Effective approximations for multiscale PDEs based on limited information

Abstract:

We address the development of effective approximations for multiscale partial differential equations (PDEs) in context of limited information.

Multiscale PDEs arise in various scientific areas (such as engineering, medicine, or physics). They are characterized by the presence of multiple scales of interest. Their simulation is thus challenging: accurately capturing the macroscopic behavior requires resolving fine-scale features, leading to prohibitive computational costs. To overcome this issue, it is often possible to rely on effective approximations, namely approximations that balance accuracy and computational efficiency. Depending on the available information about the system, several strategies can be used to design such approximations. Our approach is inspired by homogenization theory: it builds effective coarse coefficients. Nevertheless, it remains practical: it exploits measurements of system observables and no assumptions are made on the underlying microstructure. A key aspect of our study is that the available information is limited, both qualitatively and quantitatively: the observables are coarse, possibly noisy, and only a small number of measurements are accessible.

In this talk, we introduce our approach and one possible extension. First, an effective coefficient for a diffusion PDE is defined. The methodology consists in an optimization problem constructed from global measurements in the form of boundary integrals. The solutions to the effective PDE approximate the solutions to the original problem. Theoretical and numerical results confirm the quality of this approximation. Second, an extension is proposed to incorporate the knowledge of a reference effective coefficient. By linearizing the quantities of interest using perturbation theory, a computationally less expensive optimization problem is formulated.

Short bio:

Simon Ruget is an engineer from École des Ponts et Chaussées (ENPC). He is currently a third-year PhD student, under the supervision of Claude Le Bris and Frédéric Legoll. He is affiliated with the MATHERIALS team at Inria Paris, and at the CERMICS and Navier laboratories at ENPC. He specializes in applied mathematics, with a particular interest in partial differential equations. His PhD research focuses on the construction of effective approximations for multiscale PDEs in contexts where the available information is limited. During his internships at CEA and EDF, he contributed to projects involving physical system modeling and simulation. Previously, he also worked on big data software development as an intern at Thales Alenia Space.