Monte Carlo and Quasi-Monte Carlo Sampling

Lemieux, Christiane

doi:10.1007/978-0-387-78165-5

Cited by 74 publications

(17 citation statements)

References 326 publications

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“…Then, the numerical model needs to be run multiple times to generate results that reflect the uncertainties of the input parameters. For this purpose, there exist two main approaches: Monte Carlo (MC) methods, which draw samples from the probability distributions and Polynomial Chaos (PC) methods, which generate a polynomial representation based on the probability distributions (a surrogate model) ( Lemieux, 2009 ; Xiu, 2010 ). MC methods often require thousands of model runs to yield a reliable estimate of the model uncertainty and are thus not suitable for realistic 3D models.…”

Section: Methodsmentioning

confidence: 99%

Using a Digital Twin of an Electrical Stimulation Device to Monitor and Control the Electrical Stimulation of Cells in vitro

Zimmermann

Budde

Arbeiter

et al. 2021

Front. Bioeng. Biotechnol.

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Electrical stimulation for application in tissue engineering and regenerative medicine has received increasing attention in recent years. A variety of stimulation methods, waveforms and amplitudes have been studied. However, a clear choice of optimal stimulation parameters is still not available and is complicated by ambiguous reporting standards. In order to understand underlying cellular mechanisms affected by the electrical stimulation, the knowledge of the actual prevailing field strength or current density is required. Here, we present a comprehensive digital representation, a digital twin, of a basic electrical stimulation device for the electrical stimulation of cells in vitro. The effect of electrochemical processes at the electrode surface was experimentally characterised and integrated into a numerical model of the electrical stimulation. Uncertainty quantification techniques were used to identify the influence of model uncertainties on relevant observables. Different stimulation protocols were compared and it was assessed if the information contained in the monitored stimulation pulses could be related to the stimulation model. We found that our approach permits to model and simulate the recorded rectangular waveforms such that local electric field strengths become accessible. Moreover, we could predict stimulation voltages and currents reliably. This enabled us to define a controlled stimulation setting and to identify significant temperature changes of the cell culture in the monitored voltage data. Eventually, we give an outlook on how the presented methods can be applied in more complex situations such as the stimulation of hydrogels or tissue in vivo.

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Section: Methodsmentioning

confidence: 99%

Using a Digital Twin of an Electrical Stimulation Device to Monitor and Control the Electrical Stimulation of Cells in vitro

Zimmermann

Budde

Arbeiter

et al. 2021

Front. Bioeng. Biotechnol.

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“…As mentioned in the introduction, Sobol' indices may be estimated by direct MC or QMC sampling of the numerical simulator [44,6,40]. Alternatively, a surrogate model (e.g.…”

Section: Sobol' Indicesmentioning

confidence: 99%

“…Among the sensitivity measures used in SA studies, Sobol' indices [49] are particularly popular and powerful, and will be the focus of this paper. In UQ studies, such sensitivity indices are commonly estimated using Monte Carlo (MC) or quasi-Monte Carlo (QMC) sampling methods [44,6,40]. Precisely, the simulator is evaluated over a design of experiments whose elements are obtained from a set of independent realizations of the input random variables in the MC case or from a deterministic sequence of variables approaching such realizations efficiently in the QMC case.…”

mentioning

confidence: 99%

Multilevel Monte Carlo Covariance Estimation for the Computation of Sobol' Indices

Mycek¹,

Lozzo²

2019

SIAM/ASA J. Uncertainty Quantification

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Crude and quasi Monte Carlo (MC) sampling techniques are common tools dedicated to estimating statistics (expectation, variance, covariance) of a random quantity of interest. We focus here on the uncertainty quantification framework where the quantity of interest is the output of a numerical simulator fed with uncertain input parameters. Then, sampling the output involves running the simulator for different samples of the inputs, which may be computationally time-consuming. To reduce the cost of sampling, a first approach consists in replacing the numerical simulator by a surrogate model that is cheaper to evaluate, thus making it possible to generate more samples of the output and therefore leading to a lower sampling error. However, this approach adds to the sampling error an unavoidable model error. Another approach, which does not introduce any model error, is the so-called multilevel MC (MLMC) method. Given a sequence of levels corresponding to numerical simulators with increasing accuracy and computational cost, MLMC combines samples obtained at different levels to construct an estimator at a reduced cost compared to standard MC sampling. In this paper, we derive and analyze multilevel covariance estimators and adapt the MLMC convergence theorem in terms of the corresponding covariances and fourth order moments. We propose a multilevel algorithm driven by a target cost as an alternative to typical algorithms driven by a target accuracy. These results are used in a sensitivity analysis context in order to derive a multilevel estimation of Sobol' indices, whose building blocks can be written as covariance terms in a pick-and-freeze formulation. These contributions are successfully tested on an initial value problem with random parameters.

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“…The derivation of these results along with some more notation and the connection between our methods and the IS and stratification techniques from Arbenz et al (2018), Glasserman et al (1999), andNeddermeyer (2011) can be found in Section 2. There, we also briefly explain how our conditional sampling step reduces the effective dimension of the problem and therefore makes quasi-Monte Carlo (QMC) particularly attractive in our setting; in QMC, pseudo-random numbers (PRNs) are replaced by more homogeneously distributed quasi-random numbers (see, e.g., see Niederreiter (1978), Lemieux (2009), and Dick and Pillichshammer (2010)).…”

Section: Introductionmentioning

confidence: 99%

Single-Index Importance Sampling with Stratification

Hintz¹,

Hofert²,

Lemieux³

et al. 2021

Preprint

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In many stochastic problems, the output of interest depends on an input random vector mainly through a single random variable (or index) via an appropriate univariate transformation of the input. We exploit this feature by proposing an importance sampling method that makes rare events more likely by changing the distribution of the chosen index. Further variance reduction is guaranteed by combining this single-index importance sampling approach with stratified sampling. The dimension-reduction effect of single-index importance sampling also enhances the effectiveness of quasi-Monte Carlo methods. The proposed method applies to a wide range of financial or risk management problems. We demonstrate its efficiency for estimating large loss probabilities of a credit portfolio under a normal and t-copula model and show that our method outperforms the current standard for these problems.

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Monte Carlo and Quasi-Monte Carlo Sampling

Cited by 74 publications

References 326 publications

Using a Digital Twin of an Electrical Stimulation Device to Monitor and Control the Electrical Stimulation of Cells in vitro

Using a Digital Twin of an Electrical Stimulation Device to Monitor and Control the Electrical Stimulation of Cells in vitro

Multilevel Monte Carlo Covariance Estimation for the Computation of Sobol' Indices

Single-Index Importance Sampling with Stratification

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