Optimization of mesh hierarchies in multilevel Monte Carlo samplers

  • Abdul-Lateef Haji-AliEmail author
  • Fabio Nobile
  • Erik von SchwerinEmail author
  • Raúl Tempone


We perform a general optimization of the parameters in the multilevel Monte Carlo (MLMC) discretization hierarchy based on uniform discretization methods with general approximation orders and computational costs. We optimize hierarchies with geometric and non-geometric sequences of mesh sizes and show that geometric hierarchies, when optimized, are nearly optimal and have the same asymptotic computational complexity as non-geometric optimal hierarchies. We discuss how enforcing constraints on parameters of MLMC hierarchies affects the optimality of these hierarchies. These constraints include an upper and a lower bound on the mesh size or enforcing that the number of samples and the number of discretization elements are integers. We also discuss the optimal tolerance splitting between the bias and the statistical error contributions and its asymptotic behavior. To provide numerical grounds for our theoretical results, we apply these optimized hierarchies together with the Continuation MLMC Algorithm (Collier et al., BIT Numer Math 55(2):399–432, 2015). The first example considers a three-dimensional elliptic partial differential equation with random inputs. Its space discretization is based on continuous piecewise trilinear finite elements and the corresponding linear system is solved by either a direct or an iterative solver. The second example considers a one-dimensional Itô stochastic differential equation discretized by a Milstein scheme.


Multilevel Monte Carlo Monte Carlo Partial differential equations with random data Stochastic differential equations Optimal discretization 

Mathematics Subject Classification

65C05 65N30 65N22 



R. Tempone is a member of the Research Center on Uncertainty Quantification (SRI-UQ), division of Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) at King Abdullah University of Science and Technology (KAUST). The authors would like to recognize the support of the following KAUST and University of Texas at Austin AEA projects: Round 2, “Predictability and Uncertainty Quantification for Models of Porous Media”, and Round 3, “Uncertainty quantification for predictive modeling of the dissolution of porous and fractured media”. F. Nobile has been partially supported by the Swiss National Science Foundation under the Project No. 140574 “Efficient numerical methods for flow and transport phenomena in heterogeneous random porous media” and by the Center for ADvanced MOdeling Science (CADMOS). E. von Schwerin has been partially supported by the aforementioned SRI-UQ and CADMOS. We would also like to acknowledge the following open source software packages that made this work possible: PETSc [4], PetIGA [10].


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Applied Mathematics and Computational SciencesKAUSTThuwalSaudi Arabia
  2. 2.MATHICSE-CSQIEPF de LausanneLausanneSwitzerland
  3. 3.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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