Gaussian process modeling for stochastic multi-fidelity simulators, with application to fire safety
To assess the possibility of evacuating a building in case of a fire, a standard method consists in simulating the propagation of fire, using finite difference methods and takes into account the random behavior of the fire, so that the result of a simulation is non-deterministic. The mesh fineness tunes the quality of the numerical model, and its computational cost. Depending on the mesh fineness, one simulation can last anywhere from a few minutes to several weeks. In this article, we focus on predicting the behavior of the fire simulator at fine meshes, using cheaper results, at coarser mesh
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Gaussian process modeling for stochastic multi-fidelity simulators, with application to fire safety
Rйmi STROH 1,2 & Julien BECT 2 & Sйverine DEMEYER 1 & Nicolas FISCHER 1 & arXiv:1605.02561v1 [stat.ME] 9 May 2016
Emmanuel VAZQUEZ 2
LNE, Service Mathйmatiques et Statistiques;
29, avenue Roger Hennequin,78190, Trappes; prenom.nom@lne.fr 2 Laboratoire des
Signaux et Systиmes
CentraleSupйlec, CNRS, Univ. Paris-Sud, Universitй Paris Saclay
3, rue Joliot-Curie,91190, Gif-sur-Yvette; prenom.nom@lss.supelec.fr
Rйsumй. Pour йvaluer les possibilitйs d’йvacuation d’un bвtiment lors d’un incendie, une mйthode standard consiste а simuler la propagation d’un incendie, au moyen de modиles de type diffйrences finies, et en prenant en compte le comportement alйatoire du feu, de sorte que le rйsultat d’une simulation est non-dйterministe. La finesse du maillage dйtermine la qualitй du modиle numйrique, ainsi que son coыt de calcul. En fonction de la taille des mailles, une seule simulation peut durer entre quelques minutes et quelques semaines. Dans cet article, nous cherchons а prйdire le comportement du simulateur а une maille fine, а partir de rйsultats moins coыteux, а des mailles plus grossiиres. Dans la littйrature de la conception et de l’analyse d’expйriences numйriques, on parle d’approche multi-fidйlitй. Notre contribution est d’йtendre au cas de simulateurs stochastiques du modиle bayйsien multi-fidйlitй proposй par Picheny et Ginsbourger (2013) et Tuo et al.
(2014).
Mots-clйs. Expйriences numйriques, Processus gaussien, Multi-fidйlitй, Sйcuritй in- cendie
Abstract. To assess the possibility of evacuating a building in case of a fire, a standard method consists in simulating the propagation of fire, using finite difference methods and takes into account the random behavior of the fire, so that the result of a simulation is non-deterministic. The mesh fineness tunes the quality of the numerical model, and its computational cost. Depending on the mesh fineness, one simulation can last anywhere from a few minutes to several weeks. In this article, we focus on predicting the behavior of the fire simulator at fine meshes, using cheaper results, at coarser meshes. In the literature of the design and analysis of computer experiments, such a problem is referred to as multi- fidelity prediction. Our contribution is to extend to the case of stochastic simulators the
Bayesian multi-fidelity model proposed by Picheny and Ginsbourger (2013) and Tuo et al.
(2014).
Keywords. Numerical experiments, Gaussian process, Multi-fidelity, Fire safety
1 Introduction
Fire Dynamics Simulator (FDS) is a numerical simulator developed by the National In- stitute of Standards and Technology, that is used to simulate the propagation of fire in a building, and assess its conformity to fire safety standards. Fire Dynamics Simulator is based on a finite difference method, that takes into account the random behavior of fire propagation. Consequently, the outputs of Fire Dynamics Simulator are stochastic.
Using smaller mesh size increases the quality of a simulation with respect to the physical reality, but also increases the computational cost (see Table 1). Thus, the mesh size con- trols a trade-off between speed and fidelity. In other words, Fire Dynamics Simulator is a multi-fidelity simulator.
Our work aims at estimating the behavior of Fire Dynamics Simulator at a very fine mesh, with a limited computational budget, using the result of simulations carried out using coarser mesh sizes. To do this, we use a Bayesian approach, where we construct a model of the output of Fire Dynamics Simulator as a function of the mesh size. Follow- ing Kennedy and O’Hagan (2000) and others, our approach is based on Gaussian process modeling.
Section 2 shows how to extend Bayesian multi-fidelity models proposed by Picheny and Ginsbourger
(2013) and Tuo et al. (2014) in the case of deterministic simulators, to deal with the case of stochastic simulator. Section 3 presents numerical results to assess the quality of our new model.
