3.1.1 Surrogate modeling for UQ

Challenges. Surrogate models are crucial to enable the solution of both forward and backward UQ problems. Several alternative approaches, such as Polynomial Chaos, Gaussian Processes, and tensor format approximation, have been proposed and developed over the last decades. These approaches have been successfully applied to many different domains. Still, surrogates models for UQ management are facing many remaining limitations that require significant research works to handle large scale simulation-based studies and account for complex dependencies. These limitations concern multiple aspects, including the complexity related to the dimensionality of the input parameter, the definition of suitable basis representations, the complexity of the surrogate construction, and the control of the surrogate error.

Proposed actions. Platon will pursue long time efforts in the continuity of previous developments, such as the improvement of advanced sparse grid methods, sparsity promoting strategies and low-rank methods. Besides these generic developments, a first research axis will focus on the construction of surrogates for multi-physics problems (fluids, structures, chemistry,...) simulated by a system of coupled solvers. Classical surrogate methods consider the system of solvers as a single entity, and their construction requires the complete simulation with a high cost as a result. In contrast, we are proposing a divide to simplify strategy, using a surrogate of each constitutive solver, which reduces the input dimensionality of the local models, enables parallel construction and more flexible control of the computational effort. We will have to derive suitable error estimates of the contributions of the individual solver and procedures to decide the new computer experiments to reduce the error optimally. A second research axis on surrogate models will concern complexity reduction using transformation methods. Transformations can act on the input or output spaces of the model. In the first case, dimensionality reduction is achieved by finding low dimensional subspaces of the input space that convey most of the output variability. Platon will extend these methodologies to incorporate non-linear subspaces and alternative importance measures, in particular, to account for the surrogate's final usage (goal-oriented reduction). For the reduction of the output, we will consider generalizations of the preconditioning approach, which transforms the model output to a form admitting a much simpler surrogate and implicit enforcement of physical constraints. Here, the main challenges will be the automatic selection of the transformation among a dictionary and the design of computer experiments in this context (see below).

3.1.2 Uncertainty model, information theory and inference

Challenges. Uncertainty management in simulation can be considered in its infancy, and the control of the whole process, from the definition of the uncertainty model to the design of new simulations or experiments for uncertainty reduction, is still facing multiple challenges. Most past works on UQ have focused on forward-propagation and inverse problems when, in contrast, input uncertainty models and uncertainty reduction strategies, in general, have received much less attention.

Proposed actions. The uncertainty model directly affects the conclusion of UQ analyses (e.g., sensitivity analyses, estimation of failure probabilities, rare events). Therefore, it is crucial to propose uncertainty models that consistently and objectively integrate all available information and expert knowledge(s). Platon will explore the application of maximum entropy principle, likelihood maximization and moment matching methods, for the construction of uncertainty models in engineering problems.

For the inverse problem, the Team will continue its efforts in Bayesian inference toward better treatment of the model error in the calibration procedure.

Concerning uncertainty reduction, a central question is the prediction of the improvement toward the specific objective brought by a new simulation (computer experiment). Platon will investigate different strategies of design of experiment (DoE) based on measures of the improvement, such as entropy reduction, besides the classical reduction of variance.

The DoE in inference consists in proposing new physical experiments to reduce the posterior uncertainty optimally. Optimizing information gain leads to expensive numerical procedures, and suitable model error and noise models are critical to ensure the robustness of these optimal DoE procedures when applied to real-life data. Platon will work on approximation and reduction methods for optimal DoE to enable applications in large-scale engineering problems; the extension of the optimization to uncertainty reduction in general model-prediction, not just the model parameter uncertainty.

3.1.3 Multi-fidelity, Multi-level and optimization under uncertainty

Challenges. Multi-fidelity and Multi-level (MF&L) methods have been proposed to reduce the cost of surrogate model construction or statistics estimations, by relying on simulators of different complexity (in the modeled physics, discretization, or both). Although these methods have proved to be effective, particularly in the context of expensive simulations, existing algorithms must be adapted to other tasks. MF&L strategies are also missing in Robust Optimization (RO) and Reliability-Based Optimization (RBO), where one has to evaluate the objective accurately, typically some statistics of the model output (moments, quantiles, ...).

Proposed action. The Team Platon will explore MF&L approaches and the design of computer experiments to obtain the best estimation at the lowest cost (or for a prescribed computational budget) for nontrivial goal, specifically optimization and reliability problems where the accuracy needed is not uniform, possibly unknown a priori and to be estimated as the construction proceeds.

In RO and RBO, our research will focus on the estimation of robustness and reliability measures with tunable fidelity to adapt the convergence of the statistics to the advancement of the optimization procedure. Platon will include MF&L in the so-called bounding-box approach to track the level of error in the statistical estimates. Another research axis will focus on alternative estimation methods, e.g. the Quantile Bayesian regression, to include MF&L features.

3.1.4 HPC and UQ problems

Challenges. Both intrusive and non-intrusive UQ methods are associated to large computational costs, ranging from several to millions of times the cost of a deterministic solution depending on the problem and task considered. This situation is a significant obstacle to the deployment of UQ analysis in large scale simulations, and computational aspects have been central for a long time. However, works concerning the exploitation of High-Performance Computing platforms with massive parallelism are still scarce, besides the trivial parallelism of some sampling methods (e.g., Monte Carlo). Further, past efforts have concerned the formulation of the stochastic problem and relied on existing advanced solution methods (e.g. Domain Decomposition, linear algebra libraries, parallelism). However, few works have fully considered exploiting stochastic structures and HPC aspects to design novel computational strategies fully dedicated to UQ problems.

