International audienceUncertainty Analysis and Sensitivity Analysis of complex models: Coping with dynamic and static input

Anstett-Collin, Floriane

Denis-Vidal, Lilianne

Goffart, Jeanne

Mara, Thierry A.

HAL Université de Savoie

UASA of complex models: Coping with dynamic andstatic inputsFloriane Anstett-Collin, Thierry A. Mara, Lilianne Denis-Vidal, JeanneGoffartTo cite this version:Floriane Anstett-Collin, Thierry A. Mara, Lilianne Denis-Vidal, Jeanne Goffart. UASA ofcomplex models: Coping with dynamic and static inputs. 7th International Conference onSensitivity Analysis of Model Output, SAMO 2013, Jul 2013, Nice, France. <hal-00926454>HAL Id: hal-00926454https://hal.archives-ouvertes.fr/hal-00926454Submitted on 9 Jan 2014HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.7th International Conference on Sensitivity Analysis of Model Output, July 1–4 2013, NiceUASA of complex models: Coping with dynamic and static inputsF. Anstett-CollinUniversite´ de Lorraine, CRAN, Nancy, FranceT. MaraUniversite´ de La Re´union, PIMENT, Re´union, FranceL. Denis-VidalUniversity of Technology of Compie`gne, LMAC, Compie`gne, FranceJ. GoffartUniversity of Savoy, Le Bourget Du Lac, FranceIn many fields, complex systems are modelled by a set of partial differential equations with initialand boundary conditions. For instance, in mechanics or thermodynamics, the PDEs are based onconservation laws. A particular problem is defined by a set of inputs that characterized the systemof interest, embedding the initial and boundary conditions. Then, numerical methods are employedto solve the problem. In practice, the system is not accurately defined due to the uncertainty aboutsome inputs. Uncertainty and sensitivity analyses (UASA) can help assess the impact of this lackof knowledge onto the model responses ([1,2]). Let y = g(ωd(x, θ),ωs(θ), x) be the response ofinterest where: x ∈ D is the spatial/time variable, ωd is a set of random fields (dynamic inputs)and ωs is a set of random variables (static inputs) (see figure 1). As an example, in building energymodelling, ωs embeds the thermophysical properties of the materials used in the building whileωd represents the weather data.In this communication, we address the issue of performing UASA with these two kinds of uncertaininputs. Indeed, in the literature, such an issue is rarely addressed (except, for instance, in [3]).For the sake of simplicity, we assume that random variables are independent and defined by theirmarginal distribution. The random fields are also assumed independent and normally distributedwith mean ω¯i(x) and covariance function Ci(x1, x2), i = 1, · · · , Nd, Nd denoting the number ofdynamic inputs. Monte Carlo based methods can be used to perform UASA of such a model. But,while generating static inputs samples is not an issue, it is not straightforward to generate samplesthat satisfy the desired random fields distribution. One possibility is to resort to the truncatedKarhunen-Loeve (KL) expansion. The former expands a random field as follows:ωdi (x, θ) ≃ ω¯di (x) +Mi∑k=1√λkiξki(θ)fki(θ), (1)where λki and fki are the deterministic eigenvalues and eigenfunctions of the covariance functionCi(x1, x2), ξi(θ) is a set of independent standard normal variables and Mi is the number of KL-terms. The eigenmodes depend on the choice of the covariance function and are determined bysolving the Fredholm integral equation of the second kind given by:∫DCi(x1, x2)fki(x1)dx1 = λkifki(x2). (2)Equation (2) can be solved using a wavelet-Galerkin scheme ([4]). The advantage of this approachis to avoid tedious quadratures by using wavelet transform, alleviating computational effort.In practice, we retain the first Mi eigenmodes that contain the 95% of the variance of the inputωdi. The number of eigenmodes retained depends on the choice of the covariance function andmay be very different from one input to another. Note that once the eigenmodes are obtainedfor all the dynamic inputs, UASA of the model output are performed through the random vectors{ξ1, · · · , ξNd︸ ︷︷ ︸ωd,ωs}. Consequently, the effect of the group of factors ξi is the one of the dynamicinput ωdi. This effect can be estimated with sampling-based methods such as Sobol’ method ([5]).7th International Conference on Sensitivity Analysis of Model Output, July 1–4 2013, NiceThe approach is applied to a building energy model. This model presents dynamic inputs as drybulb temperature, direct and diffuse radiations, humidity, speed and direction of wind, and staticinputs as the thermal properties of the materials. The model response of interest is the energyconsumption (scalar output).scalar{ωs1✲...ωdNd✲Complex modely1✲...ym✲}scalarFig. 1 - Complex model with static and dynamic inputsReferences:[1] A. Saltelli and K. Chan and E. M. Scott, “Sensitivity analysis”,Wiley, 2000.[2] E. De Rocquigny and N. Devictor and S. Tarantola, “Uncertainty in industrial practice”,Wiley,2008.[3] L. Lilburne and S. Tarantola , “Sensitivity analysis of spatial models”, International Journalof Geographical Information Science, 23:151–168, 2009.[4] K. K. Phoon and S. P. Huang, “Implementation of Karhunen-Loeve expansion for simulationusing a Wavelet-Galerkin scheme”, Probabilistic Engineering Mechanics, 17:293–303, 2002.[5] I. M. Sobol’ , “Global sensitivity indices for nonlinear mathematical models and their MonteCarlo estimates”, Mathematics and Computers in Simulation, 55:271–280, 2001.[ F. Anstett-Collin; Universite´ de Lorraine, Centre de Recherche en Automatique de Nancy (CRAN)- CNRS UMR 7039. ESSTIN, 2 Rue Jean Lamour - 54519 Vandoeuvre Le`s Nancy, France. ][ floriane.collin@univ-lorraine.fr – ]

UASA of complex models: Coping with dynamic and static inputs

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UASA of complex models: Coping with dynamic and static inputs

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