{"id":903,"date":"2012-01-03T04:35:54","date_gmt":"2012-01-03T09:35:54","guid":{"rendered":"http:\/\/mseas.net16.net\/?p=903"},"modified":"2021-07-06T13:23:14","modified_gmt":"2021-07-06T17:23:14","slug":"dynamical-criteria-for-the-evolution-of-the-stochastic-dimensionality-in-flows-with-uncertainty-2","status":"publish","type":"post","link":"https:\/\/mseas.mit.edu\/?p=903","title":{"rendered":"Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty"},"content":{"rendered":"We estimate and study the evolution of the dominant dimensionality of\r\ndynamical systems with uncertainty governed by stochastic partial differential\r\nequations, within the context of dynamically orthogonal (DO) field equations.\r\nTransient nonlinear dynamics, irregular data and non-stationary statistics are\r\ntypical in a large range of applications such as oceanic and atmospheric flow\r\nestimation. To efficiently quantify uncertainties in such systems, it is\r\nessential to vary the dimensionality of the stochastic subspace with time. An\r\nobjective here is to provide criteria to do so, working directly with the\r\noriginal equations of the dynamical system under study and its DO\r\nrepresentation. We first analyze the scaling of the computational cost of\r\nthese DO equations with the stochastic dimensionality and show that unlike\r\nmany other stochastic methods the DO equations do not suffer from the curse of\r\ndimensionality. Subsequently, we present the new adaptive criteria for the\r\nvariation of the stochastic dimensionality based on instantaneous i) stability\r\narguments and ii) Bayesian data updates. We then illustrate the capabilities\r\nof the derived criteria to resolve the transient dynamics of two 2D stochastic\r\nfluid flows, specifically a double-gyre wind-driven circulation and a\r\nlid-driven cavity flow in a basin. In these two applications, we focus on the\r\ngrowth of uncertainty due to internal instabilities in deterministic flows. We\r\nconsider a range of flow conditions described by varied Reynolds numbers and\r\nwe study and compare the evolution of the uncertainty estimates under these\r\nvaried conditions.\r\n","protected":false},"excerpt":{"rendered":"<p>We estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations, within the context of dynamically orthogonal (DO) field equations. Transient nonlinear dynamics, irregular data and non-stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation. To efficiently [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[31,182,33,34,5,62,64],"tags":[],"class_list":["post-903","post","type-post","status-publish","format-standard","hentry","category-uncertainty-quantification-and-reduced-order-modeling","category-learning-and-data-assimilation","category-uncertainty-quantification-and-predictions","category-data-assimilation","category-publications","category-papers-in-refereed-journals-data-assimilation","category-papers-in-refereed-journals-uncertainty-quantification-and-predictions"],"_links":{"self":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/903","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=903"}],"version-history":[{"count":6,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/903\/revisions"}],"predecessor-version":[{"id":1966,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/903\/revisions\/1966"}],"wp:attachment":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=903"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=903"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=903"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}