{"id":898,"date":"2009-09-06T04:30:53","date_gmt":"2009-09-06T08:30:53","guid":{"rendered":"http:\/\/mseas.net16.net\/?p=898"},"modified":"2021-08-16T20:21:23","modified_gmt":"2021-08-17T00:21:23","slug":"dynamically-orthogonal-field-equations-for-continuous-stochastic-dynamical-systems","status":"publish","type":"post","link":"https:\/\/mseas.mit.edu\/?p=898","title":{"rendered":"Dynamically orthogonal field equations for continuous stochastic dynamical systems"},"content":{"rendered":"In this work we derive an exact, closed set of evolution equations for general continuous stochastic fields\r\ndescribed by a Stochastic Partial Differential Equation (SPDE). By hypothesizing a decomposition of the\r\nsolution field into a mean and stochastic dynamical component, we derive a system of field equations\r\nconsisting of a Partial Differential Equation (PDE) for the mean field, a family of PDEs for the orthonormal\r\nbasis that describe the stochastic subspace where the stochasticity `lives&#8217; as well as a system of Stochastic\r\nDifferential Equations that defines how the stochasticity evolves in the time varying stochastic subspace.\r\nThese new evolution equations are derived directly from the original SPDE, using nothing more than\r\na dynamically orthogonal condition on the representation of the solution. If additional restrictions are\r\nassumed on the form of the representation, we recover both the Proper Orthogonal Decomposition\r\nequations and the generalized Polynomial Chaos equations. We apply this novel methodology to two\r\ncases of two-dimensional viscous fluid flows described by the Navier\u0015Stokes equations and we compare\r\nour results with Monte Carlo simulations.","protected":false},"excerpt":{"rendered":"<p>In this work we derive an exact, closed set of evolution equations for general continuous stochastic fields described by a Stochastic Partial Differential Equation (SPDE). By hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component, we derive a system of field equations consisting of a Partial Differential Equation (PDE) for [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[31,33,5,64],"tags":[],"class_list":["post-898","post","type-post","status-publish","format-standard","hentry","category-uncertainty-quantification-and-reduced-order-modeling","category-uncertainty-quantification-and-predictions","category-publications","category-papers-in-refereed-journals-uncertainty-quantification-and-predictions"],"_links":{"self":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/898","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=898"}],"version-history":[{"count":2,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/898\/revisions"}],"predecessor-version":[{"id":1184,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/898\/revisions\/1184"}],"wp:attachment":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=898"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=898"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=898"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}