A continuation multilevel Monte Carlo algorithm

• Published in: BIT Vol. 55; no. 2; pp. 399 - 432
• Main Authors: Collier, Nathan, Haji-Ali, Abdul-Lateef, Nobile, Fabio, von Schwerin, Erik, Tempone, Raúl
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.06.2015
• Subjects: ADAPTIVE WEAK APPROXIMATION; Bayesian inference; COMPLEXITY; Computational Mathematics and Numerical Analysis; DIFFERENTIAL-EQUATIONS; FINANCE; Mathematics; MATHEMATICS AND COMPUTING; Mathematics and Statistics; Monte Carlo; Multilevel Monte Carlo; Numeric Computing; Partial differential equations with random data; SIMULATION; Stochastic differential equations; Partial differential equations with random data; Stochastic differential equations; Monte Carlo; Multilevel Monte Carlo; 65C05; Bayesian inference; 65N22
• ISSN: 0006-3835; 1572-9125; 1572-9125
• DOI: 10.1007/s10543-014-0511-3

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.