Proposed actions. Platon will continue to develop solvers for the resolution of multiple large systems resulting from the discretization of sampled stochastic problems. In particular, we shall focus on linear and non-linear (Newton-like) solvers, exchanging information (Krylov spaces) between successive solves to improve convergence rates of iterative methods. Besides the extension to non-linear problems, the work will focus on the implementation aspect and consider communication strategies when several instances of the random system are solved in parallel.

Platon will continue to develop specific domain decomposition methods for stochastic problems, and to propose effective stochastic preconditioners exploiting the independence of the local (uncertain) sub-problems. An additional but critical point concerns the association of adaptive mesh refinement (AMR, in space) with multiple resolution analysis (MRA, in the parameters) methods. Few works have solved UQ problems with deterministic AMR, and combining the two adaptive approaches within a parallel framework remains challenging; progress in this direction would enable efficient intrusive solvers for conservation laws.

Pollution reduction in commercial aircrafts

In EASA's 2019 annual report, in-flight icing was identified as a priority 1 issue for large aeroplanes. Therefore, to comply with certification rules, airframes and engine manufacturers must demonstrate safe operation under icing conditions which leads to significant costs before the new product is put into service. Wind tunnel tests and flight tests in icing conditions are usually required due to the low confidence of certification authorities place in simulations, due to the complexity of the icing process.

A breakthrough, leading to a reduction of time-to-market and certification costs, would be obtained by creating a consensus among certification authorities about the reliability of simulation tools for predicting in-flight ice accretion and the operation of IPS.

TRACES is a European Joint Doctorate network whose main goal is to provide high-level training in the field of in-flight icing to deliver a new generation of high achieving Doctoral Researchers (DR) in the diverse disciplines necessary for mastering the complexity of ice accretion and its mitigation in aircraft and aeroengines. In TRACES, Platon is developing novel methods to assess the calibration procedure by detecting potential inaccuracies of icing model and to perform an uncertainty quantification study propagating systematically the posterior distribution of each model's parameter.

Renewable energy sources

Platon is involved in the development of advanced numerical tools to simulate Organic Rankine Cycles (ORCs), which are of key-importance in renewable energy systems. Specifically, we are working on the inference of thermodynamic models parameters for complex molecular compounds, using experimental data of the worldwide first facility at Politecnico di Milano. Secondly, we are developing a robust optimization framework for the shape design of ORC turbines.

7.1.1 Stocholm

• Name:
  Stocholm
• Keyword:
  Uncertainty quantification
• Functional Description:
  Stocholm is a numerical library permitting to respond to potential partners swiftly and draft UQ solutions addressing new questions. It includes Polynomial Chaos construction, manipulation, and algebra, adaptive sparse grid methods for integration, interpolation, and projection in high-dimension, stochastic multi-resolution analysis tools with error estimators, advanced regression methods with regularization techniques and Gaussian process modeling, sampling methods with LHS, QMC and Markov Chain Monte Carlo algorithms, Bayesian inference framework and fast density estimation methods, Bayesian optimization algorithms with robust and multi-objective strategies, ...
• Release Contributions:
  We will continue integrating existing tools and new ones into the library StochOlm (C++), the most general one, to allow for maximum interoperability of the constitutive utilities. Having a unique library shared by the whole group also presents some interest for students and new researchers joining the Team, as they can benefit from the others’ experience.
• Contact:
  Olivier Le Maitre
• Partners:
  CNRS, Ecole Polytechnique

Project-team positioning

Many research groups are presently working on Uncertainty Quantification (UQ) and inference problems over the world and in France. For instance, the US has created and continues to expand large multi-disciplinary groups to address UQ challenges in energy and military domains through their national laboratories (SANDIA, Oak-Ridge, LLNL,...). These groups aim at providing generic methods and tools (mostly software) for the resolution of UQ problems (for example, the Dakota code from Sandia-Albuquerque) faced by other research groups from diverse application domains. Other countries are supporting smaller initiatives, including the CEA (civil and military) in France. Several large industrial groups, such as Bosch, EADS, or EdF, are also deploying UQ methodologies and tools (for example, the OpenTurns code from EADS/EDF) through dedicated RD units or services, responding to the demands of other services. These UQ activities have often emerged in well-established groups working in specific application domains (e.g., fluid dynamics, solid mechanics, electromagnetics, chemistry, material sciences, earth sciences, life sciences, ...), in response to some UQ aspects related to these particular domains. We cite G. Iaccarino (Uncertainty Quantification Lab within the Center for Turbulence Research, Stanford University), Y. Marzouk (Aerospace Computational Design Laboratory, MIT) and K. Wilcox (Institute for Computational Engineering and Sciences, University of Texas). The situation is globally similar in applied mathematics, where several groups develop advanced UQ methods within a broader research area (e.g., stochastic numerics, statistics, numerical analysis,...), sometimes with only a distant connection to engineering domains. For example, we can mention the research groups of M. Giles (Oxford), I. Bilionis (Purdue University), J. Garnier (Ecole Polytechnique), R. Abgrall (University of Zurich).

The objective of Platon is to team-up participants with the main interest in the development of UQ methodologies. While primarily targeting our current applications, our objective is to propose new applications through collaborations and progressive team development while maintaining the UQ as the project's identity. This strategy gives a somehow unique position of the Team within the national and international research landscapes. As far as computational mechanics and engineering are concerned, no group has been created with UQ management as the principal working area.

Then, the identity of Platon is to be contrasted with initiatives, including within Inria, which may have a UQ component, but within different methodological contexts and not as a central activity. For instance, some teams (e.g. SIERRA, TAO, SELECT, MODAL) develop statistical methods for data analysis, machine learning, and the treatment of large databases. Overall, the problems targeted in Platon are usually too costly, with high parametric dimension, and with few experimental data, so existing statistical methods can not be reused "as is", and require dedicated approaches.

