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Mathematical Specification of a PDMP Problem

Vector field

To define a PDMP Problem, you first need to give the function $F$ and the initial condition $x_{c,0}$ which define an ODE:

\[\frac{dx_c}{dt} = F(x_c(t),x_d(t),p,t)\]

where $F$ should be specified in-place as F(dxc,xc,xd,p,t), and xc0 should be an AbstractArray (or number) whose geometry matches the desired geometry of xc. Note that we are not limited to numbers or vectors for xc0; one is allowed to provide u₀ as arbitrary matrices / higher dimension tensors as well.

Jumps

Jumps are defined as a Jump process which changes states at some rate $R$ which is a scalar function of the type

\[R(x_c(t),x_d(t),p,t).\]

Note, that in between jumps, $x_d(t)$ is constant but $x_c(t)$ is allowed to evolve. $R$ should be specified in-place as R(rate,xc,xd,p,t,issum::Bool) where it mutates rate. Note that a boolean issum is provided and the behavior of R should be as follows

  • if issum == true, we only require R to return the total rate, e.g. sum(rate). We use this formalism because sometimes you can compute the sum without mutating rate.
  • if issum == false, R must populate rate with the updated rates

We then need to provide the way the jumps affect the state variable. There are two possible ways here:

  • either give a transition matrix nu: it will only affect the discrete component xd and leave xc unaffected.
  • give a function to implement jumps Delta(xc, xd, parms, t, ind_reaction::Int64) where you can mutate xc,xd or parms. The argument ind_reaction is the index of the reaction at which the jump occurs. See examples/pdmp_example_eva.jl for an example.

Problem Type

Constructors

  • PDMPProblem(F,R,Delta,nu,xc0,xd0,p,tspan)
  • PDMPProblem(F,R,nu,xc0,xd0,p,tspan) when ones does not want to provide the function Delta
  • PDMPProblem(F,R,Delta,reaction_number::Int64,xc0,xd0,p,tspan) when ones does not want to provide the transition matrix nu. The length reaction_number of the rate vector must then be provided.

We also provide a wrapper to DiffEqJump.jl. This is quite similar to how a JumpProblem would be created.

  • PDMPProblem(prob, jumps...) where prob can be an ODEProblem. For an example, please consider example/examplediffeqjumpwrapper.jl.

Fields

  • F: the function of the ODE
  • R: the function to compute the transition rates
  • Delta [Optional]: the function to effect the jumps
  • nu [Optional]: the transition matrix
  • xc0: the initial condition of the continuous part
  • xd0: the initial condition of the discrete part
  • p: the parameters to be provided to the functions F, R, Delta
  • tspan: The timespan for the problem.