Multilevel Hybrid Chernoff Tau-leap

Moraes, Álvaro; Tempone, Raúl; Vilanova, Pedro

doi:10.48550/arxiv.1403.2943

Cited by 2 publications

(13 citation statements)

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“…It is important to mention that the MNRM is used to simulate correlated exact/tauleap paths as well as nested tau-leap/tau-leap paths, as in [17,2]. In Section 2.5, we use this feature for coupling two mixed paths.…”

Section: The Modified Next Reaction Methods (Mnrm)mentioning

confidence: 99%

“…, d}. Using the exact pre-leap method we developed in [16,17] for single-level and multilevel hybrid schemes, allows us to switch adaptively between the tau-leap and an exact method. By construction, the probability that one hybrid path exits the lattice, Z d + , can be estimated by…”

Section: The Modified Next Reaction Methods (Mnrm)mentioning

confidence: 99%

“…A drawback of the tau-leap method is that the simulated paths may take negative values, which is a nonphysical consequence of the approximation and not a qualitative feature of the original process. For that reason, we proposed in [16,17] a Chernoff-based hybrid method that switches adaptively between the tau-leap and an exact method. This allows us to control the probability of reaching negative values while keeping the computational work substantially smaller than the work of an exact method.…”

mentioning

confidence: 99%

“…This allows us to control the probability of reaching negative values while keeping the computational work substantially smaller than the work of an exact method. The hybrid method developed in [16,17] can be successfully applied to systems where the state space, Z d + , can be decomposed into two regions according to the activity of the system; where all the propensities are uniformly low or uniformly high, i.e., non-stiff systems. To handle stiff systems, we first measure the total activity of the system at a certain state by the total sum of the propensity functions evaluated at this state.…”

mentioning

confidence: 99%

“…Next, in Section 4, we present a novel control variate technique to reduce the variance of the quantity of interest at level 0. In Section 5, the numerical examples illustrate the advantages of the mixed MLMC method over the hybrid MLMC method presented in [17] and to the SSA. Finally, Section 6 presents our conclusions.…”

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confidence: 99%

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A Multilevel Adaptive Reaction-splitting Simulation Method for Stochastic Reaction Networks

Moraes¹,

Tempone²,

Vilanova³

2016

SIAM J. Sci. Comput.

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Stochastic modeling of reaction networks is a framework used to describe the time evolution of many natural and artificial systems, including, biochemical reactive systems at the molecular level, viral kinetics, the spread of epidemic diseases, and wireless communication networks, among many other examples. In this work, we present a novel multilevel Monte Carlo method for kinetic simulation of stochastic reaction networks that is specifically designed for systems in which the set of reaction channels can be adaptively partitioned into two subsets characterized by either "high" or "low" activity. Adaptive in this context means that the partition evolves in time according to the states visited by the stochastic paths of the system. To estimate expected values of observables of the system at a prescribed final time, our method bounds the global computational error to be below a prescribed tolerance, T OL, within a given confidence level. This is achieved with a computational complexity of order O T OL −2 , the same as with an exact method, but with a smaller constant. We also present a novel control variate technique based on the stochastic time change representation by Kurtz, which may dramatically reduce the variance of the coarsest level at a negligible computational cost. Our numerical examples show substantial gains with respect to the standard Stochastic Simulation Algorithm (SSA) by Gillespie and also our previous hybrid Chernoff tau-leap method.

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Section: The Modified Next Reaction Methods (Mnrm)mentioning

confidence: 99%

Section: The Modified Next Reaction Methods (Mnrm)mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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A Multilevel Adaptive Reaction-splitting Simulation Method for Stochastic Reaction Networks

Moraes¹,

Tempone²,

Vilanova³

2016

SIAM J. Sci. Comput.

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Multi-index Monte Carlo: when sparsity meets sampling

2015

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Abstract. We propose and analyze a novel Multi-Index Monte Carlo (MIMC) method for weak approximation of stochastic models that are described in terms of differential equations either driven by random measures or with random coefficients. The MIMC method is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Inspired by Giles's seminal work, we use in MIMC high-order mixed differences instead of using first-order differences as in MLMC to reduce the variance of the hierarchical differences dramatically. This in turn yields new and improved complexity results, which are natural generalizations of Giles's MLMC analysis and which increase the domain of the problem parameters for which we achieve the optimal convergence, O(TOL −2 ). Moreover, in MIMC, the rate of increase of required memory with respect to TOL is independent of the number of directions up to a logarithmic term which allows far more accurate solutions to be calculated for higher dimensions than what is possible when using MLMC.We motivate the setting of MIMC by first focusing on a simple full tensor index set. We then propose a systematic construction of optimal sets of indices for MIMC based on properly defined profits that in turn depend on the average cost per sample and the corresponding weak error and variance. Under standard assumptions on the convergence rates of the weak error, variance and work per sample, the optimal index set turns out to be the total degree (TD) type. In some cases, using optimal index sets, MIMC achieves a better rate for the computational complexity than the corresponding rate when using full tensor index sets. We also show the asymptotic normality of the statistical error in the resulting MIMC estimator and justify in this way our error estimate, which allows both the required accuracy and the confidence level in our computational results to be prescribed. Finally, we include numerical experiments involving a partial differential equation posed in three spatial dimensions and with random coefficients to substantiate the analysis and illustrate the corresponding computational savings of MIMC.

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Multilevel Hybrid Chernoff Tau-leap

Cited by 2 publications

References 0 publications

A Multilevel Adaptive Reaction-splitting Simulation Method for Stochastic Reaction Networks

A Multilevel Adaptive Reaction-splitting Simulation Method for Stochastic Reaction Networks

Multi-index Monte Carlo: when sparsity meets sampling

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