A low-rank control variate for multilevel Monte Carlo simulation of high-dimensional uncertain systems

We describe and analyze a variance reduction approach for Monte Carlo (MC) sampling that accelerates the estimation of statistics of computationally expensive simulation models using an ensemble of models with lower cost. These lower cost models -which are typically lower fidelity with unknown statistics -are used to reduce the variance in statistical estimators relative to a MC estimator with equivalent cost. We derive the conditions under which our proposed approximate control variate framework recovers existing multi-model variance reduction schemes as special cases. We demonstrate that these existing strategies use recursive sampling strategies, and as a result, their maximum possible variance reduction is limited to that of a control variate algorithm that uses only a single low-fidelity model with known mean. This theoretical result holds regardless of the number of low-fidelity models and/or samples used to build the estimator. We then derive new sampling strategies within our framework that circumvent this limitation to make efficient use of all available information sources. In particular, we demonstrate that a significant gap can exist, of orders of magnitude in some cases, between the variance reduction achievable by using a single low-fidelity model and our non-recursive approach. We also present initial sample allocation approaches for exploiting this gap. They yield the greatest benefit when augmenting the high-fidelity model evaluations is impractical because, for instance, they arise from a legacy database. Several analytic examples and an example with a hyperbolic PDE describing elastic wave propagation in heterogeneous media are used to illustrate the main features of the methodology.

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“…The sets z i \ z 1 i can be chosen in several ways. Here we choose the same sampling strategy as ACV-MF 6 :…”

Section: Approximate Control Variatesmentioning

confidence: 99%

A generalized approximate control variate framework for multifidelity uncertainty quantification

Gorodetsky

Geraci

Eldred

et al. 2020

Journal of Computational Physics

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“…Therefore, the simulation experiment needs to be designed and optimized to obtain more confident and accurate statistical results within limited simulation times. [35][36][37] Due to the effect of a launch delay, the results of each simulation test are different even with the same initial conditions. Therefore, confidence intervals are used to compress the results to a certain extent.…”

Section: Simulation Experimental Designmentioning

confidence: 99%

Simulation and Evaluation of a Space Station Operational Plan Considering Launch Delay of Cargo Vehicles

Guo

Zhang

Luo

2021

Trans. Japan Soc. Aero. S Sci.

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“…Achieving small sampling errors, requires having access to large enough number of BF -and therefore LF -samples N . On the other hand, when the BF approximation does not meet the accuracy requirements but leads to estimates well correlated with the HF data, the BF approximation may serve as a control variate to MC in a single-level [26] or in a multilevel setting as in [36].…”

Section: Using Bi-fidelity Approximation To Estimate Qoi Statisticsmentioning

confidence: 99%

Bi-fidelity approximation for uncertainty quantification and sensitivity analysis of irradiated particle-laden turbulence.

Fairbanks

Jofre

Geraci

et al. 2018

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Efficiently performing predictive studies of irradiated particle-laden turbulent flows has the potential of providing significant contributions towards better understanding and optimizing, for example, concentrated solar power systems. As there are many uncertainties inherent in such flows, uncertainty quantification is fundamental to improve the predictive capabilities of the numerical simulations. For large-scale, multiphysics problems exhibiting high-dimensional uncertainty, characterizing the stochastic solution presents a significant computational challenge as many methods require a large number of high-fidelity solves. This requirement results in the need for a possibly infeasible number of simulations when a typical converged high-fidelity simulation requires intensive computational resources. To reduce the cost of quantifying highdimensional uncertainties, we investigate the application of a non-intrusive, bi-fidelity approximation to estimate statistics of quantities of interest associated with an irradiated particle-laden turbulent flow. This method relies on exploiting the low-rank structure of the solution to accelerate the stochastic sampling and approximation processes by means of cheaper-to-run, lower fidelity representations. The application of this bi-fidelity approximation results in accurate estimates of the QoI statistics while requiring a small number of high-fidelity model evaluations.Turbulent flow laden with inertial particles, or droplets, in the presence of thermal radiation is encountered in a wide range of natural phenomena and industrial applications. For instance, it is well established that turbulence-driven particle inhomogeneity plays a fundamental role in determining the rate of droplet coalescence and evaporation in ocean sprays [2] and atmospheric clouds [3]. Another example is found when studying fires, in which turbulence, soot particles, and radiation are strongly interconnected resulting in very complex physical processes [4]. From an industrial point of view, important applications include the atomization of liquid fuels in combustion chambers [5], soot formation in rocket engines [6], and more recently, volumetric particle-based solar receivers for energy harvesting [7].Even in the simplest configuration, e.g., homogeneous isotropic turbulence, particle-laden turbulent flow is known to exhibit complex interactions between the carrier and dispersed phases in the form of preferential concentration and turbulence modulation [8]. Preferential concentration is the phenomenon by which heavy particles tend to avoid intense vorticity regions and accumulate in regions of high strain rate, while turbulence modulation refers to the alteration of fluid flow characteristics in the near-field region of particle clusters as a result of two-way coupling effects, e.g., enhanced dissipation, kinetic energy transfer, or formation of wakes and vortexes. The physical complexity is further increased by the simple addition of solid walls as turbophoresis [9], i.e., tendency of particles to migr...

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A low-rank control variate for multilevel Monte Carlo simulation of high-dimensional uncertain systems

Cited by 48 publications

References 35 publications

A generalized approximate control variate framework for multifidelity uncertainty quantification

A generalized approximate control variate framework for multifidelity uncertainty quantification

Simulation and Evaluation of a Space Station Operational Plan Considering Launch Delay of Cargo Vehicles

Bi-fidelity approximation for uncertainty quantification and sensitivity analysis of irradiated particle-laden turbulence.

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