Browsing by Subject "Backward differentiation"

Now showing 1 - 3 of 3

Open Access
Computational methods for CTMCs
(John Wiley & Sons, 2011) Dayar, Tuğrul; Stewart, W. J.; Cochran, J. J.; Cox, L. A.; Keskinocak, P.; Kharoufeh, J. P.; Smith, J. C.
This article concerns the computation of stationary and transient distributions of continuous‐time Markov chains (CTMCs). Once the problem has been formulated, it is shown how computational methods for computing stationary distributions of discrete‐time Markov chains can be applied in the continuous‐time case. This is not so for the case of transient distributions, which turns out to be a much more difficult problem in general. Different approaches to computing transient distributions of CTMCs are explored, from the simple and efficient uniformization method, through matrix decomposition and powering techniques, to ordinary differential equation (ODE) solvers. This latter approach is the only one currently available for nonhomogeneous CTMCs. The basic concept is explained using simple Euler methods, but formulae for more advanced and efficient single‐step Runge–Kutta and implicit multistep BDF methods are provided.
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
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.