GDR Mascot-Num organizes a day of discussion on two research themes: **Optimization under uncertainty** compte-rendu and **Stochastic spectral methods** compte-rendu.

May 11th, 2012, Amphithéâtre Hermite, Institut Henri Poincaré, Paris.

Registration is free but appreciated.

**Organizers:** Fabrice Gamboa (U. Toulouse III), Rodolphe Le Riche (Ecole des Mines de St Etienne) and Bruno Sudret (Ecole Nationale des Ponts et Chaussées).

Most optimization problems contain parameters or models which are inherently uncertain. Therefore, optimization accounting for uncertainties has become an important issue from both a theoretical and a practical point of view. Robust optimization, chance constraints, reliability based design and optimization are emerging research directions. Although the practical interest in being able to account for uncertainties in optimization is clear (e.g., optimal yet safe structural design, model identification under perturbed measures, optimal decision making accounting for risk), many mathematical and computational challenges remain to be addressed (e.g., quantile estimation and optimization, uncertainty modelling in optimization, avoiding the curse of the double loop – uncertainty propagation and optimization –).

This GDR working day proposes to review some recent developments about optimization under uncertainty. Students and researchers are invited to share their views about current challenges and future research directions.

**Program**

**Morning talks: (10am to 11.20am) **

**10am-10.40am Michel de Lara. CERMICS, Ecole des Ponts ParisTech, Université Paris-Est**

*Viabilité stochastique pour la gestion durable des ressources naturelles*Slides

*Résumé*

Nous présenterons des applications de la théorie du contrôle et de la viabilité en temps discret à des problèmes de gestion durable des ressources naturelles. Lorsque des incertitudes affectent le système, nous montrerons comment adapter le cadre déterministe en présentant la viabilité stochastique. Il s'agit d'exhiber des stratégies de gestion qui maximisent la probabilité de satisfaire les objectifs (seuils à ne pas dépasser). Des applications à la gestion des pêches et de l'énegie seront présentés.

**10.40am-11am Vincent Dubourg.**

*Reliability-based design optimization using adaptive kriging surrogates.*Slides

*Abstract*

Design under uncertainty regroups a large number of problem formulations. The talk will focus on reliability-based design optimization for which the optimal design is the one that minimizes a deterministic objective function under multiple probabilistic constraints with respect to feared failure scenarii. The resolution of this problem requires a large number of computer simulations in order to predict failure for a given set of outcomes of the uncertain parameters, and evaluate the probabilistic constraints. Hence, it becomes intractable as soon as a single computer simulation is expensive to evaluate. The talk will attempt to provide a concise overview of the numerical strategy that has been developped in order to reduce the overall computational time in such cases. It is based on the use of adaptive kriging surrogates for the expensive-to-evaluate computer simulations coupled with reduced variance Monte Carlo simulation techniques.

**11am-11.20am Vincent Baudoui. ONERA Toulouse.**

*Optimisation robuste par modèles de substitution pour un problème de combustion.*Slides

*Résumé*

Cet exposé traite des problèmes d'optimisation sous incertitude de fonctions coûteuses que l'on peut rencontrer lors de la conception de systèmes aéronautiques où certains paramètres ne sont pas connus avec précision. Nous nous intéressons ici à la minimisation des émissions polluantes d'une chambre de combustion dont les injecteurs peuvent s'obstruer de manière aléatoire. Pour cela, nous développons une approche basée sur des modèles de substitution qui permettent de faciliter le calcul de la robustesse des solutions par rapport aux incertitudes du problème. L’erreur de ces modèles est maîtrisée grâce à une stratégie originale d'enrichissement de plan d'expériences nommée PareBRO (« Pareto Band Robust Optimization »).

**Afternoon discussion: 2.15pm to 5pm**

*Moderator: Rodolphe Le Riche-Secretary Fabrice Gamboa*

**Introductive speedy overview (15 minutes each):**

- Introduction slides
- Victor Picheny (CERFACS Toulouse): Sequential approaches to reliability estimation and optimization based on kriging Slides
- Younes AOUES (INSA Rouen): Decoupled approach for time-variant reliability-based design optimization

*Abstract*

The reliability-based design optimization (RBDO) is a powerful tool for robust and cost-effective design. The RBDO methodology seeks for the best compromise between cost and safety assurance, by considering system uncertainties. In aggressive environment, the structural performance can be significantly reduced due to different deteriorations (corrosion, fatigue, cracking…). The design optimization should take into account the time-dependent of the probability of failure. To ensure appropriate safety level during the whole structure lifetime, the RBDO is carried out on the basis of the time-variant reliability analysis. The time-variant reliability-based design optimization (TV-RBDO) approach aims to find a balanced design by reducing the expected total cost considering the whole lifetime of the structure. The total cost is defined in terms of the initial cost (i.e. including design, manufacturing, transport and construction costs), and the failure cost. A new methodology for TV-RBDO is proposed by transforming the initial problem to a sequence of equivalent deterministic design optimization sub-problems. The proposed approach is applied to find the optimal design of corroded beam, which the results show the validity and the efficiency of the proposed approach.