On the application side, there are already Inria teams working on CFD applications, some even incorporating uncertainty quantification and sensitivity analysis activities. We mention here AIRSEA, which focuses on oceanic and atmospheric flows, CARDAMOM on free-surface hydraulics, and ACUMES on unsteady models in traffic flow and biology. In contrast to our project, all these efforts primarily address challenges in their respective application areas.

Scientific achievements

Our research activity features two main axes. The first is related to methodological developments, while the second is oriented to UQ problems with industrial interests.

The first contribution concerns a computer model calibration technique considering model error. Building upon our previous work 11, we formulate a new methodology called the Complete Maximum a posteriori method. Such improvement is embodied in two key properties. First and foremost, we have removed the point mass distribution hypothesis, upon which the KOH and FMP methods rely. Some examples and calculations show that such a hypothesis is never valid. The reformulated hypothesis is more general and less stringent; for this reason, we expect the method to provide a better approximation of the parameters' posterior distribution in a wider range of cases. The second improvement is the inclusion of the Hessian in the estimated parameters' posterior. In the case where the posterior has a single mode, both KOH and FMP will correctly identify such mode but tend to underestimate the weight of the tails, leading to a false certitude effect. We have also illustrated that the correction introduced by the CMP method becomes larger as the residuals grow, showing that this can lead to an undesired waterbed effect that assigns more mass to the peaks instead of the tails. When the posterior has multiple modes, the correction allows for recovering the correct weight of such modes.

The second contribution concerns the construction of globally accurate surrogates for bayesian inference 8. We present an iterative data-driven algorithm for the construction of polynomial chaos surrogates whose accuracy is localized in regions of high posterior probability. This method is applied to the identification of the source of an oil spill. Two synthetic oil spill experiments, in which the construction of prior-based surrogates is not feasible, are conducted to assess the performance of the proposed algorithm in estimating five source parameters. The algorithm successfully provided a good approximation of the posterior distribution and accelerated the estimation of the oil spill source parameters and their uncertainties by an order of 100 folds.

The third contribution 10 introduces a cost-efficient surrogate-based methodology employing an unequal allocation scheme to model a stochastic solver. This method is then applied to estimate material properties at a mesoscopic level using the PuMA Software from an uncertainty quantification perspective. An additional contribution of this work is the uncertainty propagation and sensitivity analysis of the material properties, which also yields a systematic assessment of the choice of the voxel resolution for both the fibers and the domain. Precisely, the convergence of the quantities of interest can be surveyed, thus identifying the minimal reference elementary volume.

The last contribution concerns the application of advanced UQ methodologies to a problem of interest in aerospace applications. In particular, we propose a holistic methodology 7 to determine the quantities of interest in an optimal manner for an under-expanded high-enthalpy jet, using both experimental measurements and high-fidelity flow simulations. Given the high computational cost of the high-fidelity simulations needed to describe the flow, we built an adaptive/multi-fidelity surrogate model to replace the estimation of the costly computer solver. A Bayesian inference method then allowed for characterizing an experiment carried out in the von Karman Institute's Plasmatron facility, for which no robust methodology currently exists. We show that the reservoir pressure and temperature and the nitrogen catalytic recombination coefficient of the copper probes can be accurately determined from the available measurements. Contrarily, the test conditions do not allow us to estimate the oxygen catalytic recombination coefficient. Finally, the characterized uncertainties are propagated through the numerical solver, yielding an uncertainty-based high-fidelity representation of the hypersonic flow's structure variability.

Collaborations

Since many years, we have several long-term partnerships with KAUST, von Karman Institute for Fluid-Dynamics (VKI), Politecnico di Milano and CEA.

With KAUST, we are working on new stochastic particle tracking methods to identify and track oil spills in open waters, combining satellite images and uncertainties in predicted currents. We also develop new assimilation schemes, inference methods for fractional diffusion models, and the selection and reduction of observations. There are several joint publications and exchanges of students.

With VKI, we work on UQ methods and inverse problems for atmospheric re-entry and ablation problems. In terms of production, there are several joint publications and one joint PhDs (M. Capriati).

With Politecnico di Milano, we have several activities in the Aeronautical and Energy fields. We work on the characterization of the thermodynamic model with Bayesian approaches, uncertainty on the turbulence model for RANS aerodynamic simulation, multi-fidelity approaches. We are currently involved in the EU TRACES project. We have also a strong collaboration with Giulio Gori, former member of Platon, and now Assistant Professor at Politecnico di Milano.

With CEA Saclay, we have a long-term collaboration since four years. Nicolas Leoni defended his thesis last year, and another student is doing her PhD (Sanae Idrissi).

External support

• MSCA Doctoral Network TRACES Project (2022-2026)
• Industrials contracts with CEA
• Industrial contract with 3DS

Self assessment

In addition to developing methods-oriented research, we proposed UQ methods tailored to specific applications in collaboration with other academic and industrial partners. This action has allowed us to position ourselves with high-impact papers in many application areas.

A weakness may be finding a balance between two different axes. The first axis concerns the development of high-level research from a methodological point of view, while the second one involves collaborations with industrial partners within research contracts and European projects. We think that the team's current size does not fit very well in the long term with this double effort. For this reason, the recruitment of new forces seems mandatory to keep sustaining a good balance between these two main axes of research.

Personnel

Participants: P.M. Congedo, O. Le Maître, E. Denimal Goy, M. Duvillard, H. Dornier, H. Masson, C. Papagiannis.

