Multilevel Richardson-Romberg Extrapolation

Lemaire, Vincent; Pagès, Gilles

doi:10.2139/ssrn.2539114

Cited by 18 publications

(52 citation statements)

References 27 publications

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“…For ε = 0.1, the ML2R run about 33 times faster than MLMC in the Barrier option pricing example and about 10 times faster in the nested Monte Carlo example. For more details on the numerical aspects we refer to [55]. 3.…”

Section: Numerical Results and Commentsmentioning

confidence: 99%

“…sequence of random variables with the same distribution as Z and independent of Y . If f is smooth enough we prove that the assumptions (Bias error) and (Strong error) are satisfied with α = β = 1 (see [55] for details). For more references on the nested Monte Carlo in the field of financial risk and actuary we refer to [12,28,40].…”

Section: Frameworkmentioning

confidence: 85%

“…We now present a specific implementation of the general Multilevel Richardson-Romberg (ML2R) estimator introduced in [55]. With this particular design the ML2R estimator is a weighted version of the classical Multilevel Monte Carlo (MLMC) estimator.…”

Section: Multilevel Estimatorsmentioning

confidence: 99%

“…In particular, it is proved that one can build a combination which outperforms MLMC. This note is based on a paper of the author with G. Pagès (see [55]). …”

Section: Introductionmentioning

confidence: 99%

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Recent advances in various fields of numerical probability

et al. 2015

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Abstract. The goal of this paper is to present a series of recent contributions on some various problems of numerical probability. Beginning with the Richardson-Romberg Multilevel Monte-Carlo method which, among other fields of applications, is a very efficient method for the approximation of diffusion processes, we focus on some adaptive multilevel splitting algorithms for rare event simulation. Then, the third part is devoted to the simulation of McKean-Vlasov forward and decoupled forwardbackward stochastic differential equations by some cubature algorithms. Finally, we tackle the problem of the weak error estimation in total variation norm for a general Markov semi-group.

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Section: Numerical Results and Commentsmentioning

confidence: 99%

Section: Frameworkmentioning

confidence: 85%

Section: Multilevel Estimatorsmentioning

confidence: 99%

“…In particular, it is proved that one can build a combination which outperforms MLMC. This note is based on a paper of the author with G. Pagès (see [55]). …”

Section: Introductionmentioning

confidence: 99%

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Recent advances in various fields of numerical probability

et al. 2015

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“…The strong convergence properties of a numerical scheme, which approximates the diffusion (1.1), are useful to control the variance of the multilevel Monte Carlo estimator based on this scheme (see [5] and [9]). This motivated our study of the strong convergence of the NinomiyaVictoir scheme in [1].…”

Section: (14)mentioning

confidence: 99%

Asymptotics for the normalized error of the Ninomiya–Victoir scheme

Gerbi

Jourdain

Clément³

2018

Stochastic Processes and their Applications

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In [1] we proved strong convergence with order 1/2 of the Ninomiya-Victoir scheme X N V,η with time step T /N to the solution X of the limiting SDE. In this paper we check that the normalized error defined by √ N X − X N V,η converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields. The limit does not depend on the Rademacher random variables η. This result can be seen as a first step to adapt to the Ninomiya-Victoir scheme the central limit theorem of Lindeberg Feller type, derived in [2] for the multilevel Monte Carlo estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes. This suggests that the rate of convergence is greater than 1/2 in this case and we actually prove strong convergence with order 1.

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Fast uncertainty quantification of tracer distribution in the brain interstitial fluid with multilevel and quasi Monte Carlo

Croci

Vinje

Rognes

2020

Numer Methods Biomed Eng

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Efficient uncertainty quantification algorithms are key to understand the propagation of uncertainty—from uncertain input parameters to uncertain output quantities—in high resolution mathematical models of brain physiology. Advanced Monte Carlo methods such as quasi Monte Carlo (QMC) and multilevel Monte Carlo (MLMC) have the potential to dramatically improve upon standard Monte Carlo (MC) methods, but their applicability and performance in biomedical applications is underexplored. In this paper, we design and apply QMC and MLMC methods to quantify uncertainty in a convection‐diffusion model of tracer transport within the brain. We show that QMC outperforms standard MC simulations when the number of random inputs is small. MLMC considerably outperforms both QMC and standard MC methods and should therefore be preferred for brain transport models.

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Multilevel Richardson-Romberg Extrapolation

Cited by 18 publications

References 27 publications

Recent advances in various fields of numerical probability

Recent advances in various fields of numerical probability

Asymptotics for the normalized error of the Ninomiya–Victoir scheme

Fast uncertainty quantification of tracer distribution in the brain interstitial fluid with multilevel and quasi Monte Carlo

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