CIRM - Videos & books Library - Multilevel and multi-index sampling methods with applications - Lecture 2: Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation

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

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Abstract : We describe and analyze the Multi-Index Monte Carlo (MIMC) and the Multi-Index Stochastic Collocation (MISC) method for computing statistics of the solution of a PDE with random data. MIMC is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Instead of using first-order differences as in MLMC, MIMC uses mixed differences to reduce the variance of the hierarchical differences dramatically. These mixed differences yield new and improved complexity results, which are natural generalizations of Giles's MLMC analysis, and which increase the domain of problem parameters for which we achieve the optimal convergence. On the same vein, MISC is a deterministic combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. Provided enough mixed regularity, MISC can achieve better complexity than MIMC. Moreover, we show that, in the optimal case, the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one-dimensional spatial problem. We propose optimization procedures to select the most effective mixed differences to include in MIMC and MISC. Such optimization is a crucial step that allows us to make MIMC and MISC computationally efficient. We show the effectiveness of MIMC and MISC in some computational tests using the mimclib open source library, including PDEs with random coefficients and Stochastic Interacting Particle Systems. Finally, we will briefly discuss the use of Markovian projection for the approximation of prices in the context of American basket options.

Keywords : Multi-index Monte Carlo; Multi-index Stochastic Collocation; optimal hierarchies; Partial Differential Equations with random inputs

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
65M70 - Spectral, collocation and related methods

Additional resources :
http://smai.emath.fr/cemracs/cemracs17/Slides/tempone2.pdf

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2017-07-21_Tempone_Part2.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.19200303
Cite this video as: Tempone, Raul (2017). Multilevel and multi-index sampling methods with applications - Lecture 2: Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19200303
URI : http://dx.doi.org/10.24350/CIRM.V.19200303

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Bibliography

Bayer, C., Häppölä, J., & Tempone, R. (2017). Implied stopping rules for American basket options from Markovian projection. - https://arxiv.org/abs/1705.00558

Haji-Ali, A.-L., Nobile, F., Tamellini, L., & Tempone, R. (2016). Multi-index stochastic collocation for random PDEs. Computer Methods in Applied Mechanics and Engineering, 306(1), 95–122 - http://dx.doi.org/10.1016/j.cma.2016.03.029

Haji-Ali, A.-L., Nobile, F., Tamellini, L., & Tempone, R. (2016). Multi-index stochastic collocation convergence rates for random PDEs with parametric regularity. Foundations of Computational Mathematics, 16(6), 1555-1605 - http://dx.doi.org/10.1007/s10208-016-9327-7

Haji-Ali, A.-L., Nobile, F., & Tempone, R. (2016). Multi-index Monte Carlo: when sparsity meets sampling. Numerische Mathematik, 132(4), 767-806 - http://dx.doi.org/10.1007/s00211-015-0734-5

Nobile, F., Tempone, R., & Wolfers, S. (2016). Sparse approximation of multilinear problems with applications to kernel-based methods in UQ. - https://arxiv.org/abs/1609.00246v2

Veretennikov, A.Y. (2006). On ergodic measures for McKean-Vlasov stochastic equations. In H. Niederreiter, & D. Talay (Eds.), Monte Carlo and quasi-Monte Carlo methods 2004 (pp. 471-486). Berlin: Springer - http://dx.doi.org/10.1007/3-540-31186-6_29

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