Multi-index Monte Carlo: when sparsity meets sampling

Haji-Ali, Abdul-Lateef; Nobile, Fabio; Tempone, Raúl

doi:10.1007/s00211-015-0734-5

Cited by 131 publications

(229 citation statements)

References 34 publications

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“…The asymptotic normality of the estimator is usually shown using some form of the central limit theorem (CLT) or the Lindeberg-Feller theorem (see, e.g., Collier et al 2015;Haji-Ali et al 2015a for CLT results for the MLMC and MIMC estimators and Fig. 3-right).…”

Section: Problem Settingmentioning

confidence: 99%

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Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation

Haji-Ali

Tempone

2017

Stat Comput

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We address the approximation of functionals depending on a system of particles, described by stochastic differential equations (SDEs), in the mean-field limit when the number of particles approaches infinity. This problem is equivalent to estimating the weak solution of the limiting McKean-Vlasov SDE. To that end, our approach uses systems with finite numbers of particles and a timestepping scheme. In this case, there are two discretization parameters: the number of time steps and the number of particles. Based on these two parameters, we consider different variants of the Monte Carlo and Multilevel Monte Carlo (MLMC) methods and show that, in the best case, the optimal work complexity of MLMC, to estimate the functional in one typical setting with an error tolerance of TOL, is O TOL −3 when using the partitioning estimator and the Milstein time-stepping scheme. We also consider a method that uses the recent Multi-index Monte Carlo method and show an improved work complexity in the same typical setting of O TOL −2 log(TOL −1 ) 2 . Our numerical experiments are carried out on the so-called Kuramoto model, a system of coupled oscillators.

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Section: Problem Settingmentioning

confidence: 99%

“…Recently, the Multi-index Monte Carlo (MIMC) method (Haji-Ali et al 2015a) was introduced to tackle highdimensional problems with more than one discretization parameter. MIMC is based on the same concepts as MLMC and improves the efficiency of MLMC even further but requires mixed regularity with respect to the discretization parameters.…”

Section: Introductionmentioning

confidence: 99%

Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation

Haji-Ali

Tempone

2017

Stat Comput

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“…Multi-level/multi-index surrogate methods are closely related to many multilevel/multifidelity sampling algorithms. 24,25,[28][29][30][31][32] These sampling algorithms leverage correlation between the outputs of multiple models to reduce the variance in statistical estimators of quantities such as expectation. This variance reduction can result in orders of magnitude reduction in the computational cost of quantifying uncertainty, but like traditional MC sampling, the error decreases slowly as the number of samples is increased which can still render this approach infeasible in some contexts.…”

Section: Introductionmentioning

confidence: 99%

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

Jakeman

Eldred

Geraci

et al. 2019

Numerical Meth Engineering

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Summary In this paper, we present an adaptive algorithm to construct response surface approximations of high‐fidelity models using a hierarchy of lower fidelity models. Our algorithm is based on multi‐index stochastic collocation and automatically balances physical discretization error and response surface error to construct an approximation of model outputs. This surrogate can be used for uncertainty quantification (UQ) and sensitivity analysis (SA) at a fraction of the cost of a purely high‐fidelity approach. We demonstrate the effectiveness of our algorithm on a canonical test problem from the UQ literature and a complex multiphysics model that simulates the performance of an integrated nozzle for an unmanned aerospace vehicle. We find that, when the input‐output response is sufficiently smooth, our algorithm produces approximations that can be over two orders of magnitude more accurate than single fidelity approximations for a fixed computational budget.

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“…The intuitive requirement is only an increasing cost and accuracy with the discretization parameter. Very interesting theoretical results and applications seem to arise when more than one discretization parameter is alternately varied in a multi-dimensional hierarchy, as proposed by Haji-Ali et al [26]. This will be subject of future works.…”

Section: Generalized Notion Of "Level"mentioning

confidence: 90%

On the predictivity of pore-scale simulations: Estimating uncertainties with multilevel Monte Carlo

Icardi

Boccardo

Tempone

2016

Advances in Water Resources

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A fast method with tunable accuracy is proposed to estimate errors and uncertainties in porescale and Digital Rock Physics (DRP) problems. The overall predictivity of these studies can be, in fact, hindered by many factors including sample heterogeneity, computational and imaging limitations, model inadequacy and not perfectly known physical parameters. The typical objective of pore-scale studies is the estimation of macroscopic effective parameters such as permeability, effective diffusivity and hydrodynamic dispersion. However, these are often non-deterministic quantities (i.e., results obtained for specific pore-scale sample and setup are not totally reproducible by another "equivalent" sample and setup). The stochastic nature can arise due to the multi-scale heterogeneity, the computational and experimental limitations in considering large samples, and the complexity of the physical models. These approximations, in fact, introduce an error that, being dependent on a large number of complex factors, can be modeled as random. We propose a general simulation tool, based on multilevel Monte Carlo, that can reduce drastically the computational cost needed for computing accurate statistics of effective parameters and other quantities of interest, under any of these random errors. This is, to our knowledge, the first attempt to include Uncertainty Quantification (UQ) in pore-scale physics and simulation. The method can also provide estimates of the discretization error and it is tested on three-dimensional transport problems in heterogeneous materials, where the sampling procedure is done by generation algorithms able to reproduce realistic consolidated and unconsolidated random sphere and ellipsoid packings and arrangements. A totally automatic workflow is developed in an opensource code [1], that include rigid body physics and random packing algorithms, unstructured mesh discretization, finite volume solvers, extrapolation and post-processing techniques. The proposed method can be efficiently used in many porous media applications for problems such as stochastic homogenization/upscaling, propagation of uncertainty from microscopic fluid and rock properties to macro-scale parameters, robust estimation of Representative Elementary Volume size for arbitrary physics.

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Multi-index Monte Carlo: when sparsity meets sampling

Cited by 131 publications

References 34 publications

Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation

Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

On the predictivity of pore-scale simulations: Estimating uncertainties with multilevel Monte Carlo

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