Browsing by Subject "Chemical master equation"

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Open Access
On compact vector formats in the solution of the chemical master equation with backward differentiation
(John Wiley and Sons, 2018) Dayar, Tuğrul; Orhan, M. C.
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. Copyright
Open Access
On the numerical analysis of infinite multi-dimensional Markov chains
(2017-07) Orhan, Muhsin Can
A system with multiple interacting subsystems that exhibits the Markov property can be represented as a multi-dimensional Markov chain (MC). Usually the reachable state space of this MC is a proper subset of its product state space, that is, Cartesian product of its subsystem state spaces. Compact storage of the infinitesimal generator matrix underlying such a MC and efficient implementation of analysis methods using Kronecker operations require the set of reachable states to be represented as a union of Cartesian products of subsets of subsystem state spaces. We first show that the problem of partitioning the reachable state space of a three or higher dimensional system with a minimum number of partitions into Cartesian products of subsets of subsystem state spaces is NP-complete. Two algorithms, one merge based the other refinement based, that yield possibly nonoptimal partitionings are presented. Results of experiments on a set of problems from the literature and those that are randomly generated indicate that, although it may be more time and memory consuming, the refinement based algorithm almost always computes partitionings with a smaller number of partitions than the merge based algorithm. When the infinitesimal generator matrix underlying the MC is represented compactly using Kronecker products, analysis methods based on vector– Kronecker product multiplication need to be employed. When the factors in the Kronecker product terms are relatively dense, vector–Kronecker product multiplication can be performed efficiently by the shuffle algorithm. When the factors are relatively sparse, it may be more efficient to obtain nonzero elements of the generator matrix in Kronecker form on-the-fly and multiply them with corresponding elements of the vector. We propose a modification to the shuffle algorithm that multiplies relevant elements of the vector with submatrices of factors in which zero rows and columns are omitted. This approach avoids unnecessary floating-point operations that evaluate to zero during the course of the multiplication. Numerical experiments on a large number of models indicate that, in many cases the modified shuffle algorithm performs a smaller number of floating-point operations than the shuffle algorithm and the algorithm that generates nonzeros on-the-fly, sometimes with minimum number of floating-point operations and amount of memory possible. Although the generator matrix is stored compactly using Kronecker products, solution vectors used in the analysis still require memory proportional to the size of the reachable state space. This becomes a bigger problem as the number of dimensions increases. We show that it is possible to use the hierarchical Tucker decomposition (HTD) to store the solution vectors during Kroneckerbased Markovian analysis relatively compactly and still carry out the basic operation of vector–matrix multiplication in Kronecker form relatively efficiently. The time evolution of a stochastic chemical system modelled as a continuoustime MC (CTMC) can be described as a system of ordinary differential equations (ODEs) known as the chemical master equation (CME). The CME can be analyzed by discretizing time and solving a linear system obtained by truncating the countably infinite state space at each time step. However, it is not trivial to choose a truncated state space that includes few states with negligible probabilities and leaves out only a small probability mass. We show that it is possible to decrease the memory requirement of the ODE solver using HTD with adaptive truncation strategies and we propose a novel approach to truncate the countably infinite state space using prediction vectors. Numerical experiments indicate that adaptive truncation strategies improve time and memory efficiency significantly when fixed truncation strategies are inefficient. Finally, we consider a multi-class multi-server retrial queueing system in which customers arrive according to a class-dependent Markovian arrival process (MAP). Service and retrial times follow class-dependent phase-type (PH) distributions with the further assumption that PH distributions of retrial times are acyclic. Here, we obtain a necessary and sufficient condition for ergodicity from criteria based on drifts. The countably infinite state space of the model is truncated with an appropriately chosen Lyapunov function. The truncated model is described as a multi-dimensional MC and a Kronecker representation of its infinitesimal generator matrix is numerically analyzed.
Open Access
A software tool for the compact solution of the chemical master equation
(Springer, Cham, 2018) Dayar Tuğrul; Orhan, M. C.
The problem of computing the transient probability distribution of countably infinite multidimensional continuous-time Markov chains (CTMCs) arising in systems of stochastic chemical kinetics is addressed by a software tool. Starting from an initial probability distribution, time evolution of the probability distribution associated with the CTMC is described by a system of linear first-order ordinary differential equations, known as the chemical master equation (CME). The solver for the CME uses the time stepping implicit backward differentiation formulae (BDF). Solution vectors in BDF can be stored compactly during transient analysis in one of the Hierarchical Tucker Decomposition, Quantized Tensor Train, or Transposed Quantized Tensor Train formats.