{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:34:29Z","timestamp":1787330069208,"version":"build-2736575974"},"reference-count":41,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["F4FGA03243G001"],"award-info":[{"award-number":["F4FGA03243G001"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2202668"],"award-info":[{"award-number":["2202668"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2024,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In this work, we introduce a new definition of observability for dynamical systems, formulated on the principles of dynamic optimization. This definition gives rise to the concept of an effective region, specifically designed for partial differential equations (PDEs). The usefulness of these concepts is demonstrated through examples of state estimation using observational information for PDEs in a limited area. The findings empower a more efficient analysis of PDE observability. By confining computations to an effective region significantly smaller than the overall region in which the PDE is defined, we demonstrate a substantial reduction in computational demand of evaluating observability. As an application of observability and effective region, we propose a learning-based surrogate data assimilation (DA) model for efficient state estimation in a limited area. Our model employs a feedforward neural network for online computation, eliminating the need for integrating high-dimensional limited-area models. This approach offers significant computational advantages over traditional DA algorithms. Furthermore, our method avoids the requirement of lateral boundary conditions for the limited-area model in both online and offline computations.<\/jats:p>","DOI":"10.1137\/23m1586690","type":"journal-article","created":{"date-parts":[[2024,5,9]],"date-time":"2024-05-09T08:25:42Z","timestamp":1715243142000},"page":"C249-C271","source":"Crossref","is-referenced-by-count":2,"title":["Observability and Effective Region of Partial Differential Equations with Application to Data Assimilation"],"prefix":"10.1137","volume":"46","author":[{"given":"Wei","family":"Kang","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943 USA, and Department of Applied Mathematics, University of California, Santa Cruz, CA 95064 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Liang","family":"Xu","sequence":"additional","affiliation":[{"name":"Marine Meteorology Division, Naval Research Laboratory, Monterey, CA 93943 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hong","family":"Zhou","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,5,9]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611976083"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1002\/qj.339"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1002\/qj.340"},{"key":"ref4","first-page":"1","volume":"120","author":"Bayen A.","year":"2020","journal-title":"Proc. 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