Project-team positioning

Research on solvers, numerical schemes, and HPC algorithms specifically dedicated to UQ problems is scarce. Indeed, advanced sampling and stochastic estimation procedures, the subject of intensive outgoing research, rely on state-of-the-art deterministic solvers to generate the solution samples. To our knowledge, there is no research group (within or outside Inria) focusing entirely on the computational aspects of UQ problems. Groups producing computational utilities for UQ (e.g., Sandia's Dakota, OpenTurns) focus on the sampling part (statistical treatment), and the efficient generation of the samples is left to the user. In recent years, few works have concerned Galerkin solvers, their preconditioning, and the adaptation of domain decomposition methods (DDM) for (usually elliptic) stochastic PDEs. We can mention some activities in Manchester (preconditioning), Munich and Lausanne (DDM), and Bath (solvers for multi-level methods). In Platon, we are trying to exploit the structure of the stochastic problems to propose new strategies for their resolution (Galerkin method) or the generation of solution samples. These strategies can consist of adapting deterministic solvers to factorize the computational effort over multiple samples or, on the contrary, the definition of entirely new solution procedures to exploit parallel methods in stochastic problems better, beyond the independent resolution of independent samples. Our objective is to produce parallel and scalable methods for large-scale stochastic problems.

It becomes more and more critical to devise solution methods tailored to the stochastic problem when the numerical complexity of the underlying deterministic problem increases. For elliptic problems, highly efficient deterministic solvers' availability has somehow limited the research on stochastic solvers. The situation is different for models based on fractional diffusion operators (in space or time), where the numerical difficulties to solve these operators have virtually prevented any work on problems with stochastic fractional and diffusion coefficients. A few years ago, KAUST (Omar Knio) and KFUPM (Kassem Mustapha) initiated a research program on fractional diffusion models. Platon is involved in this program to deal with the stochastic extensions. Several new numerical schemes and algorithms to solve deterministic fractional diffusion equations have been designed. These schemes are suitable for an extension to stochastic problems (e.g., allowing for spatially variable coefficients and achieving efficient -scalability- enabling sampling methods and inverse problems).

Scientific achievements

When numerically solving partial differential equations, for a given problem and operating condition, adaptive mesh refinement (AMR) has proven its efficiency to automatically build a discretization achieving a prescribed accuracy at low cost. However, with continuously varying operating conditions, such as those encountered in uncertainty quantification, adapting a mesh for each evaluated condition becomes complex and computationally expensive. To enable more effective error and cost control, we introduce a novel approach to mesh adaptation 23. The method consists in building a unique adapted mesh that aims at minimizing the average error for a continuous set operating conditions. In the proposed implementation, this unique mesh is built iteratively, informed by an estimate of the local average error over a reduced set of sample conditions. The effectiveness and performance of the method are demonstrated on a one-dimensional Burgers equation and a two-dimensional Euler scramjet shocked flow configurations.

Secondly, a preconditioning strategy is proposed for the iterative solve of large numbers of linear systems with variable matrix and right-hand side which arise during the computation of solution statistics of stochastic elliptic partial differential equations with random variable coefficients sampled by Monte Carlo 26. Building on the assumption that a truncated Karhunen-Loeve expansion of a known transform of the random variable coefficient is known, we introduce a compact representation of the random coefficient in the form of a Voronoi quantizer. The number of Voronoi cells is set to the prescribed number of preconditioners. Upon sampling the random variable coefficient, the linear system assembled with a given realization of the coefficient is solved with the preconditioner whose centroidal variable coefficient is the closest to the realization. We consider different ways to define and obtain the centroidal variable coefficients, and we investigate the properties of the induced preconditioning strategies in terms of average number of solver iterations for sequential simulations, and of load balancing for parallel simulations. Another approach, which is based on deterministic grids on the system of stochastic coordinates of the truncated representation of the random variable coefficient, is proposed with a stochastic dimension which increases with the number P of preconditioners. This approach allows to bypass the need for preliminary computations in order to determine the optimal stochastic dimension of the truncated approximation of the random variable coefficient for a given number of preconditioners.

The third achievement focuses on accelerating numerical simulations when complex physical models need to be systematically evaluated 13. We focus on a hypersonic planetary reentry problem whose simulation involves coupling fluid dynamics and chemical reactions. Simulating chemical reactions takes most of the computational time but, on the other hand, cannot be avoided to obtain accurate predictions. We face a trade-off between cost-efficiency and accuracy: the numerical scheme has to be sufficiently efficient to be used in an operational context but accurate enough to predict the phenomenon faithfully. To tackle this trade-off, we design a hybrid numerical scheme coupling a traditional fluid dynamic solver with a neural network approximating the chemical reactions. We rely on their power in terms of accuracy and dimension reduction when applied in a big data context and on their efficiency stemming from their matrixvector structure to achieve important acceleration factors (×10 to ×18.6). This paper aims to explain how we design such cost-effective hybrid numerical schemes in practice. Above all, we describe methodologies to ensure accuracy guarantees, allowing us to go beyond traditional surrogate modeling and to use these schemes as references.

Another contributions concerns a comprehensive review of recent advancements in modelling approaches, design strategies, and testing techniques applied to friction damping in turbomachinery 14. It critically evaluates experimental testing, design processes, and optimisation studies, along with the latest developments in numerical modelling techniques. The review begins with an overview of vibration mitigation methods and the historical development of friction dampers for bladed disk systems. Subsequent sections explore research efforts aimed at enhancing numerical and simulation modelling capabilities, encompassing contact friction models, reduced-order modelling methods, and numerical solvers suitable for real-world applications and industrial high-fidelity models. The paper also delves into available testing rigs for experimental validation and characterisation of various friction damper types, as well as the literature on uncertainty quantification in friction damping. It concludes by highlighting recent trends in novel concepts, modelling techniques, and testing technologies shaping the design of next-generation friction dampers.

