Drift-Implicit Multi-Level Monte Carlo Tau-Leap Methods for Stochastic Reaction Networks

Handle URI:
http://hdl.handle.net/10754/552677
Title:
Drift-Implicit Multi-Level Monte Carlo Tau-Leap Methods for Stochastic Reaction Networks
Authors:
Ben Hammouda, Chiheb ( 0000-0002-8386-0406 )
Abstract:
In biochemical systems, stochastic e↵ects can be caused by the presence of small numbers of certain reactant molecules. In this setting, discrete state-space and stochastic simulation approaches were proved to be more relevant than continuous state-space and deterministic ones. These stochastic models constitute the theory of stochastic reaction networks (SRNs). Furthermore, in some cases, the dynamics of fast and slow time scales can be well separated and this is characterized by what is called sti↵ness. For such problems, the existing discrete space-state stochastic path simulation methods, such as the stochastic simulation algorithm (SSA) and the explicit tau-leap method, can be very slow. Therefore, implicit tau-leap approxima- tions were developed to improve the numerical stability and provide more e cient simulation algorithms for these systems. One of the interesting tasks for SRNs is to approximate the expected values of some observables of the process at a certain fixed time T. This is can be achieved using Monte Carlo (MC) techniques. However, in a recent work, Anderson and Higham in 2013, proposed a more computationally e cient method which combines multi-level Monte Carlo (MLMC) technique with explicit tau-leap schemes. In this MSc thesis, we propose new fast stochastic algorithm, particularly designed 5 to address sti↵ systems, for approximating the expected values of some observables of SRNs. In fact, we take advantage of the idea of MLMC techniques and drift-implicit tau-leap approximation to construct a drift-implicit MLMC tau-leap estimator. In addition to accurately estimating the expected values of a given observable of SRNs at a final time T , our proposed estimator ensures the numerical stability with a lower cost than the MLMC explicit tau-leap algorithm, for systems including simultane- ously fast and slow species. The key contribution of our work is the coupling of two drift-implicit tau-leap paths, which is the basic brick for constructing our proposed drift-implicit MLMC tau-leap estimator. As an example of sti↵ problem, we used the decaying-dimerizing reaction as a test example to show the advantage of our drift-implicit method over the explicit one. Through our numerical experiments, we checked the convergence properties of our coupling algorithm and showed that our proposed estimator is outperforming the explicit MLMC estimator about three times in terms of computational work. We also illustrated in a second example how our drift-implicit MLMC tau-leap estimator can be forty times faster than the explicit MLMC.
Advisors:
Tempone, Raul ( 0000-0003-1967-4446 )
Committee Member:
Bisetti, Fabrizio ( 0000-0001-5162-7805 ) ; Knio, Omar; Scavino, Marco
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Program:
Applied Mathematics and Computational Science
Issue Date:
12-May-2015
Type:
Thesis
Appears in Collections:
Applied Mathematics and Computational Science Program; Theses; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.advisorTempone, Raulen
dc.contributor.authorBen Hammouda, Chiheben
dc.date.accessioned2015-05-13T12:03:08Zen
dc.date.available2015-05-13T12:03:08Zen
dc.date.issued2015-05-12en
dc.identifier.urihttp://hdl.handle.net/10754/552677en
dc.description.abstractIn biochemical systems, stochastic e↵ects can be caused by the presence of small numbers of certain reactant molecules. In this setting, discrete state-space and stochastic simulation approaches were proved to be more relevant than continuous state-space and deterministic ones. These stochastic models constitute the theory of stochastic reaction networks (SRNs). Furthermore, in some cases, the dynamics of fast and slow time scales can be well separated and this is characterized by what is called sti↵ness. For such problems, the existing discrete space-state stochastic path simulation methods, such as the stochastic simulation algorithm (SSA) and the explicit tau-leap method, can be very slow. Therefore, implicit tau-leap approxima- tions were developed to improve the numerical stability and provide more e cient simulation algorithms for these systems. One of the interesting tasks for SRNs is to approximate the expected values of some observables of the process at a certain fixed time T. This is can be achieved using Monte Carlo (MC) techniques. However, in a recent work, Anderson and Higham in 2013, proposed a more computationally e cient method which combines multi-level Monte Carlo (MLMC) technique with explicit tau-leap schemes. In this MSc thesis, we propose new fast stochastic algorithm, particularly designed 5 to address sti↵ systems, for approximating the expected values of some observables of SRNs. In fact, we take advantage of the idea of MLMC techniques and drift-implicit tau-leap approximation to construct a drift-implicit MLMC tau-leap estimator. In addition to accurately estimating the expected values of a given observable of SRNs at a final time T , our proposed estimator ensures the numerical stability with a lower cost than the MLMC explicit tau-leap algorithm, for systems including simultane- ously fast and slow species. The key contribution of our work is the coupling of two drift-implicit tau-leap paths, which is the basic brick for constructing our proposed drift-implicit MLMC tau-leap estimator. As an example of sti↵ problem, we used the decaying-dimerizing reaction as a test example to show the advantage of our drift-implicit method over the explicit one. Through our numerical experiments, we checked the convergence properties of our coupling algorithm and showed that our proposed estimator is outperforming the explicit MLMC estimator about three times in terms of computational work. We also illustrated in a second example how our drift-implicit MLMC tau-leap estimator can be forty times faster than the explicit MLMC.en
dc.language.isoenen
dc.subjectstochastic reaction networksen
dc.subjectMultilevel Monte Carloen
dc.subjectdrift-implicit tau-leapen
dc.titleDrift-Implicit Multi-Level Monte Carlo Tau-Leap Methods for Stochastic Reaction Networksen
dc.typeThesisen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
thesis.degree.grantorKing Abdullah University of Science and Technologyen_GB
dc.contributor.committeememberBisetti, Fabrizioen
dc.contributor.committeememberKnio, Omaren
dc.contributor.committeememberScavino, Marcoen
thesis.degree.disciplineApplied Mathematics and Computational Scienceen
thesis.degree.nameMaster of Scienceen
dc.person.id127050en
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.