On $L^\infty$ stability for wave propagation and for linear inverse problems

TL;DR

This paper investigates $L^ Infty$ stability for wave equations and inverse problems, proposing new regularization methods that ensure stability in the $L^ Infty$ norm, which is crucial for detecting localized phenomena.

Contribution

It introduces an $L^ Infty$-stable regularization technique for linear wave equations and inverse problems, extending stability analysis beyond traditional $L^2$ frameworks.

Findings

Proposes a Fourier multiplier regularization method stable in $L^ Infty$.

Extends $L^ Infty$ stability concepts to inverse problems involving compact operators.

Discusses implications for the stability of hyperbolic PDE-based neural networks.

Abstract

Stability is a key property of both forward models and inverse problems, and depends on the norms considered in the relevant function spaces. For instance, stability estimates for hyperbolic partial differential equations are often based on energy conservation principles, and are therefore expressed in terms of $L^2$ norms. The focus of this paper is on stability with respect to the $L^\infty$ norm, which is more relevant to detect localized phenomena. The linear wave equation is not stable in $L^\infty$, and we design an alternative solution method based on the regularization of Fourier multipliers, which is stable in $L^\infty$. Furthermore, we show how these ideas can be extended to inverse problems, and design a regularization method for the inversion of compact operators that is stable in $L^\infty$. We also discuss the connection with the stability of deep neural networks modeled by hyperbolic PDEs.