The final contribution concerns a novel approach to applying data assimilation techniques for particle-based simulations using the Ensemble Kalman Filter 24. While data assimilation methods have been effectively applied to Eulerian simulations, their application in Lagrangian solution discretizations has not been properly explored. We introduce two specific methodologies to address this gap. The first methodology employs an intermediary Eulerian transformation that combines a projection with a remeshing process. The second is a purely Lagrangian scheme designed for situations where remeshing is not appropriate. The second is a purely Lagrangian scheme that is applicable when remeshing is not adapted. These methods are evaluated using a one-dimensional advection-diffusion model with periodic boundaries. Performance benchmarks for the one-dimensional scenario are conducted against a grid-based assimilation filter Subsequently, assimilation schemes are applied to a non-linear two-dimensional incompressible flow problem, solved via the Vortex-In-Cell method. The results demonstrate the feasibility of applying these methods in more complex scenarios, highlighting their effectiveness in both the one-dimensional and two-dimensional contexts.

Collaborations

With KAUST, we worked with Omar Knio on numerical schemes for fractional diffusion equation and their extension to the stochastic case.

With CEA-CESTA, we worked on scientific machine learning techniques within the thesis of Paul Novello.

External support

• Industrial contracts with CEA
• Industrial contract with 3DS
• Industrial contract with Framatome

Self assessment

Concerning the fractional diffusion models, we are already engaged in the extension of the hierarchical matrix method to solve the spatial stochastic fractional diffusion equation with a Galerkin method and to develop sparse storage strategies to reduce the complexity of the stochastic time-fractional problem. These are promising and very original researches. We (Platon) are dependent on the collaboration to access some of the numerical utilities (H-matrices).

Personnel

Participants: P.M. Congedo, O. Le Maître, E. Denimal Goy, Z. Jones, H. Nicolas.

Project-team positioning

Optimization Under Uncertainty is an important axis of research, due to both the evergrowing computational power available and the need for efficiency, reliability and cost optimality. The presence of uncertainty could make the solution of a deterministic optimization problem suboptimal or even infeasible. Since this behavior could impact strongly the design performances, both academia and industry focused their effort to developing optimization under uncertanty methodologies. Optimization under uncertainty is a broad domain including several modeling paradigms, such as for example stochastic programming, Reliability-Based Design Optimization (RBDO, that deals with probabilistic and worst-case feasibility constraints), and Robust Design Optimization (RDO, where the deterministic objectives are replaced with averaged or worst-case ones, possibly in a multi-objective context such as the classical Taguchi optimization).

Note that most of the groups active in optimization under uncertainty also have strong activities in uncertainty quantification. Thus there is an overlap with the state of the art presented in Section 8.1. The Optimization & Uncertainty Quantification Group of Sandia-Albuquerque aim at providing advanced methods for the resolution of optimization under uncertainty problems. We mention as well optimization under uncertainties activities emerged in well-established groups working in specific application domains. We cite the Aerospace Computational Design Laboratory from MIT and the Institute for Computational Engineering and Sciences from University of Texas. In France, we can mention the OQUAIDO Chair ( Optimization and QUAntification of Uncertainties), hosted by the École des Mines de Saint-Étienne from 2016 to 2021, aiming to bring together academic and industrial partners to solve problems related to uncertainty quantification, inversion and optimization.

In the context of the Optimization under Uncertainty, Platon is devoted to developing novel methods to tackle constrained multi-objective optimization, with specific attention on cost-efficient and mainly derivative-free strategies. Specifically, we look for an optimal trade-off between computational cost and accuracy in the case of problems involving complex and expensive numerical solvers. Platon is exploring also dedicated representation and the design of computer experiments to obtain the best estimation at the lowest cost (or for a prescribed computational budget) for nontrivial goal, specifically optimization and reliability problems where the accuracy needed is not uniform, possibly unknown a priori and to be estimated as the construction proceeds. More recently, we have worked also on sample average approximation methods using a risk-averse stochastic programming formulations.

Several Inria Teams have the optimization problem as core activity, such as for example BONUS, EDGE, INOCS, POLARIS, RANDOPT. Main difference is that we are not interested in working on generic optimization algorithms, as mentioned before. In our past and current works, we use standard optimization algorithms, mainly for continuous optimization. We focus our attention on dedicated representations to efficiently estimate uncertainty-based metrics within an optimization problem. The Inria teams POLARIS and INOCS work on innovative methods for stochastic optimization that are quite different from those proposed by Platon.

Scientific achievements

The first contribution concerns a computational framework for controlling the location of isolated response curves, i.e. responses that are not connected to the main solution branch and form a closed curve in parameter space 12. The methodology relies on bifurcation tracking to follow the evolution of fold bifurcations in a codimension-2 parameter space. Singularity theory is used to distinguish points of isola formation and merger from codimension-2 bifurcations and an optimization problem is formulated to delay or advance the onset or merger of isolated response curves or control their position in the state/parameter space. We illustrate the methodology on three examples: a finite element model of a cantilever beam with cubic nonlinearity at its tip, a two-degree-of-freedom oscillator with asymmetry and a two-degree-of-freedom base-excited oscillator exhibiting multiple isolas. Our results show that the location of points of isola formation and mergers can effectively be controlled through structural optimization.

