If the reachable state space is large it may come in handy to consider a finite state projection [ 29], in which matrix A is replaced by A k, a k × k submatrix of the true operator A, the corresponding indexed system states form the finite state projection and p (t f) ≈ exp(t f A k ) p k (0) is the approximation to p (t) = exp(t A ) p (0) at time t f.
In this case a finite state projection using Krylov subspaces, such as [ 34] would be preferable.
Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection.
This idea was recently explored in a study that used an approach called the Finite State Projection to compute stochastic distributions that can result from different models of the hyperosmolarity pathway in yeast.
In the second example we demonstrate how URDME can aid in the development of efficient simulation algorithms by explaining how a novel method, the Diffusive Finite State Projection (DFSP) [ 32], was integrated into URDME as a plug-in solver.
Therefore, the system is truncated to a rectangle Ω ¯ : = { 0, …, 150 } × { 0, …, 150 } such that x (on ), x (off ) ∈ Ω ¯, yielding a good approximation in the finite state projection to the infinite-dimensional chemical master equation.
Although only a few classes of CME problems are known to have exact and computationally tractable analytical solutions, the recently proposed finite state projection (FSP) technique provides a systematic reduction of the CME with guaranteed accuracy bounds.
As a more direct approach, Munsky and Khammash [ 3] have proposed the finite state projection (FSP), where the CME is solved on a finite subset of the state space.
Each state projection, lighted up on giant screens, was like ice cracking on a frozen lake: tiny fissures about to collapse into a giant swoosh of water.
We show that the ground state projection and the ground state energy are analytic functions of the coupling constant in a cone with apex at the origin, provided a mild infrared assumption holds.
