Computer simulation models a.k.a. simulators are used nowadays in virtually all fields of applied science and engineering. Usually, simulators that predict quantities of interests (QoI) as a function of input parameters are deterministic, i.e. they can be considered as a mapping from an input- to an output space. Running the simulator twice with the same input values provides identical outputs.
In contrast, so-called stochastic simulators contain hidden sources of uncertainty (e.g. latent variables) or uncontrollable inputs, on top of the well-identified and controllable inputs, meaning that repeated runs with the same inputs provides different results. Of interest is the resulting distribution of the QoI conditioned by the input (controllable) parameters. This distribution can be characterized in a rough way by replicating the runs of the simulator for the same controllable inputs. Unfortunately, in the context of uncertainty propagation and sensitivity analysis, handling stochastic simulators may be highly demanding due to these replications. One appropriate solution can be to use surrogate models (also referred to metamodels) to approximate the conditional expectation of the model, from a limited number of simulations.
In this MS, we will present recent developments in the field of surrogate models for stochastic simulators, be it for uncertainty propagation, sensitivity analysis or robust design.
14:00
Multiscale global sensitivity analysis for stochastic reaction networks
Alen Alexanderian | North Carolina State University | United States
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Authors:
Alen Alexanderian | North Carolina State University | United States
Pierre Gremaud | North Carolina State University | United States
Michael Merritt | North Carolina State University | United States
Microscopic approaches to chemical reaction networks (CRNs) lead to discrete-in-time stochastic models where the state variables are the numbers of molecules for each species. In addition to intrinsic stochasticity, these systems include rate parameters that are often subject to uncertainty. Global sensitivity analysis (GSA) enables determining rate parameters that are most influential to uncertainty in model output. However, such an analysis is challenging for stochastic models. This is due to the high computational cost of simulating the model and the need to sample the model in a product probability space defined by the product of the probability spaces carrying the parametric uncertainty and intrinsic model stochasticity. On the other hand, it is cheaper to perform GSA on the corresponding mean-field model given by the reaction rate equations (RREs). We propose a multiscale GSA framework for such systems that relates the GSA measures corresponding to RREs to the stochastic GSA measure corresponding to the stochastic system. We will also discuss stochastic models in general and use of surrogates for GSA in such systems
14:30
Sensitivity analysis in general metric spaces
Agnès Lagnoux | Université de Toulouse | France
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Authors:
Agnès Lagnoux | Université de Toulouse | France
Fabrice Gamboa | Université de Toulouse | France
Thierry Klein | Ecole Nationale de l'Aviation Civile / Université de Toulouse | France
Leonardo Moreno | Universidad de la República | Uruguay
In the last decades, the use of computer code experiments to model some physical phenomenon has increased considerably. Due to the expensive computational cost of such black box models it has become crucial to understand the global influence of one or several inputs on the output. Sobol index is well tailored when the output space is R. It compares using the Hoeffding decomposition the conditional variance of the output knowing some of the input variables to the global variance of the output. Since Sobol indices are based on the variance through the Hoeffding decomposition, they only quantify the influence of the inputs with respect to the mean behavior of the computer code. Many authors proposed another way to compare the conditional distribution of the output knowing some of the inputs to the distribution of the output. For instance, a sensitivity index based on the whole distribution of the output thanks to the Cramér-von-Mises distance is defined by Gamboa et al. in 2018. The authors showed that the Pick-Freeze estimation procedure can be used providing an asymptotically Gaussian estimator of the index. The goal of this work is twofold. We want to work with codes valued in metric spaces and to build sensitivity indices in such a framework no longer based on the Hoeffding decomposition but rather on the paper of Gamboa et al. in 2018. We provide asymptotically Gaussian estimators based on U-statistics. In addition, we prove their asymptotic normality straightforwardly.
15:00
Stochastic computer codes and sensitivity analysis
Thierry Klein | Ecole Nationale de l'Aviation Civile / Université de Toulouse | France
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Authors:
Thierry Klein | Ecole Nationale de l'Aviation Civile / Université de Toulouse | France
Fabrice Gamboa | Université de Toulouse | France
Agnès Lagnoux | Université de Toulouse | France
Leonardo Moreno | Universidad de la República | Uruguay
In practice it may occur that the output of the computer code is a probability distribution function. In this framework,
the output space can be considered as a space of probability distribution functions, that we choose to endow with the Wasserstein type distance. Then we use the sensitivity indices dedicated to general metric spaces. In some other applications, we deal with stochastic codes in the sense that two evaluations of the code for the same input x lead to different outputs. The practitioner is interested in the distribution f_x of the output for a given x. This type of codes can be traduced in terms of a deterministic code by considering an extra input which is not chosen by the practitioner but which is a latent variable generated randomly by the computer code at each evaluation.
15:30
Global sensitivity analysis for models described by stochastic differential equations
Pierre Etoré | Université Grenoble Alpes | France
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Authors:
Pierre Etoré | Université Grenoble Alpes | France
Clémentine Prieur | Université Grenoble Alpes | France
Khoi Pham Dang | Université Grenoble Alpes | France
Long Li | Université de Rennes | France
Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest. One of the statistical tools used to quantify the influence of each input variable on the quantity of interest are the Sobol' sensitivity indices. In this paper, we consider stochastic models described by stochastic differential equations (SDE). We focus the study on mean quantities, defined as the expectation with respect to the Wiener measure of a quantity of interest related to the solution of the SDE itself. Our approach is based on a Feynman-Kac representation of the quantity of interest, from which we get a parametrized partial differential equation (PDE) representation of our initial problem. We then handle the uncertainty on the parametrized PDE using polynomial chaos expansion and a stochastic Galerkin projection.