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23
Error bound for piecewise deterministic processes modeling stochastic reaction systems
, 2012
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An efficient finite difference method for parameter sensitivities of continuous time Markov chains
, 2012
"... We present an efficient finite difference method for the computation of parameter sensitivities that is applicable to a wide class of continuous time Markov chain models. The estimator for the method is constructed by coupling the perturbed and nominal processes in a natural manner, and the analys ..."
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Cited by 26 (13 self)
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We present an efficient finite difference method for the computation of parameter sensitivities that is applicable to a wide class of continuous time Markov chain models. The estimator for the method is constructed by coupling the perturbed and nominal processes in a natural manner, and the analysis proceeds by utilizing a martingale representation for the coupled processes. The variance of the resulting estimator is shown to be an order of magnitude lower due to the coupling. We conclude that the proposed method produces an estimator with a lower variance than other methods, including the use of common random numbers, in most situations. Often the variance reduction is substantial. The method is no harder to implement than any standard continuous time Markov chain algorithm, such as “Gillespie’s algorithm.” The motivating class of models, and the source of our examples, are the stochastic chemical kinetic models commonly used in the biosciences, though other natural application areas include population processes and queuing networks.
Weak error analysis of numerical methods for stochastic models of population processes
, 2012
"... The simplest, and most common, stochastic model for population processes, including those from biochemistry and cell biology, are continuous time Markov chains. Simulation of such models is often relatively straightforward, as there are easily implementable methods for the generation of exact sampl ..."
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Cited by 8 (3 self)
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The simplest, and most common, stochastic model for population processes, including those from biochemistry and cell biology, are continuous time Markov chains. Simulation of such models is often relatively straightforward, as there are easily implementable methods for the generation of exact sample paths. However, when using ensemble averages to approximate expected values, the computational complexity can become prohibitive as the number of computations per path scales linearly with the number of jumps of the process. When such methods become computationally intractable, approximate methods, which introduce a bias, can become advantageous. In this paper, we provide a general framework for understanding the weak error, or bias, induced by different numerical approximation techniques in the current setting. The analysis takes into account both the natural scalings within a given system and the step size of the numerical method. Examples are provided to demonstrate the main analytical results as well as the reduction in computational complexity achieved by the approximate methods.
Stochastic representations of ion channel kinetics and exact stochastic simulation of neuronal dynamics
, 2014
"... In this paper we provide two representations for stochastic ion channel kinetics, and compare the performance of exact simulation strategies with different, commonly used, approximate strategies. The first representation we present is a random time change representation, popularized by Thomas Kurtz, ..."
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Cited by 3 (3 self)
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In this paper we provide two representations for stochastic ion channel kinetics, and compare the performance of exact simulation strategies with different, commonly used, approximate strategies. The first representation we present is a random time change representation, popularized by Thomas Kurtz, with the second being analogous to a “Gillespie” representation. Stochastic models of ion channel kinetics typically consist of an ordinary differential equation, governing the voltage, coupled with a stochastic jump process, governing the number of open ion channels. The processes are coupled because the parameters of the ODE for the voltage depend upon the number of open ion channels, and the propensity for the opening and closing of the channels depends explicitly upon the timevarying voltage. Exact stochastic algorithms are provided for the different representations, which are preferable to either (a) fixed time step or (b) piecewise constant propensity algorithms, which still appear in the literature. As examples, we provide versions of the exact algorithms for the MorrisLecar conductance based model, and detail the error induced, both in a weak and a strong sense, by the use of approximate algorithms on this model. We include readytouse implementations of the random time change algorithm in both XPP and Matlab. Finally, through the consideration of parametric sensitivity analysis, we show how the representations presented here are useful in the development of further computational methods. The general representations and simulation strategies provided here are known in other parts of the sciences, but less so in the present setting.
An asymptotic relationship between coupling methods for stochastically modeled population processes
, 2014
"... This paper is concerned with elucidating a relationship between two common coupling methods for the continuous time Markov chain models utilized in the cell biology literature. The couplings considered here are primarily used in a computational framework by providing reductions in variance for diff ..."
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Cited by 3 (3 self)
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This paper is concerned with elucidating a relationship between two common coupling methods for the continuous time Markov chain models utilized in the cell biology literature. The couplings considered here are primarily used in a computational framework by providing reductions in variance for different Monte Carlo estimators, thereby allowing for significantly more accurate results for a fixed amount of computational time. Common applications of the couplings include the estimation of parametric sensitivities via finite difference methods and the estimation of expectations via multilevel Monte Carlo algorithms. While a number of coupling strategies have been proposed for the models considered here, and a number of articles have experimentally compared the different strategies, to date there has been no mathematical analysis describing the connections between them. Such analyses are critical in order to determine the best use for each. In the current paper, we show a connection between the common reaction path (CRP) method and the split coupling (SC) method, which is termed coupled finite differences (CFD) in the parametric sensitivities literature. In particular, we show that the two couplings are both limits of a third coupling strategy we call the “localCRP” coupling, with the split coupling method arising as a key parameter goes to infinity, and the common reaction path coupling arising as the same parameter goes to zero. The analysis helps explain why the split coupling method often provides a lower variance than does the common reaction path method, a fact previously shown experimentally.
Multilevel Monte Carlo methods
"... An outline history inspired by undergraduate numerical projects course at Cambridge, and summer projects at RollsRoyce this was one of my first textbooks after 25 years working on CFD, 10 years ago I switched to Monte Carlo methods for computational finance and other application areas ..."
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Cited by 2 (1 self)
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An outline history inspired by undergraduate numerical projects course at Cambridge, and summer projects at RollsRoyce this was one of my first textbooks after 25 years working on CFD, 10 years ago I switched to Monte Carlo methods for computational finance and other application areas
Nonnested Adaptive Timesteps in Multilevel Monte Carlo
"... Abstract This paper shows that it is relatively easy to incorporate adaptive timesteps into multilevel Monte Carlo simulations without violating the telescoping sum on which multilevel Monte Carlo is based. The numerical approach is presented for both SDEs and continuoustime Markov processes. Numer ..."
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Abstract This paper shows that it is relatively easy to incorporate adaptive timesteps into multilevel Monte Carlo simulations without violating the telescoping sum on which multilevel Monte Carlo is based. The numerical approach is presented for both SDEs and continuoustime Markov processes. Numerical experiments are given for each, with the full code available for those who are interested in seeing the implementation details. 1 Multilevel Monte Carlo and Adaptive Simulations Multilevel Monte Carlo methods [8, 4, 6] are a very simple and general approach to improving the computational efficiency of a wide range of Monte Carlo applications. Given a set of approximation levels ℓ = 0,1,...,L giving a sequence of approximations Pℓ of a stochastic output P, with the cost and accuracy both increasing as ℓ increases, then a trivial telescoping sum gives E[PL] = E[P0]+ L ℓ=1