The second contribution illustrates a framework for the robust optimization of the heat flux distribution for an anti-ice electrothermal ice protection system under uncertain conditions 9. The considered uncertainty regards a lack of knowledge concerning the characteristics of the cloud, i.e., the liquid water content and the median volume diameter of water droplets, and the accuracy of measuring devices, i.e., the static temperature probe. Uncertain parameters are modeled as random variables, and two sets of bounds are investigated. A forward uncertainty propagation analysis is carried out using a Monte Carlo approach exploiting a surrogate models. The optimization framework relies on a gradient-free algorithm (mesh adaptive direct search), and two different objective functions are considered, namely, the 95 quantile of the freezing mass rate and the statistical frequency of the fully evaporative operating regime. The framework is applied to a reference test case, revealing a potential to improve the heat flux distribution of the baseline design. A new heat flux distribution is proposed, and it presents a more efficient use of the thermal power, increasing flight safety even at nonnominal environmental conditions.

Collaborations

We worked with DLR (German Aerospace Center) within the EU NEXTAIR project for robust optimization, and the thesis of Zachary Jones.

Several collaborations are with Politecnico di Milano within the TRACES project.

The collaboration with KAUST has brought specific stochastic optimization problems with structures that considerably differ from our other researches (e.g. two-stages optimization, introduction of recourse, discrete optimization, ...). These problems also involve different risk mitigation approaches. Working on these problems we have learn alternative formulations and uncertainty treatments that we plan to apply to engineering applications. Similarly, we have contributed with sampling and uncertainty modelling strategies that are original for this types of problems.

External support

• MSCA Doctoral Network TRACES Project (2022-2026)
• EU NEXTAIR Project (2022-2026)
• Industrial contract with Bañulsdesign

Self assessment

Concerning the strong point, we proposed advanced state-of-the-art methods in different aspects of optimization under uncertainty, which are topics of great interest in academia. At the same time, we consolidated industrial collaborations that have allowed us to develop high-impact projects with a relevant societal impact.

Concerning a potential weakness, we think it is particularly challenging, given the size of the team, to keep proposing innovative methods and, at the same time, to contribute to projects at the industrial and European scale. New recruitments seem necessary to ensure this twofold effort.

9.1.1 3DS

Since 2022, the team benefits from a "contrat d'accompagnement" for the thesis of Meryem Benmahdi.

9.1.2 CEA

Since 2022, the team benefits from a "contrat d'accompagnement" for the thesis of Marius Duvillard, and from a "contrat d'accompagnement" for the thesis of Sanae Idrissi.

9.1.3 Framatome

Since 2023, the team benefits from a "research contrat" in the context of the Pré-Defi project with Framatome.

10.1.1 Inria associate team not involved in an IIL or an international program

10.2.1 Visits of international scientists

10.2.2 Visits to international teams

10.3.1 Horizon Europe

10.4.1 ANR LabCom

10.4.2 ANR JCJC

10.4.3 France 2030

11.1.1 Scientific events: organisation

• Enora Denimal Goy was in the organisation and scientific committee of the GDR EX-MODELI 2024 workshop.
• Pietro Marco Congedo and Enora Denimal Goy have been the organizers and chairmen of the Winter of Codes, TRACES EU Project, Inria Center of Saclay, December 9-14 2024.

11.1.2 Scientific events: selection

11.1.3 Journal

11.1.4 Invited talks

• Enora Denimal Goy has given an invited talk at the CSMA Junior 2024 workshop, Porquerolles, the 17th of May 2024.
• Pietro Marco Congedo has given an invited talk at the Opening Training School, TRACES EU Project, Politecnico di Milano, January 18 2024.
• Pietro Marco Congedo has given an invited talk at the Second Training School, TRACES EU Project, TU Braunschweig, September 25 2024.
• Pietro Marco Congedo has given an invited talk at the Advanced Computational Fluid Dynamics Methods for Hypersonic Flows - VKI LS and STO LS AVT-358, von Karman Institut for Fluid Dynamics, March 25 2024.
• Pietro Marco Congedo has given an invited talk (Plenary) at the NICFD conference, Ecole Centrale de Lyon, October 31 2024.
• Pietro Marco Congedo has given an invited talk at the "Journées Scientifiques 2024 du réseau thématique RT Terre et Energies", Nouan-le-Fuzelier, November 4-8 2024.
• Olivier Le Maitre gave an invited talk at the International workshop on Practical Data Assimilation and Uncertainty Quantification (PAUQ), Orleans, June 2024.

11.1.5 Leadership within the scientific community

• Enora Denimal Goy is a board member of the French GDR EX-MODELI.
• Enora Denimal Goy is an elected board member of the CSMA Junior (French Computational Structural Mechanics Association)

11.1.6 Research administration

• Pietro Marco Congedo is the Scientific Director of the Inria International Lab CWI-Inria.
• Olivier Le Maître is the Scientific Director of the MATritime Labcom.
• Enora Denimal Goy is the head of the Associate Team HYPATIE in collaboration with Imperial College London.
• Enora Denimal Goy is the PI of the ANR JCJC MeMoRa.
• Enora Denimal Goy was the PI of the Exploratory Action NO-BIF.

11.2.1 Teaching

• E. Denimal Goy, 2024: INSA Rennes, Graduate Level (10h/y), academic advisor of an apprentice.
• O. Le Maitre, 2024: Doctoral School SMEMAG, Graduate Level (22h/y), Course on Uncertainty Quantification Methods.
• O. Le Maître, 2024: Ecole Polytechnique, Dept. Applied Math, last year Engineering Degree, PC on Uncertainty and Risk Management (22h/y).

