TY - JOUR

T1 - A continuation multilevel Monte Carlo algorithm

AU - Collier, Nathan

AU - Haji Ali, Abdul Lateef

AU - Nobile, Fabio

AU - von Schwerin, Erik

AU - Tempone, Raul

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: Raul Tempone is a member of the Strategic Research Initiative on Uncertainty Quantification in Computational Science and Engineering at KAUST (SRI-UQ). The authors would like to recognize the support of King Abdullah University of Science and Technology (KAUST) AEA project "Predictability and Uncertainty Quantification for Models of Porous Media" and University of Texas at Austin AEA Round 3 "Uncertainty quantification for predictive modeling of the dissolution of porous and fractured media". We would also like to acknowledge the use of the following open source software packages: PETSc [4], PetIGA [8], NumPy, matplotlib [21].

PY - 2014/9/5

Y1 - 2014/9/5

N2 - We propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for weak approximation of stochastic models. The CMLMC algorithm solves the given approximation problem for a sequence of decreasing tolerances, ending when the required error tolerance is satisfied. CMLMC assumes discretization hierarchies that are defined a priori for each level and are geometrically refined across levels. The actual choice of computational work across levels is based on parametric models for the average cost per sample and the corresponding variance and weak error. These parameters are calibrated using Bayesian estimation, taking particular notice of the deepest levels of the discretization hierarchy, where only few realizations are available to produce the estimates. The resulting CMLMC estimator exhibits a non-trivial splitting between bias and statistical contributions. We also show the asymptotic normality of the statistical error in the MLMC estimator and justify in this way our error estimate that allows prescribing both required accuracy and confidence in the final result. Numerical results substantiate the above results and illustrate the corresponding computational savings in examples that are described in terms of differential equations either driven by random measures or with random coefficients. © 2014, Springer Science+Business Media Dordrecht.

AB - We propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for weak approximation of stochastic models. The CMLMC algorithm solves the given approximation problem for a sequence of decreasing tolerances, ending when the required error tolerance is satisfied. CMLMC assumes discretization hierarchies that are defined a priori for each level and are geometrically refined across levels. The actual choice of computational work across levels is based on parametric models for the average cost per sample and the corresponding variance and weak error. These parameters are calibrated using Bayesian estimation, taking particular notice of the deepest levels of the discretization hierarchy, where only few realizations are available to produce the estimates. The resulting CMLMC estimator exhibits a non-trivial splitting between bias and statistical contributions. We also show the asymptotic normality of the statistical error in the MLMC estimator and justify in this way our error estimate that allows prescribing both required accuracy and confidence in the final result. Numerical results substantiate the above results and illustrate the corresponding computational savings in examples that are described in terms of differential equations either driven by random measures or with random coefficients. © 2014, Springer Science+Business Media Dordrecht.

UR - http://hdl.handle.net/10754/563752

UR - http://link.springer.com/10.1007/s10543-014-0511-3

UR - http://www.scopus.com/inward/record.url?scp=84930003384&partnerID=8YFLogxK

U2 - 10.1007/s10543-014-0511-3

DO - 10.1007/s10543-014-0511-3

M3 - Article

VL - 55

SP - 399

EP - 432

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 2

ER -