Section: New Results

Quantifying Uncertainty

Sensitivity analysis for forecasting ocean models

Participants : Anestis Antoniadis, Eric Blayo, Gaëlle Chastaing, Céline Helbert, Alexandre Janon, François-Xavier Le Dimet, Simon Nanty, Maëlle Nodet, Clémentine Prieur, Federico Zertuche, Simon Nanty, Laurent Gilquin.

Scientific context

Forecasting ocean systems require complex models, which sometimes need to be coupled, and which make use of data assimilation. The objective of this project is, for a given output of such a system, to identify the most influential parameters, and to evaluate the effect of uncertainty in input parameters on model output. Existing stochastic tools are not well suited for high dimension problems (in particular time-dependent problems), while deterministic tools are fully applicable but only provide limited information. So the challenge is to gather expertise on one hand on numerical approximation and control of Partial Differential Equations, and on the other hand on stochastic methods for sensitivity analysis, in order to develop and design innovative stochastic solutions to study high dimension models and to propose new hybrid approaches combining the stochastic and deterministic methods.

Estimating sensitivity indices

A first task is to develop tools for estimated sensitivity indices. In variance-based sensitivity analysis, a classical tool is the method of Sobol' [100] which allows to compute Sobol' indices using Monte Carlo integration. One of the main drawbacks of this approach is that the estimation of Sobol' indices requires the use of several samples. For example, in a $d$-dimensional space, the estimation of all the first-order Sobol' indices requires $d+1$ samples. Some interesting combinatorial results have been introduced to weaken this defect, in particular by Saltelli [98] and more recently by Owen [95] but the quantities they estimate still require $O\left(d\right)$ samples. In a recent work [104] we introduce a new approach to estimate for any $k$ all the $k$-th order Sobol' indices by using only two samples based on replicated latin hypercubes. We establish theoretical properties of such a method for the first-order Sobol' indices and discuss the generalization to higher-order indices. As an illustration, we propose to apply this new approach to a marine ecosystem model of the Ligurian sea (northwestern Mediterranean) in order to study the relative importance of its several parameters. The calibration process of this kind of chemical simulators is well-known to be quite intricate, and a rigorous and robust — i.e. valid without strong regularity assumptions — sensitivity analysis, as the method of Sobol' provides, could be of great help. The computations are performed by using CIGRI, the middleware used on the grid of the Grenoble University High Performance Computing (HPC) center. We are also applying these estimates to calibrate integrated land use transport models. It is the first step in the PhD of Laurent Gilquin (started in October 2013). Laurent Gilquin is supervised by Clémentine Prieur and Elise Arnaud (EPI STEEP) and his PhD is funded by the ANR project CITIES.

We can now wonder what are the asymptotic properties of these new estimators, or also of more classical ones. In [67] , the authors deal with asymptotic properties of the estimators. In [70] , the authors establish also a multivariate central limit theorem and non asymptotic properties.

Intrusive sensitivity analysis, reduced models

Another point developed in the team for sensitivity analysis is model reduction. To be more precise regarding model reduction, the aim is to reduce the number of unknown variables (to be computed by the model), using a well chosen basis. Instead of discretizing the model over a huge grid (with millions of points), the state vector of the model is projected on the subspace spanned by this basis (of a far lesser dimension). The choice of the basis is of course crucial and implies the success or failure of the reduced model. Various model reduction methods offer various choices of basis functions. A well-known method is called “proper orthogonal decomposition" or “principal component analysis". More recent and sophisticated methods also exist and may be studied, depending on the needs raised by the theoretical study. Model reduction is a natural way to overcome difficulties due to huge computational times due to discretizations on fine grids. In [10] , the authors present a reduced basis offline/online procedure for viscous Burgers initial boundary value problem, enabling efficient approximate computation of the solutions of this equation for parametrized viscosity and initial and boundary value data. This procedure comes with a fast-evaluated rigorous error bound certifying the approximation procedure. The numerical experiments in the paper show significant computational savings, as well as efficiency of the error bound. When a metamodel is used (for example reduced basis metamodel, but also kriging, regression, $...$) for estimating sensitivity indices by Monte Carlo type estimation, a twofold error appears : a sampling error and a metamodel error. Deriving confidence intervals taking into account these two sources of uncertainties is of great interest. We obtained results particularly well fitted for reduced basis metamodels [90] . We are now considering problems related to more general models such as Shallow-Water models. In [15] , the authors provide asymptotic confidence intervals in the double limit where the sample size goes to infinity and the metamodel converges to the true model.

