Jump Problems

Jump Problems

Mathematical Specification of an problem with jumps

Jumps are defined as a Poisson process which changes states at some rate. When there are multiple possible jumps, the process is a compound Poisson process. On their own, a jump equation is continuous-time Markov Chain where the time to the next jump is exponentially distributed as calculated by the rate. This type of process, known in biology as "Gillespie discrete stochastic simulations" and modeled by the Chemical Master Equation (CME), is the same thing as adding jumps to a DiscreteProblem. However, any differential equation can be extended by jumps as well. For example, we have an ODE with jumps, denoted by

\[\frac{du}{dt} = f(u,p,t) + Σ c_i(u,p,t)dp_i\]

where $dp_i$ is a Poisson counter of rate $\lambda_i(u,p,t)$. Extending a stochastic differential equation to have jumps is commonly known as a Jump Diffusion, and is denoted by

\[\frac{du}{dt} = f(u,p,t) + Σgᵢ(u,t)dWⁱ + Σ c_i(u,p,t)dp_i\]

Types of Jumps: Regular, Variable, Constant Rate and Mass Action

A RegularJump is a set of jumps that do not make structural changes to the underlying equation. These kinds of jumps only change values of the dependent variable (u) and thus can be treated in an inexact manner. Other jumps, such as those which change the size of u, require exact handling which is also known as time-adaptive jumping. These can only be specified as a ConstantRateJump, MassActionJump, or a VariableRateJump.

We denote a jump as variable rate if its rate function is dependent on values which may change between constant rate jumps. For example, if there are multiple jumps whose rates only change when one of them occur, than that set of jumps is a constant rate jump. If a jump's rate depends on the differential equation, time, or by some value which changes outside of any constant rate jump, then it is denoted as variable.

A MassActionJump is a specialized representation for a collection of constant rate jumps that can each be interpreted as a standard mass action reaction. For systems comprised of many mass action reactions, using the MassActionJump type will offer improved performance. Note, only one MassActionJump should be defined per JumpProblem; it is then responsible for handling all mass action reaction type jumps. For systems with both mass action jumps and non-mass action jumps, one can create one MassActionJump to handle the mass action jumps, and create a number of ConstantRateJumps to handle the non-mass action jumps.

RegularJumps are optimized for regular jumping algorithms like tau-leaping and hybrid algorithms. ConstantRateJumps and MassActionJumps are optimized for SSA algorithms. ConstantRateJumps, MassActionJumps and VariableRateJumps can be added to standard DiffEq algorithms since they are simply callbacks, while RegularJumps require special algorithms.

Defining a Regular Jump

The constructor for a RegularJump is:

RegularJump(rate,c,c_prototype;mark_dist = nothing,constant_c = false)

dc is an n x m matrix, where n is the number of Poisson processes and m is the number of dependent variables (should match length(u)). rate is a vector equation which should compute the rates in to out which is a length n vector.

Defining a Constant Rate Jump

The constructor for a ConstantRateJump is:

ConstantRateJump(rate,affect!)

Defining a Mass Action Jump

The constructor for a MassActionJump is:

MassActionJump(rate_consts, reactant_stoich, net_stoich; scale_rates = true)

Notes for Mass Action Jumps

Defining a Variable Rate Jump

The constructor for a VariableRateJump is:

VariableRateJump(rate,affect!;
                   idxs = nothing,
                   rootfind=true,
                   save_positions=(true,true),
                   interp_points=10,
                   abstol=1e-12,reltol=0)

Note that this is the same as defining a ContinuousCallback, except that instead of the condition function, you provide a rate(u,p,t) function for the rate at a given time and state.

Defining a Jump Problem

To define a JumpProblem, you must first define the basic problem. This can be a DiscreteProblem if there is no differential equation, or an ODE/SDE/DDE/DAE if you would like to augment a differential equation with jumps. Denote this previously defined problem as prob. Then the constructor for the jump problem is:

JumpProblem(prob,aggregator::Direct,jumps::JumpSet;
            save_positions = typeof(prob) <: AbstractDiscreteProblem ? (false,true) : (true,true))

The aggregator is the method for aggregating the constant jumps. These are defined below. jumps is a JumpSet which is just a gathering of jumps. Instead of passing a JumpSet, one may just pass a list of jumps themselves. For example:

JumpProblem(prob,aggregator,jump1,jump2)

and the internals will automatically build the JumpSet. save_positions is the save_positions argument built by the aggregation of the constant rate jumps.

Note that a JumpProblem/JumpSet can only have 1 RegularJump (since a RegularJump itself describes multiple processes together). Similarly, it can only have one MassActionJump (since it also describes multiple processes together).

Constant Rate Jump Aggregators

Constant rate jump aggregators are the methods by which constant rate jumps, including MassActionJumps, are lumped together. This is required in all algorithms for both speed and accuracy. The current methods are:

To pass the aggregator, pass the instantiation of the type. For example:

JumpProblem(prob,Direct(),jump1,jump2)

will build a problem where the constant rate jumps are solved using Gillespie's Direct SSA method.

Recommendations for Constant Rate Jumps

For representing and aggregating constant rate jumps