Fenrir

fenrir_nll(prob::ODEProblem, data::NamedTuple{(:t, :u)}, observation_noise_var::Real,
    diffusion_var::Union{Real,Vector{<:Real}};
    adaptive=false, dt=false,  proj=I, order=3::Int, tstops=[])

Compute the "Fenrir" approximate negative log-likelihood (NLL) of the data.

This is a convenience function that

1. Solves the ODE with a ProbNumDiffEq.EK1 of the specified order and with a diffusion as provided by the diffusion_var argument, and
2. Fits the ODE posterior to the data via Kalman filtering and thereby computes the negative log-likelihood on the way.

By default, the solver steps exactly through the time points data.t. In addition, you can provide a step size with dt or time stops with tstops. Or, set adaptive=true for adaptive step-size selection - use at your own risk!

Returns a tuple (nll::Real, times::Vector{Real}, states::StructVector{Gaussian}), where states contains the filtering posterior. Its mean and covariance can be accessed with states.μ and states.Σ.

Arguments

• prob::ODEProblem: the initial value problem of interest
• data::NamedTuple{{(:t, :u)}}: the data to be fitted
• observation_noise_var::Real: the scalar observation noise variance
• diffusion_var: the diffusion parameter for the integrated Wiener process prior; this plays a similar role as kernel hyperparamers in Gaussian-process regression
• dt=false: step size parameter, passed to OrdinaryDiffEq.init
• adaptive=false: flag to determine if adaptive step size selection should be used; use at your own risk!
• tstops=[]: additional time stops the algorithm should step through; passed to OrdinaryDiffEq.solve
• order::Int=3: the order of the ProbNumDiffEq.EK1 solver
• proj=I: the matrix which maps the ODE state to the measurements; typically a projection