Let us come back to the output of interest. Is it possible to get better error certification when the output is specified. A work in this sense has been submitted, dealing with goal oriented uncertainties assessment [89] .

Sensitivity analysis with dependent inputs

An important challenge for stochastic sensitivity analysis is to develop methodologies which work for dependent inputs. For the moment, there does not exist conclusive results in that direction. Our aim is to define an analogue of Hoeffding decomposition [88] in the case where input parameters are correlated. Clémentine Prieur supervised Gaëlle Chastaing's PhD thesis on the topic (defended in September 2013) [2] . We obtained first results [81] , deriving a general functional ANOVA for dependent inputs, allowing defining new variance based sensitivity indices for correlated inputs. We then adapted various algorithms for the estimation of these new indices. These algorithms make the assumption that among the potential interactions, only few are significant. Two papers have been submitted [64] , [66] .

Céline Helbert and Clémentine Prieur supervise the PhD thesis of Simon Nanty (funded by CEA Cadarache). The subject of the thesis is the analysis of uncertainties for numerical codes with temporal and spatio-temporal input variables, with application to safety and impact calculation studies. This study implies functional dependent inputs. A first step is the modeling of these inputs.

Multy-fidelity modeling for risk analysis

Federico Zertuche's PhD concerns the modeling and prediction of a digital output from a computer code when multiple levels of fidelity of the code are available. A low-fidelity output can be obtained, for example on a coarse mesh. It is cheaper, but also much less accurate than a high-fidelity output obtained on a fine mesh. In this context, we propose new approaches to relieve some restrictive assumptions of existing methods ( [91] , [97] ) : a new estimating method of the classical cokriging model when designs are not nested and a nonparametric modeling of the relationship between low-fidelity and high-fidelity levels. The PhD takes place in the REDICE consortium and in close link with industry. The first year was also dedicated to the development of a case study in fluid mechanics with CEA in the context of the study of a nuclear reactor.

The second year of the thesis was dedicated to the development of a new sequential approach based on a course to fine wavelets algorithm.

Evaluation of a posteriori covariance errors

In the context of data assimilation, taking into account the a priori covariance error on the prediction and on the observations, the model and the observations, an analysis can be obtained followed by a prediction. This one makes sense only if an estimation of the error can be provided. The tendency is to use "ensemble methods" i.e. to realize a large number of predictions and estimate statistics on the results. This method raises two problems: the high computational cost and the weak theoretical justification. We have proposed a new method based on the fact that in the linear case the covariance is the inverse of the Hessian. The principle of our method is to add a correcting term to the Hessian in the non linear case. This work has been published in 2013 [14] . This paper has also be presented at the 6th WMO Symposium on Data Assimilation held in College Park, MD, USA in October 2013 [73] .

Second Order Information in Variational Data Assimilation

This theme is centered around sensitivity analysis with respect to the observations. The link between data and models is made only in the Optimality System. Therefore a sensitivity analysis on the observations must be carried out on the Optimality System thus using second order information. This research is done in cooperation with Victor Shutyaev (Institute of Numerical Mathematics, Moscow), Tran Thu Ha (Institute of Mechanics, Ha Noi, Vietnam). One paper is published in the Russ. J. Of Numerical Analysis [18] . Another application to identification of parameters in a hydrological model is submitted [105] .