**Organizer: **Anthony Nouy (Ecole Centrale de Nantes).

In the last two decades, a growing attention has been given to functional approaches for uncertainty quantification, where uncertain quantities are seen as functionals of random parameters characterizing the uncertainties. This functional view, combined with approximation theory and numerical analysis, has led to the development of a family of numerical methods, the so-called stochastics spectral methods, for the identification of uncertainties and their propagation through numerical models.

In the morning session, presentations will recall the bases of these methods and introduce some recent advances in their use for probabilistic inference and for the propagation of uncertainties through numerical models.

In the afternoon, the discussion will emphasize on recent advances and remaining challeging issues for addressing complex problems encountered in science and engineering. In particular, these issues are related to the handling of very high dimensional problems, the robust control of approximation errors and the development of efficient numerical methods for computational efficiency.

**Preliminary program**

**Morning talks: (11.40am to 1.00pm) **

**Olivier Le Maître, CNRS, LIMSI.**

*Stochastic Spectral Methods for Parametric Uncertainty Propagation*
Slides

*Abstract*

In this talk, I will review the use of stochactic spectral methods for the propagation of parametric uncertainty through numerical models. After introducing the spectral polynomial chaos (PC) expansions of random quantities, the presentation will discuss different numerical strategies for the determination of the expansion coefficients of the uncertain model output. These strategies include non-intrusive methods and the stochastic Galerkin projection method. For the latter, I will discuss the structure of the resulting deterministic problem to be solved and the treatment of model non-linearities. I will the introduce stochastic multi-resolution framework and adaptive strategies for piecewise polynomial approximations which are needed in presence of complex dependences of the model output with respect to its uncertain parameters. Numerical examples will be shown to illustrate the various methods.

**Christophe Desceliers, Université Paris-Est Marne-la-Vallée, MSME.**

*Maximum Likelihood Estimation of Stochastic Chaos Representations from Experimental Data* Slides

*Abstract*

This talk deals with the identification of probabilistic models of the random coefficients in stochastic boundary value problems (SBVP). The data used in the identification correspond to experimental measurements of the displacement field of some specimens submitted to specified external forces. Starting with a particular mathematical model for the mechanical behavior of the specimen, the unknown field to be identified is projected on an adapted functional basis. For each set of measurements of the displacement, an inverse problem is formulated to calculate the corresponding optimal realization of the coefficients of the unknown random field on the adapted basis. Realizations of these coefficients are then used, in conjunction with the maximum likelihood principle, to set-up and solve an optimization problem for the estimation of the coefficients in a polynomial chaos representation of the parameters of the SBVP. Since the Monte Carlo numerical method is used to calculate the likelihood, then the efficiency of such identification relies on the effective orthogonality of the computed realizations of the polynomial chaos. Thus, a new method is used such that an effective orthogonality of the realizations of the polynomial chaos is reached.

**Mathilde Chevreuil, GeM, Université de Nantes.**

*Tensor approximation methods based on regression for parametric uncertainty propagation*
Slides

*Abstract*

In this talk I will present a tensor product approximation method based on regression techniques in order to deal with high dimensional uncertainty propagation problems. The underlying assumption is that the model output functional can be well represented in a low dimensional basis composed of rank-one functions. The proposed method consists in constructing this basis using a greedy algorithm and sparse regularization techniques. The latter are used in order to retain only the most significant basis functions, which results in an improvement of robustness of the regression-based tensor approximation method when dealing with a limited number of samples. As a result, the proposed technique allows to approximate the response of models with a large number of random inputs even with a limited number of model evaluations.

**Afternoon discussion: 2.15pm to 5pm**

*Moderator: Anthony Nouy*

*Introductive talks (15 minutes each):*

- Gael Poette (CEA)
- Virginie Ehrlacher (CERMICS): Greedy algorithms for high-dimensional non-symmetric problems Slides