An optimal Finite State Projection Method

Sunkara, Vikram; Hegland, Markus

doi:10.1016/j.procs.2010.04.177

Cited by 23 publications

(27 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In applications where the factors that generate A are non-linear, predicting the state space is difficult, at times requires brute force. This problem led to the work by Sunkara and Hegland [7] giving the Optimal Finite State Projection method (ofsp), which is known to be order optimal [7]. The ofsp gives control over where the state space is and where it might be drifting, helping us tackle bigger problems where the location of the states with probabilities away from zero is unknown.…”

Section: The Finite State Projection Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Parallelising the finite state projection method

Sunkara¹,

Hegland²

2011

ANZIAMJ

Self Cite

View full text Add to dashboard Cite

Many realistic mathematical models of biological and chemical systems, such as enzyme cascades and gene regulatory networks, need to include stochasticity. These systems are described as Markov processes and are modelled using the Chemical Master Equation. The Chemical Master Equation is a differential-difference equation (continuous in time and discrete in the state space) for the probability of a certain state at a given time. The state space is the population count of species in the system. A successful method for computing the Chemical Master Equation is the Finite State Projection Method. We give a new algorithm to distribute the Finite State Projection Method method onto multi-core systems. This method is called the Parallel Finite State Projection method. This article also analyses the theory needed for parallelisation of the Chemical Master Equation.

show abstract

Section: The Finite State Projection Methodsmentioning

confidence: 99%

“…Until 2005, it was considered only feasible to take stochastic simulation methods where we average over realisations to find the solution to the cme. Then we had the introduction of the Finite State Projection method, which helped compute problems with larger state spaces [5,2,6,7].…”

Section: Introductionmentioning

confidence: 99%

Parallelising the finite state projection method

Sunkara¹,

Hegland²

2011

ANZIAMJ

Self Cite

View full text Add to dashboard Cite

show abstract

“…The significance of algorithm 7 is that it is possible to impose an accuracy of tol on the truncated vector by choosing the rank rt in nodet based on dropping the smallest singular values whose squared sum is less than or equal to tol 2 ∕(2d − 3) (see pp. [18][19] in the work of Kressner et al 47 ) when adding two vectors. The number of flops for adding J vectors with this algorithm becomes O(dJ 2 r 2 max (m max + r 2 max + Jr max )).…”

Section: Addition Of Vectors In Htd Formatmentioning

confidence: 99%

On compact vector formats in the solution of the chemical master equation with backward differentiation

Dayar

Orhan

2018

Numerical Linear Algebra App

View full text Add to dashboard Cite

Summary A stochastic chemical system with multiple types of molecules interacting through reaction channels can be modeled as a continuous‐time Markov chain with a countably infinite multidimensional state space. Starting from an initial probability distribution, the time evolution of the probability distribution associated with this continuous‐time Markov chain is described by a system of ordinary differential equations, known as the chemical master equation (CME). This paper shows how one can solve the CME using backward differentiation. In doing this, a novel approach to truncate the state space at each time step using a prediction vector is proposed. The infinitesimal generator matrix associated with the truncated state space is represented compactly, and exactly, using a sum of Kronecker products of matrices associated with molecules. This exact representation is already compact and does not require a low‐rank approximation in the hierarchical Tucker decomposition (HTD) format. During transient analysis, compact solution vectors in HTD format are employed with the exact, compact, and truncated generated matrices in Kronecker form, and the linear systems are solved with the Jacobi method using fixed or adaptive rank control strategies on the compact vectors. Results of simulation on benchmark models are compared with those of the proposed solver and another version, which works with compact vectors and highly accurate low‐rank approximations of the truncated generator matrices in quantized tensor train format and solves the linear systems with the density matrix renormalization group method. Results indicate that there is a reason to solve the CME numerically, and adaptive rank control strategies on compact vectors in HTD format improve time and memory requirements significantly.

show abstract

“…Even though the dynamics of the derivative are fairly simple, solving the CME is a hard problem [5][6][7][8][9][10][11][12]. A special case of the CME which is truly unyielding to approximation is when a system is close to its population boundary (for example, close to zero).…”

Section: Introductionmentioning

confidence: 99%

On the Properties of the Reaction Counts Chemical Master Equation

Sunkara

2019

Entropy

View full text Add to dashboard Cite

The reaction counts chemical master equation (CME) is a high-dimensional variant ofthe classical population counts CME. In the reaction counts CME setting, we count the reactionswhich have fired over time rather than monitoring the population state over time. Since a reactioneither fires or not, the reaction counts CME transitions are only forward stepping. Typically thereare more reactions in a system than species, this results in the reaction counts CME being higher indimension, but simpler in dynamics. In this work, we revisit the reaction counts CME frameworkand its key theoretical results. Then we will extend the theory by exploiting the reactions counts’forward stepping feature, by decomposing the state space into independent continuous-time Markovchains (CTMC). We extend the reaction counts CME theory to derive analytical forms and estimatesfor the CTMC decomposition of the CME. This new theory gives new insights into solving hittingtimes-, rare events-, and a priori domain construction problems.

show abstract

An optimal Finite State Projection Method

Cited by 23 publications

References 5 publications

Parallelising the finite state projection method

Parallelising the finite state projection method

On compact vector formats in the solution of the chemical master equation with backward differentiation

On the Properties of the Reaction Counts Chemical Master Equation

Contact Info

Product

Resources

About