2 Model for multi-fidelity
To formalize, consider n input-output pairs of a stochastic simulator with a tuning param- eter, ((xi , ti ) , Zi )1≤i≤n ∈ (X Ч T) Ч R, where X ⊂ Rd and T ⊂ R+ . The outputs Zi are supposed to be realizations of random variables, following distributions Pxi ,ti . The results are mutually independent. In order to simplify, the distributions Pxi ,ti are assumed to be
Gaussian distributions, with means ξ (xi , ti ), and variances λ (xi , ti ) :
Zi ∼ N (ξ (xi , ti ) , λ (xi , ti )) . (1)
To simplify, λ is supposed to depend only on t : λ (x, t) = λ (t). Besides, we add a
Gaussian prior distribution on the mean process, ξ. We assumed that ξ ∼ GP (m, k). The prior distributions of ξ and λ are independent.
Mesh size t (cm) 100 50 33.33 25 20 Duration of one simulation (h) 1/12 1 6 20 54
Table 1: Approximate duration of one run of Fire Dynamics Simulator, on the example pre- sented in Section 3, as a function of the tuning parameter.
A popular model for the process ξ was developed by Kennedy and O’Hagan (2000) : it is a recursive model, built in the case of finite number of levels, which links two successive
Gaussian processes by a autoregressive relationship, AR (1). However, this model is not well-suited to a simulator with a continuous tuning parameter. Indeed, even if this model can be extended for any number of levels (Le Gratiet, 2013), the number of covariance parameters increases strongly with the number of levels. Also, this model does not actually use the value of the tuning parameter, t.
For these reasons, another modeling was recently developed by Picheny and Ginsbourger
(2013) and Tuo et al. (2014). They supposed that an ideal simulator can be thought up by setting the tuning parameter to an extreme value (t = 0). Then, the process ξ can be written as ξ (x, t) = ξ0 (x) + ε (x, t) , (2) where ξ0 models this ideal simulator, and ε represents a deterministic h numerical i error, independent of ξ0 , which decreases when t tends to 0 : limt→0 E ε (x, t) = 0.
Following this decomposition, the covariance function of ξ, k, is the sum of two covari- ance functions : kξ0 (x, x′ ) + kε ((x, t) , (x′ , t′ )). Two assumptions are made : first, the covariance function kε is separable, kε ((x, t) , (x′ , t′ )) = r (t, t′ ) kXε (x, x′ ), then, the two spatial covariance functions, kξ0 and kXε are stationary. We choose anisotropic Matйrn covariance functions for both. The correlation r is chosen as a function of Brownian covariance function: r (t, t′ ) = min {t, t′ }L , L a positive real parameter.
The mean of ξ, m, is supposed constant, with an improper uniform prior distribution on R.
Finally, in order to improve the estimation of the observation variances, particularly on costly levels, we add a prior distribution on (λ (t))t∈T . This prior distribution describes two ideas : values of λ (t) are not precisely-known a priori, so Var [λ (t)] are large; but the variances are alike, so Var [λ (t) λ (t′ )] are small. Finally, a log-normal prior is chosen :
(ln [λ (t)])t∈T ∼ N ln (λprior ) 1, ς 2 Id + s2 1 , (3) with λprior equals to 1% of the range of the output; 1 is the vector of ones; Id the identity matrix; 1 the square matrix of ones; s2 = ln (10)2 ≫ ς 2 = (ln (2) /3)2 .
Finally, our multi-fidelity non-stationary model is built as follows :
n o
(Zi )1≤i≤n ξ, (λ (t))t∈T ∼ N (ξ (xi , ti ))1≤i≤n , diag (λ (ti ))1≤i≤n ;
ξ ∼ GP (m, k) ;
m (x, t) = m ∼ UR ; (4)
′ ′ ′ ′ L ′ k ((x, t) , (x , t )) = kξ0 (x − x ) + min {t, t } kXε (x − x ) ;
(ln [λ (t)])t∈T ∼ N ln (λprior ) 1, ς 2 Id + s2 1 .
The parameter m is integrated out analytically. All other parameters, (λ (t))t∈T , L and the hyper-parameters of kξ0 and kXε , are estimated by maximization of the joint posterior density (MAP estimation).
Kind of design Multi-fidelity design High-fidelity design Property Nested Latin Hypercube t (cm) 100 50 33 25 20 t (cm) 100 50 33 25 20 Observations npoints 270 90 30 10 0 npoints 0 0 0 0 100 Speed ≈ 11 faster 1 (reference)
Table 2: Summary of designs used for comparison. npoints : number of points.
3 Numerical results
We consider a parallelepiped building, with two doors and two windows, simulated with
Fire Dynamics Simulator. We study the maximal temperature in the building, Ttc (x), as function of d = 8 inputs (external temperature, fire area. . . ).
The model presented in Section 2 is called multi-fidelity non-stationary model and denoted by M-F1 . Our objective here is to compare it to two other models. The first model is similar to (4), but uses a stationary anisotropic Matйrn covariance function on
X Ч T. This model, called multi-fidelity stationary model and denoted by M-F2 , is a simplification of the multi-fidelity non-stationary model. The second model is built only from the most accurate level of the simulator. This model, called high-fidelity model and noted H-F., serves as a reference value.
