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

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 $L^{\mathrm{\infty }}$ norm, which is more relevant to detect localized phenomena. The linear wave equation is not stable in $L^{\mathrm{\infty }}$, and we design an alternative solution method based on the regularization of Fourier multipliers, which is stable in $L^{\mathrm{\infty }}$. 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 $L^{\mathrm{\infty }}$. We also discuss the connection with the stability of deep neural networks modeled by hyperbolic PDEs.