A continuation multilevel Monte Carlo algorithm

Abstract

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 weak and strong errors. 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.

Multilevel Monte Carlo methods for computing failure probability of porous media flow systems

“…Some way of balancing the error sources is necessary to make the error asymptotically vanish. One might want the errors to converge with different rates, for example to make the multilevel Monte Carlo estimator asymptotically a normal distribution, which requires the stochastic error to decrease with any strictly greater rate than the numerical error (see [8,Lemma 7.1]). However, in practice, an even balance between the errors is justified since it prevents one source of error to dominate the other.…”

“…The sample size N L for the deepest level L in the MLMC estimator is typically too small for estimating Var [Y L ] reliably using sample variance. See [8,12] for elaborate discussions about this. The approach used here is to estimate the variance for the lower levels and extrapolate it to the deeper levels.…”

“…An improvement of the MC estimator for application on numerical simulations of controllable numerical accuracy is the multilevel Monte Carlo (MLMC) estimator [20] first introduced in the context of differential equations in [15]. It has since then been applied to elliptic PDEs in [3,7,25,29] and has been further analyzed and extended in [8,18,19]. It exploits the convergence of numerical solutions with respect to some discretization parameter h (typically mesh size) and uses a series of corrector estimators of increasing cost but decreasing variance to allow for redistribution of the variance reduction effort to cheap low-accuracy problems.…”

Gaussian process modelling for uncertainty quantification in convectively-enhanced dissolution processes in porous media

“…Equation (14) shows that the total mass of solute in the domain increases through solute injection across the upper boundary and decreases through the first order reaction. We also note that in this particular problem, the more CO 2 is absorbed through the upper boundary, the more the process is considered to be e cient.…”

“…For modelling the correlation of Z, i.e., expression (21), we use the following exponential covariance function [29,54,14,13]:…”