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

[2410.11467] On $L^\infty$ stability for wave propagation and for linear inverse problems Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Analysis of PDEs arXiv:2410.11467 (math) [Submitted on 15 Oct 2024 ( v1 ), last revised 10 Apr 2026 (this version, v2)] Title: On $L^\infty$ stability for wave propagation and for linear inverse problems Authors: Rima Alaifari , Giovanni S. Alberti , Tandri Gauksson View a PDF of the paper titled On $L^\infty$ stability for wave propagation and for linear inverse problems, by Rima Alaifari and 1 other authors View PDF HTML (experimental) 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. Comments: 29 pages, 8 figures Subjects: Analysis of PDEs (math.AP) ; Numerical Analysis (math.NA) MSC classes: 35L05, 47A52, 65J20 Cite as: arXiv:2410.11467 [math.AP] (or arXiv:2410.11467v2 [math.AP] for this version) https://doi.org/10.48550/arXiv.2410.11467 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Journal of Hyperbolic Differential Equations, Vol. 23, No. 01, pp. 3-33 (2026) Related DOI : https://doi.org/10.1142/S0219891626400011 Focus to learn more DOI(s) linking to related resources Submission history From: Giovanni S. Alberti [ view email ] [v1] Tue, 15 Oct 2024 10:15:16 UTC (3,247 KB) [v2] Fri, 10 Apr 2026 12:58:34 UTC (1,642 KB) Full-text links: Access Paper: View a PDF of the paper titled On $L^\infty$ stability for wave propagation and for linear inverse problems, by Rima Alaifari and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2024-10 Change to browse by: cs cs.NA math math.NA References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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