CIRM - Videos & books Library - Multilevel and multi-index sampling methods with applications - Lecture 1: Adaptive strategies for Multilevel Monte Carlo

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Authors : Tempone, Raul (Author of the conference)
CIRM (Publisher )

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Abstract : We will first recall, for a general audience, the use of Monte Carlo and Multi-level Monte Carlo methods in the context of Uncertainty Quantification. Then we will discuss the recently developed Adaptive Multilevel Monte Carlo (MLMC) Methods for (i) It Stochastic Differential Equations, (ii) Stochastic Reaction Networks modeled by Pure Jump Markov Processes and (iii) Partial Differential Equations with random inputs. In this context, the notion of adaptivity includes several aspects such as mesh refinements based on either a priori or a posteriori error estimates, the local choice of different time stepping methods and the selection of the total number of levels and the number of samples at different levels. Our Adaptive MLMC estimator uses a hierarchy of adaptively refined, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform discretization MLMC method introduced independently by M. Giles and S. Heinrich. In particular, we show that our adaptive MLMC algorithms are asymptotically accurate and have the correct complexity with an improved control of the multiplicative constant factor in the asymptotic analysis. In this context, we developed novel techniques for estimation of parameters needed in our MLMC algorithms, such as the variance of the difference between consecutive approximations. These techniques take particular care of the deepest levels, where for efficiency reasons only few realizations are available to produce essential estimates. Moreover, we show the asymptotic normality of the statistical error in the MLMC estimator, justifying in this way our error estimate that allows prescribing both the required accuracy and confidence level in the final result. We present several examples to illustrate the above results and the corresponding computational savings.

Keywords : Multilevel Monte Carlo; continuation Multilevel Monte Carlo; optimal hierarchies for Multilevel Monte Carlo; adaptive algorithms; Partial Differential Equations with random inputs; It Stochastic Differential Equations; Stochastic Reaction Networks

MSC Codes :
35R60 - PDEs with randomness, stochastic PDE
60H15 - Stochastic partial differential equations
60H35 - Computational methods for stochastic equations
65C05 - Monte Carlo methods
65C30 - Stochastic differential and integral equations

Additional resources :
http://smai.emath.fr/cemracs/cemracs17/Slides/tempone1.pdf

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2017-07-20_Tempone_part1.mp4

Information on the Event

Event Title : CEMRACS - Summer school: Numerical methods for stochastic models: control, uncertainty quantification, mean-field / CEMRACS - École d'été : Méthodes numériques pour équations stochastiques : contrôle, incertitude, champ moyen
Event Organizers : Bouchard, Bruno ; Chassagneux, Jean-François ; Delarue, François ; Gobet, Emmanuel ; Lelong, Jérôme
Dates : 17/07/17 - 25/08/17
Event Year : 2017
Event URL : http://conferences.cirm-math.fr/1556.html

Citation Data

DOI : 10.24350/CIRM.V.19200003
Cite this video as: Tempone, Raul (2017). Multilevel and multi-index sampling methods with applications - Lecture 1: Adaptive strategies for Multilevel Monte Carlo. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19200003
URI : http://dx.doi.org/10.24350/CIRM.V.19200003

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Bibliography

Ben Hammouda, C., Moraes, A., & Tempone, R. (2017). Multilevel hybrid split-step implicit tau-leap. Numerical Algorithms, 74(2), 527-560 - http://dx.doi.org/10.1007/s11075-016-0158-z

Collier, N., Haji-Ali, A.-L., Nobile, F., von Schwerin, E., & Tempone, R. (2015). A Continuation multilevel Monte Carlo algorithm. BIT Numerical Mathematics, 55(2), 399-432 - http://dx.doi.org/10.1007/s10543-014-0511-3

Haji-Ali, A.-L., Nobile, F., von Schwerin, E., & Tempone, R. (2016). Optimization of mesh hierarchies in multilevel Monte Carlo samplers. Stochastic and Partial Differential Equations. Analysis and Computations, 4(1), 76-112 - http://dx.doi.org/10.1007/s40072-015-0049-7

Hoel, H., Häppölä, J., & Tempone, R. (2016). Construction of a mean square error adaptive Euler-Maruyama method with applications in multilevel Monte Carlo. In R. Cools, & D. Nuyens (Eds.), Monte Carlo and quasi-Monte Carlo methods (pp. 29-86). Cham: Springer - http://dx.doi.org/10.1007/978-3-319-33507-0_2

Hoel, H., von Schwerin, E., Szepessy, A., & Tempone, R. (2014). Implementation and analysis of an adaptive multilevel Monte Carlo algorithm. Monte Carlo Methods and Applications, 20(1), 1-41 - http://dx.doi.org/10.1515/mcma-2013-0014

Moraes, A., Tempone, R., & Vilanova, P. (2016). Multilevel hybrid Chernoff tau-leap. BIT Numerical Mathematics, 56(1), 189-239 - http://dx.doi.org/10.1007/s10543-015-0556-y

Moraes, A., Tempone, R., & Vilanova, P. (2016). A multilevel adaptive reaction-splitting simulation method for stochastic reaction networks. SIAM Journal on Scientific Computing, 38(4), A2091-A2117 - http://dx.doi.org/10.1137/140972081

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