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

Multi angle

Auteurs : Tempone, Raul (Auteur de la Conférence)
CIRM (Editeur )

Loading the player...

Résumé : 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

Codes MSC :
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

Ressources complémentaires :
http://smai.emath.fr/cemracs/cemracs17/Slides/tempone1.pdf

Informations sur la Vidéo

Réalisateur :

Hennenfent, Guillaume

Langue :

Date de publication :

Date de captation :

Sous collection :

arXiv category :

Domaine :

Format :

Durée :

Audience :

Download :

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

Informations sur la Rencontre

Nom de la rencontre : 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
Organisateurs de la rencontre : Bouchard, Bruno ; Chassagneux, Jean-François ; Delarue, François ; Gobet, Emmanuel ; Lelong, Jérôme
Dates : 17/07/17 - 25/08/17
Année de la rencontre : 2017
URL Congrès : http://conferences.cirm-math.fr/1556.html

Données de citation

DOI : 10.24350/CIRM.V.19200003
Citer cette vidéo: 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

Voir aussi

[Multi angle] Forward and backward simulation of Euler scheme / Auteur de la Conférence Gobet, Emmanuel.
[Multi angle] Stochastic variational inequalities for random mechanics / Auteur de la Conférence Mertz, Laurent.
[Multi angle] Mean field type control with congestion / Auteur de la Conférence Laurière, Mathieu.
[Multi angle] On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications / Auteur de la Conférence Silva Álvarez, Francisco José.
[Multi angle] Project evaluation under uncertainty / Auteur de la Conférence Zubelli, Jorge P..
[Multi angle] Some asymptotic results about American options and volativity / Auteur de la Conférence De Marco, Stefano.
[Multi angle] Projected particle methods for solving McKean Vlasov SDEs / Auteur de la Conférence Belomestny, Denis.
[Multi angle] Splitting algorithm for nested events / Auteur de la Conférence Goudenège, Ludovic.
[Multi angle] Lecture 2: Introduction to HPC - MPI: design of parallel program and MPI / Auteur de la Conférence Lelong, Jérôme.
[Multi angle] Lecture 1: Introduction to HPC, random generation, and OpenMP / Auteur de la Conférence Lelong, Jérôme.
[ Post-edited] Numerical methods for mean field games - Lecture 2: Monotone finite difference schemes / Auteur de la Conférence Achdou, Yves.
[Multi angle] Numerical methods for mean field games - Lecture 3: Variational MFG and related algorithms for solving the discrete system of nonlinear equations / Auteur de la Conférence Achdou, Yves.
[Multi angle] Numerical methods for mean field games - Lecture 1: Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions / Auteur de la Conférence Achdou, Yves.
[Multi angle] Metamodels for uncertainty quantification and reliability analysis / Auteur de la Conférence Marelli, Stefano.
[Multi angle] Global sensitivity analysis in stochastic systems / Auteur de la Conférence Le Maître, Olivier.
[Multi angle] Subsurface flow with uncertainty : applications and numerical analysis issues / Auteur de la Conférence Charrier, Julia.
[Multi angle] Dynamic formulations of optimal transportation and variational MFGs / Auteur de la Conférence Benamou, Jean-David.
[Multi angle] Least squares regression Monte Carlo for approximating BSDES and semilinear PDES / Auteur de la Conférence Turkedjiev, Plamen.
[Multi angle] Bandits in auctions (& more) / Auteur de la Conférence Perchet, Vianney.
[Multi angle] Multilevel and multi-index sampling methods with applications - Lecture 2: Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation / Auteur de la Conférence Tempone, Raul.
[Multi angle] Model-free control and deep learning / Auteur de la Conférence Bellemare, Marc.
[Multi angle] Optimal vector quantization: from signal processing to clustering and numerical probability / Auteur de la Conférence Pagès, Gilles.
[Multi angle] The Metropolis Hastings algorithm: introduction and optimal scaling of the transient phase / Auteur de la Conférence Jourdain, Benjamin.
[Multi angle] Branching for PDEs / Auteur de la Conférence Warin, Xavier.
[Multi angle] On the interplay between kinetic theory and game theory / Auteur de la Conférence Degond, Pierre.
[Multi angle] Particle algorithm for McKean SDE: a short review on numerical analysis / Auteur de la Conférence Bossy, Mireille.
[Multi angle] Cubature methods and applications / Auteur de la Conférence Crisan, Dan.
[Multi angle] Capacity expansion games with application to competition in power generation investments / Auteur de la Conférence Aïd, René.
[Multi angle] An introduction to BSDE / Auteur de la Conférence Imkeller, Peter.
[Multi angle] Mean field games with major and minor players / Auteur de la Conférence Carmona, René.

Bibliographie

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

Tempone.jpg

Sélection Signaler une erreur

TUTELLES

PARTENAIRES

Destination de la recherche

Raccourcis

Voir aussi