Contenu

Reduced Order Models: Bridging PDE Models and Simulation Data

Scientific Context

This school aims to introduce hybrid numerical modeling methodologies that combine partial differential equation models with high-fidelity data. These methodologies focus on achieving explainability, accuracy, and robustness while maintaining speed and efficiency for performance prediction, parametric optimization, control, and data assimilation in complex industrial systems.

The preferred methodology in this school is Reduced Order Models (ROM). These parametric mathematical models are derived from partial differential equations by utilizing offline-calculated and stored solutions.

In certain applications, the solution space has low dimensionality, allowing for minimal degradation of accuracy against computational speed and model scalability. Thus, ROMs can address the computational complexity associated with the curse of dimensionality by significantly reducing it.

While ROMs have matured to a certain extent in the past decade, challenges persist. Parametric problems governed by advection fields or solutions with substantially compact support features, such as shockwaves, face difficulties in limited dimensional reduction and insufficient model generalization for out-of-sample solutions. These issues often result from a linear or affine approximation of the solution space, and the school will focus on specific techniques to address them.

Description of the Summer School

Initially, a general introduction to reduced modeling will be provided, emphasizing “projection-based” methods for elliptic or nonlinear hyperbolic problems.

Subsequently, challenges related to approximating solutions with essentially local spatial and temporal characteristics will be explored in detail. Various methodologies to overcome these challenges will be outlined, including nonlinear approximation techniques for dealing with fields where the movement of coherent structures is dominant.

Moreover, hybrid and multiscale formulations, combining reduced and full-order models, will be presented to optimally address locally complex solutions and complex geometric parameterizations.

Techniques such as Dynamic Mode Decomposition (DMD) and subspace interpolation will be discussed in this context.

Provisional Program

—> Day 1

Morning: Introduction to model reduction by Charbel Farhat. Empirical quadratures ECSW / Quadratic/nonlinear approximation.

Afternoon: Hands on by Jean-Philippe Argaud.

—> Day 2

Morning: Lecture by Angelo Iollo, Damiano Lombardi, Virginie Erlacher on “Non-linear Interpolations/Mappings.”

Afternoon: Hands on by Angelo Iollo, Damiano Lombardi, Virginie Erlacher.

—> Day 3

Morning: Lecture by Angelo Iollo, Damiano Lombardi, Virginie Erlacher on “Non-linear Interpolations/Mappings.”

Afternoon: Hands on by Angelo Iollo, Damiano Lombardi, Virginie Erlacher.

—> Day 4

Morning: Lecture by Tommaso Taddei on “Component-Based Model Reduction / Domain Decomposition.”

Afternoon: Hands on by Tommaso Taddei and Angélique Ponçot on “Application of the method to the elasticity equation.”

—> Day 5

Morning: Lecture by Michel Bergmann and Denis Sipp on “Dynamic Mode Decomposition (DMD). Subspace Interpolation.”

Afternoon: Hands on by Michel Bergmann and Denis Sipp, and Gael Poette on “Application of reduction to quasi-periodic non-stationary problems.”

Invited conference at the end of each afternoon.

Practical informations

Date

01 juillet – 05 juillet 2024

Place

EDF Lab
Paris-Saclay
Boulevard Gaspard Monge
91120 Palaiseau

Registration

If you wish to participate, please fill the registration form and send it before 12 mai 2024 at Régis Vizet.

Contacts

Summer schools secretary
Régis Vizet – CEA
tel: 01 69 26 47 45
Fax: 01 69 26 70 05

Coordinators of the numerical analysis 2024 school
Angelo Iollo
Tommaso Taddei
Michel Bergmann
Gaël Poette
Jean-Philippe Argaud
Angélique Poncot