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Title: Multi-level methods and approximating distribution functions

Abstract

Biochemical reaction networks are often modelled using discrete-state, continuous-time Markov chains. System statistics of these Markov chains usually cannot be calculated analytically and therefore estimates must be generated via simulation techniques. There is a well documented class of simulation techniques known as exact stochastic simulation algorithms, an example of which is Gillespie’s direct method. These algorithms often come with high computational costs, therefore approximate stochastic simulation algorithms such as the tau-leap method are used. However, in order to minimise the bias in the estimates generated using them, a relatively small value of tau is needed, rendering the computational costs comparable to Gillespie’s direct method. The multi-level Monte Carlo method (Anderson and Higham, Multiscale Model. Simul. 10:146–179, 2012) provides a reduction in computational costs whilst minimising or even eliminating the bias in the estimates of system statistics. This is achieved by first crudely approximating required statistics with many sample paths of low accuracy. Then correction terms are added until a required level of accuracy is reached. Recent literature has primarily focussed on implementing the multi-level method efficiently to estimate a single system statistic. However, it is clearly also of interest to be able to approximate entire probability distributions of species counts.more » We present two novel methods that combine known techniques for distribution reconstruction with the multi-level method. We demonstrate the potential of our methods using a number of examples.« less

Authors:
;  [1]
  1. Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG (United Kingdom)
Publication Date:
OSTI Identifier:
22611442
Resource Type:
Journal Article
Resource Relation:
Journal Name: AIP Advances; Journal Volume: 6; Journal Issue: 7; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; ACCURACY; APPROXIMATIONS; COMPARATIVE EVALUATIONS; CORRECTIONS; DISTRIBUTION; DISTRIBUTION FUNCTIONS; MARKOV PROCESS; MONTE CARLO METHOD; PROBABILITY; REDUCTION; SIMULATION

Citation Formats

Wilson, D., E-mail: daniel.wilson@dtc.ox.ac.uk, and Baker, R. E.. Multi-level methods and approximating distribution functions. United States: N. p., 2016. Web. doi:10.1063/1.4960118.
Wilson, D., E-mail: daniel.wilson@dtc.ox.ac.uk, & Baker, R. E.. Multi-level methods and approximating distribution functions. United States. doi:10.1063/1.4960118.
Wilson, D., E-mail: daniel.wilson@dtc.ox.ac.uk, and Baker, R. E.. 2016. "Multi-level methods and approximating distribution functions". United States. doi:10.1063/1.4960118.
@article{osti_22611442,
title = {Multi-level methods and approximating distribution functions},
author = {Wilson, D., E-mail: daniel.wilson@dtc.ox.ac.uk and Baker, R. E.},
abstractNote = {Biochemical reaction networks are often modelled using discrete-state, continuous-time Markov chains. System statistics of these Markov chains usually cannot be calculated analytically and therefore estimates must be generated via simulation techniques. There is a well documented class of simulation techniques known as exact stochastic simulation algorithms, an example of which is Gillespie’s direct method. These algorithms often come with high computational costs, therefore approximate stochastic simulation algorithms such as the tau-leap method are used. However, in order to minimise the bias in the estimates generated using them, a relatively small value of tau is needed, rendering the computational costs comparable to Gillespie’s direct method. The multi-level Monte Carlo method (Anderson and Higham, Multiscale Model. Simul. 10:146–179, 2012) provides a reduction in computational costs whilst minimising or even eliminating the bias in the estimates of system statistics. This is achieved by first crudely approximating required statistics with many sample paths of low accuracy. Then correction terms are added until a required level of accuracy is reached. Recent literature has primarily focussed on implementing the multi-level method efficiently to estimate a single system statistic. However, it is clearly also of interest to be able to approximate entire probability distributions of species counts. We present two novel methods that combine known techniques for distribution reconstruction with the multi-level method. We demonstrate the potential of our methods using a number of examples.},
doi = {10.1063/1.4960118},
journal = {AIP Advances},
number = 7,
volume = 6,
place = {United States},
year = 2016,
month = 7
}
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  • Approximations of functions by nonorthogonal basis functions are examined, and criteria for best fits for several types of convergence are discussed. In particular, expansions in Gaussian and Breit--Wigner functions are examined and some specific numerical examples with Gaussian functions are given, illustrating how the expansion parameters can be calculated analytically, rather than by searching for a best fit in a multidimensional space, as is conventionally done. 4 figures, 2 tables.