724 research outputs found
Efficient simulation of stochastic chemical kinetics with the Stochastic Bulirsch-Stoer extrapolation method
No full textBackgroundBiochemical systems with relatively low numbers of components must be simulated stochastically in order to capture their inherent noise. Although there has recently been considerable work on discrete stochastic solvers, there is still a need for numerical methods that are both fast and accurate. The Bulirsch-Stoer method is an established method for solving ordinary differential equations that possesses both of these qualities.ResultsIn this paper, we present the Stochastic Bulirsch-Stoer method, a new numerical method for simulating discrete chemical reaction systems, inspired by its deterministic counterpart. It is able to achieve an excellent efficiency due to the fact that it is based on an approach with high deterministic order, allowing for larger stepsizes and leading to fast simulations. We compare it to the Euler ?-leap, as well as two more recent ?-leap methods, on a number of example problems, and find that as well as being very accurate, our method is the most robust, in terms of efficiency, of all the methods considered in this paper. The problems it is most suited for are those with increased populations that would be too slow to simulate using Gillespie’s stochastic simulation algorithm. For such problems, it is likely to achieve higher weak order in the moments.ConclusionsThe Stochastic Bulirsch-Stoer method is a novel stochastic solver that can be used for fast and accurate simulations. Crucially, compared to other similar methods, it better retains its high accuracy when the timesteps are increased. Thus the Stochastic Bulirsch-Stoer method is both computationally efficient and robust. These are key properties for any stochastic numerical method, as they must typically run many thousands of simulations
High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula
No full textIn recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the difficulty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage [High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics 22 (1996) 81-101] were able to construct strong local order 1.5 stochastic Runge-Kutta methods for certain cases, it is known that all extant stochastic Runge-Kutta methods suffer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler-Maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques. An additional difficulty in solving SODEs arises even in the linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper first the work of Magnus [On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (1954) 649-673] (applied to deterministic non-commutative linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed - hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number (d) of Wiener processes it is shown that strong local order 1 Runge-Kutta methods with d+1 stages can be constructed by evaluated a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the efficacy of these approaches
The Burrage mansion, Redlands, San Bernardino County, ca.1900
No full textPhotograph of the Burrage mansion, Redlands, San Bernardino County, ca.1900. Two adjoined(?) Mission Revival style buildings are visible beyond the trees in the foreground. The buildings feature multicurved gables, Spanish tiled roofs, and two dome towers. Lush forests of trees surround the mansion. A man (gardener?) is working near a tree in the foreground. At right, in the foreground, is a tree bearing fruits (orange?). Mountains are visible in the distance.; "Copper king Albert Burrage built the mansion in 1900. The 37,000-square-foot home is a full-scale copy of a Spanish mission. It rests on 17 acres. There are beautiful trees and flowers from around the world. The Burrage Mansion is a fine example of Mission Revival, a style that may be better represented in Redlands than any other city in Southern California. " -- unknown author
Parallel implementation of stochastic simulation for large-scale cellular processes
No full textExperimental and theoretical studies have shown the importance of stochastic processes in genetic regulatory networks and cellular processes. Cellular networks and genetic circuits often involve small numbers of key proteins such as transcriptional factors and signaling proteins. In recent years stochastic models have been used successfully for studying noise in biological pathways, and stochastic modelling of biological systems has become a very important research field in computational biology. One of the challenge problems in this field is the reduction of the huge computing time in stochastic simulations. Based on the system of the mitogen-activated protein kinase cascade that is activated by epidermal growth factor, this work give a parallel implementation by using OpenMP and parallelism across the simulation. Special attention is paid to the independence of the generated random numbers in parallel computing, that is a key criterion for the success of stochastic simulations. Numerical results indicate that parallel computers can be used as an efficient tool for simulating the dynamics of large-scale genetic regulatory networks and cellular processes
