Multilevel Monte Carlo Methods and Uncertainty Quantication

Aretha L Teckentrup

Research output: ThesisDoctoral Thesis

Abstract

We consider the application of multilevel Monte Carlo methods to elliptic partial
dierential equations with random coecients. Such equations arise, for example,
in stochastic groundwater
ow modelling. Models for random coecients
frequently used in these applications, such as log-normal random elds with exponential
covariance, lack uniform coercivity and boundedness with respect to
the random parameter and have only limited spatial regularity.
To give a rigorous bound on the cost of the multilevel Monte Carlo estimator
to reach a desired accuracy, one needs to quantify the bias of the estimator. The
bias, in this case, is the spatial discretisation error in the numerical solution of the
partial dierential equation. This thesis is concerned with establishing bounds
on this discretisation error in the practically relevant and technically demanding
case of coecients which are not uniformly coercive or bounded with respect to
the random parameter.
Under mild assumptions on the regularity of the coecient, we establish new
results on the regularity of the solution for a variety of model problems. The
most general case is that of a coecient which is piecewise Holder continuous with
respect to a random partitioning of the domain. The established regularity of the
solution is then combined with tools from classical discretisation error analysis to
provide a full convergence analysis of the bias of the multilevel estimator for nite
element and nite volume spatial discretisations. Our analysis covers as quantities
of interest several spatial norms of the solution, as well as point evaluations of the
solution and its gradient and any continuously Frechet dierentiable functional.
Lastly, we extend the idea of multilevel Monte Carlo estimators to the framework
of Markov chain Monte Carlo simulations. We develop a new multilevel
version of a Metropolis Hastings algorithm, and provide a full convergence analysis.
LanguageEnglish
QualificationPh.D.
Awarding Institution
  • University of Bath
Supervisors/Advisors
  • Scheichl, Robert, Supervisor
Award date19 Jun 2013
StatusUnpublished - Jun 2013

Fingerprint

Multilevel Methods
Monte Carlo method
Discretization Error
Regularity
Estimator
Uncertainty
Random Parameters
Convergence Analysis
Markov Chain Monte Carlo Simulation
Partial
Metropolis-Hastings Algorithm
Coercivity
Ground Water
Error Analysis
Boundedness
Partitioning
Quantify
Discretization
Numerical Solution
Cover

Keywords

  • pdes with random coecients
  • non-uniformly coercive

Cite this

Multilevel Monte Carlo Methods and Uncertainty Quantication. / Teckentrup, Aretha L.

2013. 150 p.

Research output: ThesisDoctoral Thesis

Teckentrup, AL 2013, 'Multilevel Monte Carlo Methods and Uncertainty Quantication', Ph.D., University of Bath.
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N2 - We consider the application of multilevel Monte Carlo methods to elliptic partialdierential equations with random coecients. Such equations arise, for example,in stochastic groundwater ow modelling. Models for random coecientsfrequently used in these applications, such as log-normal random elds with exponentialcovariance, lack uniform coercivity and boundedness with respect tothe random parameter and have only limited spatial regularity.To give a rigorous bound on the cost of the multilevel Monte Carlo estimatorto reach a desired accuracy, one needs to quantify the bias of the estimator. Thebias, in this case, is the spatial discretisation error in the numerical solution of thepartial dierential equation. This thesis is concerned with establishing boundson this discretisation error in the practically relevant and technically demandingcase of coecients which are not uniformly coercive or bounded with respect tothe random parameter.Under mild assumptions on the regularity of the coecient, we establish newresults on the regularity of the solution for a variety of model problems. Themost general case is that of a coecient which is piecewise Holder continuous withrespect to a random partitioning of the domain. The established regularity of thesolution is then combined with tools from classical discretisation error analysis toprovide a full convergence analysis of the bias of the multilevel estimator for niteelement and nite volume spatial discretisations. Our analysis covers as quantitiesof interest several spatial norms of the solution, as well as point evaluations of thesolution and its gradient and any continuously Frechet dierentiable functional.Lastly, we extend the idea of multilevel Monte Carlo estimators to the frameworkof Markov chain Monte Carlo simulations. We develop a new multilevelversion of a Metropolis Hastings algorithm, and provide a full convergence analysis.

AB - We consider the application of multilevel Monte Carlo methods to elliptic partialdierential equations with random coecients. Such equations arise, for example,in stochastic groundwater ow modelling. Models for random coecientsfrequently used in these applications, such as log-normal random elds with exponentialcovariance, lack uniform coercivity and boundedness with respect tothe random parameter and have only limited spatial regularity.To give a rigorous bound on the cost of the multilevel Monte Carlo estimatorto reach a desired accuracy, one needs to quantify the bias of the estimator. Thebias, in this case, is the spatial discretisation error in the numerical solution of thepartial dierential equation. This thesis is concerned with establishing boundson this discretisation error in the practically relevant and technically demandingcase of coecients which are not uniformly coercive or bounded with respect tothe random parameter.Under mild assumptions on the regularity of the coecient, we establish newresults on the regularity of the solution for a variety of model problems. Themost general case is that of a coecient which is piecewise Holder continuous withrespect to a random partitioning of the domain. The established regularity of thesolution is then combined with tools from classical discretisation error analysis toprovide a full convergence analysis of the bias of the multilevel estimator for niteelement and nite volume spatial discretisations. Our analysis covers as quantitiesof interest several spatial norms of the solution, as well as point evaluations of thesolution and its gradient and any continuously Frechet dierentiable functional.Lastly, we extend the idea of multilevel Monte Carlo estimators to the frameworkof Markov chain Monte Carlo simulations. We develop a new multilevelversion of a Metropolis Hastings algorithm, and provide a full convergence analysis.

KW - pdes with random coecients

KW - non-uniformly coercive

M3 - Doctoral Thesis

ER -