Research reports

Years: 2025 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

On L-infinity stability for wave propagation and for linear inverse problems

by R. Alaifari and G. Alberti and T. Gauksson

(Report number 2025-06)

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.

Keywords: Stability, localized artifacts, well-posedness, hyperbolic PDEs

@Techreport{AAG25_1127,
  author = {R. Alaifari and G. Alberti and T. Gauksson},
  title = {On L-infinity stability for wave propagation and for linear inverse problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-06},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-06.pdf },
  year = {2025}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).