WAN Approximation of PDEs: Best Approximation, Stabilization and Boundary Conditions

Overview

Abstract

We present a theoretical analysis of the Weak Adversarial Networks (WAN) method, recently proposed in [1, 2], as a method for approximating the solution of partial differential equations in high dimensions and tested in the framework of inverse problems. In a very general abstract framework, we will address issues related with the existence and stability of the discrete solution, as well as approximation bounds, in the spirit of Cea’s Lemma. We will also propose two new WAN-based abstract formulations that, by avoiding the need for direct normalization, result in more favorable versions of the optimization problem to be solved for obtaining the approximate solution. Furthermore, we analyze the method’s effectiveness for the solution of Dirichlet boundary problems with implicit representation of the geometry. This is a joint work with E. Burman and C. He.

Brief Biography

Silvia Bertoluzza is the Research Director at the Institute for Numerical Analysis and Information Technologies Enrico Magenes of the Italian National Research Council, in Pavia. Her research activity is mainly focused on the theoretical analysis of numerical methods for the solution of partial differential equations. In this framework, she studies several aspects of different approaches, such as domain decomposition and fictitious domain methods, stabilization techniques, wavelet methods, polytopal methods.  She has been Network coordinator for the EC TMR Network "Wavelets in numerical simulation" and of the ECIHP Network "Breaking Complexity", and unit responsible for the ERC Advanced Grant "New CHallenges for (adaptive) PDE solvers: the interplay of ANalysis and GEometry" (PI: Annalisa Buffa), active from October 2016 to September 2022.