Effective simulation techniques for biological systems
No full textIn this paper we give an overview of some very recent work on the stochastic simulation of systems involving chemical reactions. In many biological systems (such as genetic regulation and cellular dynamics) there is a mix between small numbers of key regulatory proteins, and medium and large numbers of molecules. In addition, it is important to be able to follow the trajectories of individual molecules by taking proper account of the randomness inherent in such a system. We describe different types of simulation techniques (including the stochastic simulation algorithm, Poisson Runge-Kutta methods and the Balanced Euler method) for treating simulations in the three different reaction regimes: slow, medium and fast. We then review some recent techniques on the treatment of coupled slow and fast reactions for stochastic chemical kinetics and discuss how novel computing implementations can enhance the performance of these simulations
Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise
No full textThere has been considerable recent work on the development of energy conserving one-step methods that are not symplectic. Here we extend these ideas to stochastic Hamiltonian problems with additive noise and show that there are classes of Runge-Kutta methods that are very effective in preserving the expectation of the Hamiltonian, but care has to be taken in how the Wiener increments are sampled at each timestep. Some numerical simulations illustrate the performance of these methods
A spectral method for time-fractional diffusion systems
No full textThe time fractional derivative of a function y(t) depends on the past history of the function
y(t), and so time fractional differential systems are naturally suitable to describe evolutionary processes with memory. Fractional models are increasingly used in many modelling situations including, for example, viscoelastic materials in mechanics, anomalous diffusion in transport dynamics of complex systems and some biological processes in rheology. Here we consider a time-fractional reaction diusion problem [2]. This is a non-local model and as the solution depends on all its past history, numerical step-by-step methods are computationally expensive. We propose a mixed method, which consists of a finite difference scheme through space and a spectral collocation method through time. The spectral method considerably reduces the computational cost with respect to step-by-step methods and is exponentially convergent [3]. Some classes of spectral bases are considered, which exhibit different convergence rates and some numerical results based on time diffusion reaction diffusion equations are given [1].
References
[1] Burrage, K., Cardone, A., D'Ambrosio, R. and Paternoster, B. 2017 Numerical solution of time fractional diffusion systems. Appl. Numer. Math. 116 8294.
[2] Gafiychuk, V., Datsko, B. and Meleshko, V. 2008 Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math. 220(1-2) 215225.
[3] Zayernouri, M. and Karniadakis, G. Em 2014 Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1) A40A62
A multi-scaled approach for simulating chemical reaction systems
No full textIn this paper we give an overview of some very recent work, as well as presenting a new approach, on the stochastic simulation of multi-scaled systems involving chemical reactions. In many biological systems (such as genetic regulation and cellular dynamics) there is a mix between small numbers of key regulatory proteins, and medium and large numbers of molecules. In addition, it is important to be able to follow the trajectories of individual molecules by taking proper account of the randomness inherent in such a system. We describe different types of simulation techniques (including the stochastic simulation algorithm, Poisson Runge–Kutta methods and the balanced Euler method) for treating simulations in the three different reaction regimes: slow, medium and fast. We then review some recent techniques on the treatment of coupled slow and fast reactions for stochastic chemical kinetics and present a new approach which couples the three regimes mentioned above. We then apply this approach to a biologically inspired problem involving the expression and activity of LacZ and LacY proteins in E. coli, and conclude with a discussion on the significance of this work
Populations of models, experimental designs and coverage of parameter space by Latin Hypercube and Orthogonal sampling
No full textIn this paper we have used simulations to make a conjecture about the coverage of a t dimensional subspace of a d dimensional parameter space of size n when performing k trials of Latin Hypercube sampling. This takes the form P(k, n, d, t) = 1 - e. We suggest that this coverage formula is independent of d and this allows us to make connections between building Populations of Models and Experimental Designs. We also show that Orthogonal sampling is superior to Latin Hypercube sampling in terms of allowing a more uniform coverage of the t dimensional subspace at the sub-block size level. These ideas have particular relevance when attempting to perform uncertainty quantification and sensitivity analyses
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