短课程：Model and  Data Reduction

授课老师：Mejdi Azaiez

上课时间：2020年4月6日-10日下午5-8节；2020年4月13日-14日下午5-8节

授课方式：网上教学

Motivation

The industrial sectors around materials, structures, buildings and complex fluids, sometimes interacting, wave propagation, structural design, are both buyers and sellers for everything related to the optimization and dimensioning, and control of manufacturing processes and production systems. Nevertheless, many problems in science and engineering are still intractable, despite the progress made in modelling, numerical analysis, algorithms and computational sciences on the one hand, and in computing power, which has been steadily increasing in recent years, on the other. In this context, the reduction of data and models (ROM) is bringing about a paradigm shift, thanks to reductions in calculation times of several orders of magnitude. These models are making it possible to solve highly complex optimization and control problems that would be beyond the reach of conventional methods in the next quarter century.

Plan

Part I: Data reduction techniques using tensorisation

I.1 ) Two-parameter data

- Presentation of techniques: Singular Value Decomposition (SVD), Proper Orthogonal Decomposition (POD), Proper Generalized Decomposition (PGD), Dynamic Mode Decomposition (DMD)

I.2 ) Multi-parameter data

- Presentation of the techniques: High Singular Value Decomposition (HSVD), High Proper Orthogonal Decomposition (HPOD), Recursive Proper Orthogonal Decomposition (RPOD) and PGD.

I.3 ) Interpolation of data

- Presentation of techniques: Empirical Interpolation Methods (EIM) and its discrete version (DEIM), Non-Uniform Rational B-Splines (NURBS) and Interpolation on Grassman varieties

Part II: Model Reduction

II.1 ) Galerkin POD

- Presentation of the GPOD method for the heat equation case

- Presentation of the GPOD method for the elasticity case

- Presentation of the GPOD method for the case of incompressible Navier-Stokes

II.2 ) Reduced Basis Method

- Presentation of the RB method for the heat equation case

- Presentation of the RB method for the diffusion advection case

II.3 ) Galerkin PGD

- Presentation of the PGD method for the Laplacian case

- Presentation of the PGD method for the diffusion advection case