Adaptive Multilevel Monte Carlo Simulation

Hoel, Håkon; Schwerin, Erik von; Szepessy, Anders; Tempone, Raúl

doi:10.1007/978-3-642-21943-6_10

Cited by 42 publications

(45 citation statements)

References 12 publications

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“…The most significant prior research on adaptive timestepping in MLMC has been by Hoel, von Schwerin, Szepessy and Tempone [9] and [10]. In their research, they construct a multilevel adaptive timestepping discretisation in which the timesteps used on level are a subdivision of those used on level −1, which in turn are a subdivision of those on level −2, and so on.…”

Section: In the Particular Case In Which |E[p ]−E[p] | ∝mentioning

confidence: 98%

Non-nested Adaptive Timesteps in Multilevel Monte Carlo Computations

Giles

Lester

Whittle

2016

Springer Proceedings in Mathematics &Amp; Statistics

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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 continuous-time Markov processes. Numerical experiments are given for each, with the full code available for those who are interested in seeing the implementation details.Keywords multilevel Monte Carlo · adaptive timestep · SDE · continuous-time Markov process Multilevel Monte Carlo and Adaptive SimulationsMultilevel Monte Carlo methods [4,6,8] 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 = 0, 1, . . . , L giving a sequence of approximations P of a stochastic output P, with the cost and accuracy both increasing as increases, then a trivial telescoping sum givesexpressing the expected value on the finest level as the expected value on the coarsest level of approximation plus a sum of expected corrections.

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Section: In the Particular Case In Which |E[p ]−E[p] | ∝mentioning

confidence: 98%

Non-nested Adaptive Timesteps in Multilevel Monte Carlo Computations

Giles

Lester

Whittle

2016

Springer Proceedings in Mathematics &Amp; Statistics

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“…] is in this setting based on constructing numerical realizations X (t) on stochastic adaptively refined meshes ∆t { } so that the 17) are asymptotically fulfilled, and by determining the number of samples M 0 to ensure that the statistical error…”

Section: Stochastic Time Steppingmentioning

confidence: 99%

“…[26]. The idea of extending the MLMC method [11] to hierarchies of adaptively refined, non uniform time discretizations that are generated by the adaptive algorithm introduced in [26,25,8] was first introduced and tested computationally by the authors in [17].…”

Section: Introductionmentioning

confidence: 99%

“…Later, the work [17] presented a MLMC method using a hierarchy of adaptively refined, non uniform time discretizations that are generated by the adaptive algorithm introduced in [26,25,8], and, as such, it may be considered a generalization of Giles' work [11]. This work improves the algorithms presented in [17] and furthermore, it provides mathematical analysis of these new adaptive MLMC algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis.…”

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confidence: 99%

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Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

Hoel

Schwerin

Szepessy

et al. 2013

McMa

Self Cite

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Abstract. We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic differential equations (SDE). The work [11] proposed and analyzed a MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, forward Euler Monte Carlo method from O TOL −3 to O (TOL −1 log(TOL −1 )) 2 for a mean square error of O TOL 2 . Later, the work [17] presented a MLMC method using a hierarchy of adaptively refined, non uniform time discretizations that are generated by the adaptive algorithm introduced in [26,25,8], and, as such, it may be considered a generalization of Giles' work [11]. This work improves the algorithms presented in [17] and furthermore, it provides mathematical analysis of these new adaptive MLMC algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis. Numerical tests include one case with singular drift and one with stopped diffusion, where the complexity of a uniform single level method is O TOL −4 . For both these cases the results confirm the theory, exhibiting savings in the computational cost for achieving the accuracy O (TOL) from O TOL −3 for the adaptive single level algorithm to essentially O TOL −2 log TOL −1 2 for the adaptive MLMC.

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“…To mitigate the known slow convergence of Monte Carlo methods, multilevel Monte Carlo methods have been recently developed; see, e.g., [2,3,4,5,6,7,8]. A hierarchy of nested finite element subspaces…”

Section: Introductionmentioning

confidence: 99%

A multilevel stochastic collocation method for SPDEs

Gunzburger

Jantsch

Teckentrup

et al. 2015

AIP Conference Proceedings

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Abstract. We present a multilevel stochastic collocation method that, as do multilevel Monte Carlo methods, uses a hierarchy of spatial approximations to reduce the overall computational complexity when solving partial differential equations with random inputs. For approximation in parameter space, a hierarchy of multi-dimensional interpolants of increasing fidelity are used. Rigorous convergence and computational cost estimates for the new multilevel stochastic collocation method are derived and used to demonstrate its advantages compared to standard single-level stochastic collocation approximations as well as multilevel Monte Carlo methods.

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Adaptive Multilevel Monte Carlo Simulation

Cited by 42 publications

References 12 publications

Non-nested Adaptive Timesteps in Multilevel Monte Carlo Computations

Non-nested Adaptive Timesteps in Multilevel Monte Carlo Computations

Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

A multilevel stochastic collocation method for SPDEs

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