Optimization of mesh hierarchies in multilevel monte carlo samplers

Abdul Lateef Haji-Ali*, Fabio Nobile, Erik von Schwerin, Raúl Tempone

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

30 Scopus citations

Abstract

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 aniterative solver. The second example considers a one-dimensional Itô stochastic differential equation discretized by a Milstein scheme.

Original languageEnglish (US)
Pages (from-to)76-112
Number of pages37
JournalStochastics and Partial Differential Equations: Analysis and Computations
Volume4
Issue number1
DOIs
StatePublished - Mar 2016

Bibliographical note

Publisher Copyright:
© Springer Science+Business Media New York 2015.

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics

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