Two datasets have been built for this numerical experiment (see Table 2). The first one is built with 400 simulations, at different mesh sizes. It is used to build both multi-fidelity models (M-F1 and M-F2 ). The second one consists of 100 simulations at t = 20 cm. It is used to build the high-fidelity model H-F., and also for validation.
The results of prediction are presented on Figure 1. On the left, figures show a com- parison between predictions (posterior means) and observations. For the high-fidelity model, predictions are made by leave-one-out cross-validation. On the right, the densities of normalized residuals are compared with the probability density function of the nor- mal distribution. Overall, the two multi-fidelity models present a goodness-of-fit similar to that of the high-fidelity model, which is our reference. On closer inspection, it ap- pears that both multi-fidelity models—and most particularly the multi-fidelity stationary model—actually underestimate T c for high values. However, the standard deviations of the densities of residuals are close to one, suggesting that posterior variances are neither too large, nor too thin.
c
Finally, we consider the problem of estimating the probability PX (T20cm (x) > 60°C) that the output temperature exceeds a critical threshold (here, 60°C), where PX is a probability distribution of the input space X. Such a probability is useful for fire engineers to assess the safety of the building. Figure 2 presents different estimations of a posterior probability density function of probability of exceeding the threshold. These densities are estimated with nsim = 1000 conditional random simulations on npts = 5000 inputs.
The high-fidelity model yields the narrowest posterior distribution. The multi-fidelity
M-F1 M-F2 H-F. M-F1 M-F2 H-F.
Probability density function
Probability density function
Probability density function
110 110
(°C)
(°C)
(°C)
0.35 0.35 0.35
100 100
90 0.3 0.3 0.3
90
Predictions of T20cm
Predictions of T20cm
Predictions of T20cm
80 80 0.25 0.25 0.25 c c
c
70 0.2 0.2 0.2 60 50 0.15 0.15 0.15 50 ag replacements 40 PSfrag40 replacements 0.1 0.1 0.1 30 30 0.05 0.05 0.05 20 0 0 0 20 40 60 80100 20 40 60 80100 20 40 60 80 100 −5 0 5 −5 0 5 −5 0 5 c c c T20cmobs T20cmobs T20cmobs ∆T20cm c ∆T20cm c ∆T20cm c
Figure 1: Results on prediction of level 20 cm, by the three models. From left to right, multi-fidelity non-stationary model (M-F1 ); multi-fidelity stationary model (M-F2 ); high-fidelity model (H-F.). Left : predictions versus observations; dashed line : first bisector (y = x). Right :
Estimation of the probability density function of the normalized residuals ∆T20cmc ; dashed line :
probability density function of normal distribution.
nbSimu = 1e3; nbPtsPerSimu = 5e3
Probability density function of the probability of failure
60 M-F1
M-F2
H-F.
PSfrag replacements
Posterior density
0.3 0.35 0.4 0.45 0.5 0.55 0.6 c p (PX (T20cm > 60°C)) c
Figure 2: Estimations of the posterior function of PX (T20cm > 60°C). These densities are estimated with nsim = 1000 conditional random simulations on npts = 5000 inputs.
stationary model also yields a small posterior uncertainty, but the support of the density of the probability of exceeding the threshold does not agree with that of the high-fidelity model. Our model has a larger posterior variance, but is compatible with the reference model, and has been obtained using less computational resources.
4 Conclusion
To conclude, we have proposed an extension of the model of Picheny and Ginsbourger
(2013) and Tuo et al. (2014) to the case of stochastic simulators. Our numerical results show that the proposed model makes it possible to predict the behavior of a stochastic multi-fidelity simulator at high fidelity, from simulations at low fidelities. We believe that this is a promising approach, particularly in the domain of fire safety. Future research will concentrate on fully Bayesian estimation of the parameters of the model, and sequential design of experiment in order to achieve a more accurate estimation of the probability of exceeding the threshold, using a limited computational budget for additional simulations.
References
Marc C Kennedy and Anthony O’Hagan. Predicting the output from a complex computer code when fast approximations are available. Biometrika, 87(1):1–13, 2000.
Loic Le Gratiet. Multi-fidelity Gaussian process regression for computer experiments. PhD thesis, Universitй Paris-Diderot-Paris VII, 2013.
Victor Picheny and David Ginsbourger. A nonstationary space-time gaussian process model for partially converged simulations. SIAM/ASA Journal on Uncertainty Quan- tification, 1(1):57–78, 2013.
Rui Tuo, C.F. Jeff Wu, and Dan Yu. Surrogate modeling of computer experiments with different mesh densities. Technometrics, 56(3):372–380, 2014.