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In addition to providing a more comprehensive discussion of the foundations of these methods, we propose a new numerical method for the adaptive Langevin\/stochastic gradient Nos\u00e9--Hoover thermostat that achieves a dramatic improvement in numerical efficiency over the most popular stochastic gradient methods reported in the literature. We also demonstrate that the newly established method inherits a superconvergence property (fourth order convergence to the invariant measure for configurational quantities) recently demonstrated in the setting of Langevin dynamics. Our findings are verified by numerical experiments.<\/jats:p>","DOI":"10.1137\/15m102318x","type":"journal-article","created":{"date-parts":[[2016,3,1]],"date-time":"2016-03-01T11:07:52Z","timestamp":1456830472000},"page":"A712-A736","source":"Crossref","is-referenced-by-count":28,"title":["Adaptive Thermostats for Noisy Gradient Systems"],"prefix":"10.1137","volume":"38","author":[{"given":"Benedict","family":"Leimkuhler","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaocheng","family":"Shang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2016,3,1]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/130935616"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/140962644"},{"key":"atypb3","first-page":"1591","volume-title":"Proceedings of the 29th International Conference on Machine Learning","author":"Ahn S.","year":"2012"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1063\/1.2810937"},{"key":"atypb5","volume-title":"Basic Probability Theory","author":"Ash R. 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