11.2.2 Supervision

• Pietro Marco Congedo has been the co-advisor of the thesis of Michele Capriati in collaboration with von Karman Institute for Fluid-dynamics (Belgium), defended the January 24 2024.
• Olivier Le Maître is the co-advisor of the thesis of Marius Duvillard in collaboration with CEA Cadarache.
• Olivier Le Maître is the co-advisor of the thesis of Nadège Polette in collaboration with CEA DAM.
• Pietro Marco Congedo and Olivier Le Maître are advisors of the thesis of Meryem Benmahdi, in collaboration with 3DS.
• Pietro Marco Congedo and Olivier Le Maître are advisors of the thesis of Sanae Idrissi Janati, in collaboration with CEA Saclay.
• Pietro Marco Congedo and Olivier Le Maître are advisors of the thesis of Zachary Jones.
• Pietro Marco Congedo and Olivier Le Maître are advisors of the thesis of Christos Papagiannis, in collaboration with LEGI Lab.
• Pietro Marco Congedo and Olivier Le Maître are advisors of the thesis of Hugo Dornier, in collaboration with ONERA.
• Pietro Marco Congedo and Olivier Le Maître are advisors of the thesis of Hugo Nicolas, in collaboration with Bañulsesign (Projet Matritime).
• Pietro Marco Congedo, Olivier Le Maître and Enora Denimal Goy are advisors of the thesis of Omar Kahol.
• Enora Denimal Goy is the co-advisor of the thesis of Hugo Masson, in collaboration with Ecole des Ponts.
• Enora Denimal Goy is the co-advisor of the thesis of Nina Delette, in collaboration with IFPEN.
• Enora Denimal Goy is the co-advisor of the thesis of Erwan Dehillerin, in collaboration with Ecole Centrale de Lyon.
• Enora Denimal Goy has been the co-supervisor of the postdoc Adrien Mélot, in collaboration with Imperial College London.
• Enora Denimal Goy has been the co-supervisor of the postdoc Vincent Mahé, in collaboration with I4S (Inria Rennes).
• Enora Denimal Goy has been the co-supervisor of the MSc thesis of Diana Stancic, in collaboration with Imperial College London.
• Pietro Marco Congedo, Olivier Le Maître and Enora Denimal Goy are the supervisors of Hanane Khatouri, research engineer in the context of the Pré-DEFI project with Framatome.

11.2.3 Juries

• Enora Denimal Goy served as an examiner for the PhD defense of Lisa Fournier at ISAE-Supaero in December 2024.
• Pietro Marco Congedo served as a reviewer for the PhD defense of Piero Favaretti at Università di Trieste in April 2024.
• Pietro Marco Congedo served as a reviewer for the HdR defense of Maria Adela Puscas at Sorbonne Université in June 2024.
• Pietro Marco Congedo served as an examiner for the HdR defense of Maria Giovanna Rodio at Institut Polytechnique de Paris in June 2024.
• Pietro Marco Congedo served as the President of the Jury for the PhD defense of Paul Lartaud at the Institut Polytechnique de Paris in October 2024.
• Pietro Marco Congedo served as the President of the Jury for the PhD defense of Camille Matar at Sorbonne Université in October 2024.
• Pietro Marco Congedo served as the President of the Jury for the PhD defense of Nathalie Nouaime at Sorbonne Université in December 2024.
• Oliver Le Maître served as reviewer and Chair of the Jury for the PhD defence of Jérémy Briant at INP-Toulouse in November 2024.
• Olivier Le Maitre served as president of the Jury for the Habilitation thesis of Frédéric Joly at Université Paris-Saclay in December 2024.
• Olivier Le Maître served as examiner for the PhD defence of Adrien Béguinet at Université Paris-Saclay.

11.2.4 Internal or external Inria responsabilities

• Enora Denimal Goy is an elected member of the "Comité de Centre" of the Inria Saclay research center.
• Enora Denimal Goy is a member of the Inria national committee on gender equality.
• Pietro Marco Congedo is a member of the BCEP (Bureau du comité des équipes-projets) of the Inria Center of Saclay.
• Pietro Marco Congedo participated in the Admission Jury for the DR2 Inria selection, June 6 2024.
• Pietro Marco Congedo coordinated the working group for the creation of the BOOST Inria team (Inria Center of Saclay).
• Pietro Marco Congedo is the coordinator of the Committee for the sustainable development of the Center Inria Saclay Île-de-France.

11.3.1 Specific official responsibilities in science outreach structures

• Pietro Marco Congedo is the Coordinator of "Maths/Engineering" Program of the Labex Mathématiques Hadamard (IPP and Paris-Saclay University), since 2022.
• Olivier Le Maître is a member of the Conseil du Laboratoire du CMAP (Ecole Polytechnique, IPP).
• Olivier Le Maître is the Adjunct Director of the Ecole Doctorale de Mathématiques Hadamard (EDMH).
• Olivier Le Maître is member of math committee of the PhD Track Program of IP-Paris.
• Pietro Marco Congedo is the coordinator of the "Pôle Analyse" of CMAP Lab (Ecole Polytechnique, IPP).
• Olivier Le Maître is the corresponding member of the Inria SIF center with the French Agency for Math and Industry (AMIES).

