Sampling transitions

transition(
    sys::CoupledSDEs,
    x_i,
    x_f;
    rad_i,
    rad_f,
    tmax,
    rad_dims,
    cut_start,
    kwargs...
) -> Tuple{Any, Any, Any}

Generates a sample transition from point x_i to point x_f.

This function simulates sys in time, starting from initial condition x_i, until entering a length(sys.u)-dimensional ball of radius rad_f around x_f.

Keyword arguments

• rad_i=0.1: radius of ball around x_i
• rad_f=0.1: radius of ball around x_f
• tmax=1e3: maximum time when the simulation stops even x_f has not been reached
• rad_dims=1:length(sys.u): the directions in phase space to consider when calculating the radii rad_i and rad_f. Defaults to all directions. To consider only a subspace of state space, insert a vector of indices of the dimensions to be included.
• cut_start=true: if false, returns the whole trajectory up to the transition
• kwargs...: keyword arguments passed to CommonSolve.solve

Output

[path, times, success]

• path (Matrix): transition path (size [dim × N], where N is the number of time points)
• times (Vector): time values (since start of simulation) of the path points (size N)
• success (bool): if true, a transition occured (i.e. the ball around x_f has been reached), else false

See also transitions, trajectory.

function transitions(sys::CoupledSDEs, x_i, x_f, N=1; kwargs...)

Generates an ensemble of N transition samples of sys from point x_i to point x_f.

This function repeatedly calls the transition function to efficiently generate an ensemble of transitions. Multi-threading is enabled.

Keyword arguments

• rad_i=0.1: radius of ball around x_i
• rad_f=0.1: radius of ball around x_f
• Nmax: number of attempts before the algorithm stops even if less than N transitions occurred.
• tmax=1e3: maximum time when the simulation stops even x_f has not been reached
• rad_dims=1:length(sys.u): the directions in phase space to consider when calculating the radii rad_i and rad_f. Defaults to all directions. To consider only a subspace of state space, insert a vector of indices of the dimensions to be included.
• cut_start=true: if false, returns the whole trajectory up to the transition
• show_progress: shows a progress bar with respect to Nmax
• kwargs...: keyword arguments passed to CommonSolve.solve

See also transition.

Output

[samples, times, idx, N_fail]

• samples (Array of Matrices): sample paths. Each path i has size (dim × Ni), where Ni is the number of path points
• times (Array of Vectors): time values (since simulation start) of path points for each path
• idx (Array): list of sample indices i that produced a transition
• N_fail (Int): number of samples that failed to produce a transition

An example script using transitions is available here.