Robustness of Kriging when interpolating in random simulation with heterogeneous variances: Some experiments

Kleijnen, J.P.C.; Beers, W.C.M. van

doi:10.1016/j.ejor.2003.09.037

Cited by 83 publications

(49 citation statements)

References 8 publications

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“…The behavior of the Lebesgue constant seems to be linear and seems to be in accordance with previous works for the classical EIM (see [4]). The linear increase is far from the theoretical exponential upper bound of (8) and suggests that the bound might not be optimal in sets F of small Kolmogorov n-width. In the example, two types of sensors have been used (of pressure and velocity) and the idea of introducing different types of sensors could be extended to make more adequate distinctions among them.…”

Section: Discussionmentioning

confidence: 81%

“…• the obtention (if possible) of a general theory on the impact of Σ on the behavior of (Λ M ) and of a tighter upper bound than the one presented in (8).…”

Section: Discussionmentioning

confidence: 99%

“…Its behavior seems linear with the dimension of interpolation and is therefore far from the crude theoretical upper bound given in formula (8). From the results presented in section 4, an idea to improve the behavior of the Lebesgue constant could be to consider a smaller value for s. However, in the present context, we have not sought the optimization of (Λ M ) as a function of the parameter s because, in a real case, the linear functionals are fixed by the filter characteristics of the sensors involved in the experiment.…”

Section: A Numerical Application Involving the Stokes Equationmentioning

confidence: 99%

“…Although several procedures have been explored in this direction (we refer to [6], [7] and also to the kriging studies in the stochastic community such as [8]), of particular interest for the present work is the Empirical Interpolation Method (EIM, [2], [3], [4]) that has been developed in the broad framework where the functions f to approximate belong to a compact set F of continuous functions (X = C(Ω)). The structure of F is supposed to make any f ∈ F be approximable by finite expansions of small size.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Generalized Empirical Interpolation Method: Stability theory on Hilbert spaces with an application to the Stokes equation

Maday

Mula

Patera

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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The Generalized Empirical Interpolation Method (GEIM) is an extension first presented in [1] of the classical empirical interpolation method (see [2], [3], [4]) where the evaluation at interpolating points is replaced by the evaluation at interpolating continuous linear functionals on a class of Banach spaces. As outlined in [1], this allows to relax the continuity constraint in the target functions and expand the application domain. A special effort has been made in this paper to understand the concept of stability condition of the generalized interpolant (the Lebesgue constant) by relating it in the first part of the paper to an inf-sup problem in the case of Hilbert spaces. In the second part, it will be explained how GEIM can be employed to monitor in real time physical experiments by combining the acquisition of measurements from the processes with their mathematical models (parameter-dependent PDE's). This idea will be illustrated through a parameter dependent Stokes problem in which it will be shown that the pressure and velocity fields can efficiently be reconstructed with a relatively low dimension of the interpolating spaces.

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Section: Discussionmentioning

confidence: 81%

“…• the obtention (if possible) of a general theory on the impact of Σ on the behavior of (Λ M ) and of a tighter upper bound than the one presented in (8).…”

Section: Discussionmentioning

confidence: 99%

Section: A Numerical Application Involving the Stokes Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Generalized Empirical Interpolation Method: Stability theory on Hilbert spaces with an application to the Stokes equation

Maday

Mula

Patera

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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“…However, in experiments with random simulation models such as queueing models, the output variances var(w i ) are not constant at all! Fortunately, [12] demonstrates that the Kriging model is not very sensitive to this variance heterogeneity.…”

Section: Kriging: New Resultsmentioning

confidence: 99%

Kriging Metamodeling in Simulation: A Review

2007

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This article reviews Kriging (also called spatial correlation modeling). It presents the basic Kriging assumptions and formulas-contrasting Kriging and classic linear regression metamodels. Furthermore, it extends Kriging to random simulation, and discusses bootstrapping to estimate the variance of the Kriging predictor. Besides classic one-shot statistical designs such as Latin Hypercube Sampling, it reviews sequentialized and customized designs. It ends with topics for future research.

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Robust tuning and sensitivity analysis of stochastic integer and fractional‐order PID control systems: application of surrogate‐based robust simulation‐optimization

Parnianifard

Fakhfakh

Kotti

et al. 2020

Int J Numerical Modelling

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This paper aims to make a trade‐off between performance and robustness in stochastic control systems with probabilistic uncertainties. For this purpose, we develop a surrogate‐based robust simulation‐optimization approach for robust tuning and analyzing the sensitivity of stochastic controllers. Kriging surrogate is combined with robust design optimization to construct a robust simulation‐optimization model in the class of dual response surfaces. Randomness in simulation experiments due to uncertainty is analyzed through bootstrapping technique by computing confidence regions for the estimation of Pareto frontier. Results confirmed a proper trade‐off between the model's performance with the measure of expected Integral Squared Error (ISE) and robustness against uncertainty in the plant's physical parameters. Finally, the proposed method is evaluated in terms of accuracy, computational cost, and simplicity particularly in comparison with some common existed techniques in the tuning of the Proportional‐Integral‐Derivative (PID) and Fractional‐Order PID (FOPID) controllers.

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Robustness of Kriging when interpolating in random simulation with heterogeneous variances: Some experiments

Cited by 83 publications

References 8 publications

The Generalized Empirical Interpolation Method: Stability theory on Hilbert spaces with an application to the Stokes equation

The Generalized Empirical Interpolation Method: Stability theory on Hilbert spaces with an application to the Stokes equation

Kriging Metamodeling in Simulation: A Review

Robust tuning and sensitivity analysis of stochastic integer and fractional‐order PID control systems: application of surrogate‐based robust simulation‐optimization

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