed and solved.

Warning

This method must be long when N become large.

Arguments

model : a Model
init_x₂ : the initial state $x_2(t_0)$
end_x₂ : the final state $x_2(t_f)$
init_x₁ : the initial state $x_1(t_0)$
N : the number of subdivision of the singular arc

Keyword arguments (optional)

verbose : if true, print the optimization process (default is false)

Return

real_sol : the Solution

Filtration.real_solve_singular — Method

real_solve_singular(
    model::Model,
    init_x₂::Real,
    end_x₂::Real,
    init_x₁::Real,
    N::Int64;
    verbose
) -> Solution

Provide a real solution, where the control is only composed by -1 and 1 values. In this method, the singular arc is subdivided into N intervals. On each subintervals, we ensure that the solution minimize the cost and satisfy that the begining and the end of the state $x\_1$ on each subintervals corresponds to the value of the singular arc $x\_s$.

Arguments

model : a Model
init_x₂ : the initial state $x_2(t_0)$
end_x₂ : the final state $x_2(t_f)$
init_x₁ : the initial state $x_1(t_0)$
N : the number of subdivision of the singular arc

Keyword arguments (optional)

verbose : if true, print the optimization process (default is false)

Return

real_sol : the Solution

Filtration.synthesis_plot — Method

synthesis_plot(
    multisynthesis::MultiSynthesis;
    label_curve,
    colors,
    kwargs...
) -> Plots.Plot

Plot a MultiSynthesis, in a feedback form.

Arguments

multisynthesis : a MultiSynthesis

Keyword arguments (optional)

label_curve : labels of the curves. Initialized to ["u₋", "u₊", "uₛ", "switching locus", "dispersal locus", "dispersal locus", "dispersal locus"].
colors : colors of the curves. Initialized to [:green, :gold, :gray, :gray30, :gray15].
kwargs... : keyword arguments for plot

Returns

plt : the plot

Filtration.synthesis_plot — Method

synthesis_plot(
    s::Synthesis;
    SL_nb_points,
    label_curve,
    colors,
    kwargs...
) -> Plots.Plot

Plot a Synthesis, in a feedback form.

Arguments

s : a Synthesis

Keyword arguments (optional)

SLnbpoints : Number of points for the switching locus curve. Initialized to nothing (points of the ODESolution)
labelcurve : labels of the curves ``[u-, u+, us, \text{switching locus}, \text{dispersal locus}, \text{dispersal locus}]$. Initialized to$["u₋", "u₊", "uₛ", "switching locus", "dispersal locus", "dispersal locus"]``.
colors : colors of the curves for $u_s$ and the switching locus. Initialized to [:green, :gold, :gray20, :gray20].
kwargs... : keyword arguments for plot

Returns

plt : the plot

Filtration.jl private functions

Index

Documentation