{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:22:19Z","timestamp":1787322139269,"version":"build-2736575974"},"reference-count":41,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/501100004147","name":"Tsinghua University","doi-asserted-by":"publisher","award":["042202008"],"award-info":[{"award-number":["042202008"]}],"id":[{"id":"10.13039\/501100004147","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["42450242"],"award-info":[{"award-number":["42450242"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12201631"],"award-info":[{"award-number":["12201631"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2025,10,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>The direct method and the Yau\u2013Yau algorithm are two crucial filtering methods in nonlinear filtering problems.\u00a0Recently, a series of research works [X.\u00a0Luo and S.-T.\u00a0Yau, IEEE Trans.\u00a0Automat.\u00a0Control, 58 (2013), pp.\u00a02563\u20132578; X.\u00a0Luo and S.-T.\u00a0Yau, IEEE Trans.\u00a0Automat.\u00a0Control, 58 (2013), pp.\u00a02495\u20132507; Z.\u00a0Wang, X.\u00a0Luo, S.\u00a0S.-T.\u00a0Yau, and Z.\u00a0Zhang, IEEE Trans.\u00a0Automat.\u00a0Control, 65 (2019), pp.\u00a01613\u20131624; W.\u00a0Dong, X.\u00a0Luo, and S.\u00a0S.-T.\u00a0Yau, IEEE Trans.\u00a0Automat.\u00a0Control, 66 (2021), pp.\u00a01559\u20131572; S.\u00a0Li, Z.\u00a0Wang, S.\u00a0S.-T.\u00a0Yau, and Z.\u00a0Zhang, IEEE Trans.\u00a0Automat.\u00a0Control, 68 (2023), pp.\u00a04405\u20134412; X.\u00a0Chen, J.\u00a0Shi, and S.\u00a0S.-T.\u00a0Yau, IEEE Trans.\u00a0Automat.\u00a0Control, 64 (2019), pp.\u00a01648\u20131654; J.\u00a0Shi, X.\u00a0Chen, W.\u00a0Dong, and S.\u00a0S.\u00a0T.\u00a0Yau, IEEE Control Systems Lett., 1 (2017), pp.\u00a0233\u2013237] in these two methods have gained great success in nonlinear numerical experiments.\u00a0Therefore, a key issue is whether a general framework of convergence analyses can be developed for these numerical algorithms based on partial differential equations. The contributions of this work consist of two parts. The first one is that we set up a convergence analysis framework for the direct method and the Yau\u2013Yau algorithm, which can show the deep connections between these two methods. They are different formulations of solving the robust DMZ equation by reducing it to solve the Yau\u2013Yau PDE [B.\u00a0Balaji, PMC Phys.\u00a0A, 3 (2009), pp.\u00a01\u201328]. The second one is that we prove the explicit convergence rates of the Yau\u2013Yau algorithm with the spectral method and the direct method with Gaussian approximation, in terms of the numbers of spectral basis and time steps.<\/jats:p>","DOI":"10.1137\/24m166704x","type":"journal-article","created":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T07:03:11Z","timestamp":1757660591000},"page":"3356-3377","source":"Crossref","is-referenced-by-count":0,"title":["Explicit Convergence Analyses of PDE-Based Filtering Algorithms"],"prefix":"10.1137","volume":"63","author":[{"given":"Jiayi","family":"Kang","sequence":"first","affiliation":[{"name":"Hetao Institute of Mathematics and Interdisciplinary Sciences (HIMIS), Shenzhen 518000, Guangdong, P. R. China, Beijing Institute of Mathematical Sciences and Applications (BIMSA), Beijing 101408, P. R. China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiuqiong","family":"Chen","sequence":"additional","affiliation":[{"name":"Corresponding author.\u00a0School of Mathematics, Renmin University of China, Beijing, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stephen S.-T.","family":"Yau","sequence":"additional","affiliation":[{"name":"Corresponding author.\u00a0Hetao Institute of Mathematics and Interdisciplinary Sciences (HIMIS), Shenzhen 518000, Guangdong, P. R. China, Beijing Institute of Mathematical Sciences and Applications (BIMSA), Beijing 101408, P. R. China, Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,9,12]]},"reference":[{"key":"ref1","volume-title":"A Primer on Spectral Theory","author":"Aupetit B.","year":"2012"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1186\/1754-0410-3-2"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1080\/17442508.2010.523466"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.5802\/afst.1257"},{"key":"ref5","first-page":"630","volume":"53","author":"Chen X.","year":"2017","journal-title":"IEEE Trans. Automat. 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