International journals

• 7 articleM.Michele Capriati, A.Alessandro Turchi, P. M.Pietro Marco Congedo and T. E.Thierry E. Magin. Holistic characterization of an under-expanded high-enthalpy jet under uncertainty.Physics of Fluids366June 2024HALDOIback to text
• 8 articleS.Samah El Mohtar, O.Olivier Le Maître, O.Omar Knio and I.Ibrahim Hoteit. Iterative data-driven construction of surrogates for an efficient Bayesian identification of oil spill source parameters from image contours.Computational Geosciences284May 2024, 681-696HALDOIback to text
• 9 articleM.Mariachiara Gallia, B.Bárbara Arizmendi Gutiérrez, G.Giulio Gori, A.Alberto Guardone and P. M.Pietro Marco Congedo. Robust Optimization of a Thermal Anti-Ice Protection System in Uncertain Cloud Conditions.Journal of Aircraft611January 2024, 1-15HALDOIback to text
• 10 articleF.Florian Girault, F.Francisco Torres Herrador, B.Bernd Helber, A.Alessandro Turchi, T.Thierry Magin and P. M.Pietro Marco Congedo. Stochastic mesoscale characterization of ablative materials for atmospheric entry.Applied Mathematical Modelling135November 2024, 745-758HALDOIback to text
• 11 articleN.Nicolas Leoni, O.Olivier Le Maitre, M.-G.Maria-Giovanna Rodio and P. M.Pietro Marco Congedo. BAYESIAN CALIBRATION WITH ADAPTIVE MODEL DISCREPANCY.International Journal for Uncertainty Quantification1412024, 19-41HALDOIback to text
• 12 articleA.Adrien Mélot, E.Enora Denimal and L.Ludovic Renson. Control of isolated response curves through optimization of codimension-1 singularities.Computers & StructuresApril 2024, 1-19HALDOIback to text
• 13 articleP.Paul Novello, G.Gaël Poëtte, D.David Lugato, S.Simon Peluchon and P. M.Pietro Marco Congedo. Accelerating hypersonic reentry simulations using deep learning-based hybridization (with guarantees).Journal of Computational Physics498February 2024, 112700HALDOIback to text
• 14 articleJ.Jie Yuan, C.Chiara Gastaldi, E.Enora Denimal and B.Benjamin Chouvion. Friction damping for turbomachinery: A comprehensive review of modelling, design strategies, and testing capabilities.Progress in Aerospace Sciences147May 2024, 101018HALDOIback to text

International peer-reviewed conferences

• 15 inproceedingsM.Meryem Benmahdi, O.Olivier Le Maitre and P. M.Pietro Marco Congedo. Artificial Neural Networks for UQ and calibration of one-dimensional arterial hemodynamics.ECCOMAS 2024 - The 9th European Congress on Computational Methods in Applied Sciences and EngineeringLisbon, PortugalJune 2024HAL
• 16 inproceedingsH.Hugo Dornier, O. P.Olivier P Le Maître, P. M.Pietro M Congedo, S.Sébastien Bourasseau, I.Itham Salah El Din and J.Julien Marty. Mean adaptive mesh refinement for efficient CFD simulations with operating conditions variability.European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2024)Lisbonne, PortugalJune 2024HAL
• 17 inproceedingsO.Olivier Le Maitre, P. M.Pietro Marco Congedo and Z.Zachary Jones. Sampling and Estimating the Set of Pareto Optimal Solutions in Stochastic Multi-Objective Optimization.ECCOMAS 2024 - The 9th European Congress on Computational Methods in Applied Sciences and EngineeringLisbon, PortugalJune 2024HAL
• 18 inproceedingsA.Adrien Mélot, E.Enora Denimal Goy and L.Ludovic Renson. Nonlinear system identification with control-based continuation of bifurcation curves.ENOC 2024 - 11th European Nonlinear Dynamics ConferenceDelft, NetherlandsJuly 2024, 1-2HAL
• 19 inproceedingsA.Adrien Mélot, E.Enora Denimal Goy and L.Ludovic Renson. On the use of bifurcation curves for system identification and model updating purposes.ECCOMAS 2024 - 9th European Congress on Computational Methods in Applied Sciences and EngineeringLisbon, PortugalJune 2024, 1-1HAL
• 20 inproceedingsA.Adrien Mélot, E.Enora Denimal Goy and L.Ludovic Renson. Structural optimization for controlling isolated response curves.ENOC 2024 - 11th European Nonlinear Dynamics ConferenceDelft, NetherlandsJuly 2024, 1-3HAL

National peer-reviewed Conferences

• 21 inproceedingsA.Adrien Mélot, E.Enora Denimal Goy and L.Ludovic Renson. Contrôle de courbes de réponses isolées par optimisation structurelle.16ème Colloque National en Calcul de Structures (CSMA 2024)Hyères, France2024HAL

Conferences without proceedings

• 22 inproceedings M.Martin Hochhausen, M.Matthieu Sacher, B.Benoît Augier, H.Hugo Nicolas, P.Patrick Bot, F.Frédéric Hauville and Y.Yves Clouet. How sailor morphology affects Olympic windfoil performance? Sports Physics 2024 Rennes, France December 2024 HAL

Reports & preprints

• 23 miscH.Hugo Dornier, O.Olivier Le Maitre, P. M.Pietro Marco Congedo, I.Itham Salah El Din, J.Julien Marty and S.Sébastien Bourasseau. Mean Mesh Adaptation for Efficient CFD Simulations with Operating Conditions Variability.November 2024HALback to text
• 24 miscM.Marius Duvillard, L.Loïc Giraldi and O.Olivier Le Maitre. Ensemble Data Assimilation for Particle-based Methods.October 2024HALback to text
• 25 miscN.Nadège Polette, O.Olivier Le Maître, P.Pierre Sochala and A.Alexandrine Gesret. Change of Measure for Bayesian Field Inversion with Hierarchical Hyperparameters Sampling.2024HALDOI
• 26 miscN.Nicolas Venkovic, P.Paul Mycek, O. L.Olivier Le Maître and L.Luc Giraud. Preconditioners based on Voronoi quantizers of random variable coefficients for stochastic elliptic partial differential equations.March 2024HALback to text

Other scientific publications

• 27 inproceedingsH.Hugo Masson, M.Michaël Peigney and E.Enora Denimal Goy. Robust topology optimization accounting for uncertain micro-structural changes.Conférence pour les 50 ans du CMAPPalaiseau, FranceSeptember 2024HAL