{"id":3992,"date":"2013-10-21T23:03:24","date_gmt":"2013-10-22T03:03:24","guid":{"rendered":"http:\/\/mseas.mit.edu\/?p=3992"},"modified":"2016-08-22T13:11:45","modified_gmt":"2016-08-22T17:11:45","slug":"uncertainty-quantification-and-prediction-for-non-autonomous-linear-and-nonlinear-systems","status":"publish","type":"post","link":"https:\/\/mseas.mit.edu\/?p=3992","title":{"rendered":"Uncertainty Quantification and Prediction for Non-autonomous Linear and Nonlinear Systems"},"content":{"rendered":"p> The science of uncertainty quantification has gained a lot of attention over recent years.\r\nThis is because models of real processes always contain some elements of uncertainty, and\r\nalso because real systems can be better described using stochastic components. Stochastic\r\nmodels can therefore be utilized to provide a most informative prediction of possible future\r\nstates of the system. In light of the multiple scales, nonlinearities and uncertainties in\r\nocean dynamics, stochastic models can be most useful to describe ocean systems.\r\n<p> Uncertainty quantification schemes developed in recent years include order reduction\r\nmethods (e.g. proper orthogonal decomposition (POD)), error subspace statistical estimation\r\n(ESSE), polynomial chaos (PC) schemes and dynamically orthogonal (DO) field\r\nequations. In this thesis, we focus our attention on DO and various PC schemes for quantifying\r\nand predicting uncertainty in systems with external stochastic forcing. We develop\r\nand implement these schemes in a generic stochastic solver for a class of non-autonomous\r\nlinear and nonlinear dynamical systems. This class of systems encapsulates most systems\r\nencountered in classic nonlinear dynamics and ocean modeling, including flows modeled\r\nby Navier-Stokes equations. We first study systems with uncertainty in input parameters\r\n(e.g. stochastic decay models and Kraichnan-Orszag system) and then with external\r\nstochastic forcing (autonomous and non-autonomous self-engineered nonlinear systems).\r\nFor time-integration of system dynamics, stochastic numerical schemes of varied order\r\nare employed and compared. Using our generic stochastic solver, the Monte Carlo, DO\r\nand polynomial chaos schemes are intercompared in terms of accuracy of solution and\r\ncomputational cost.<\/p>\r\n<p> To allow accurate time-integration of uncertainty due to external stochastic forcing, we\r\nalso derive two novel PC schemes, namely, the reduced space KLgPC scheme and the modified\r\nTDgPC (MTDgPC) scheme. We utilize a set of numerical examples to show that the\r\ntwo new PC schemes and the DO scheme can integrate both additive and multiplicative\r\nstochastic forcing over significant time intervals. For the final example, we consider shallow\r\nwater ocean surface waves and the modeling of these waves by deterministic dynamics\r\nand stochastic forcing components. Specifically, we time-integrate the Korteweg-de Vries\r\n(KdV) equation with external stochastic forcing, comparing the performance of the DO\r\nand Monte Carlo schemes. We find that the DO scheme is computationally efficient to integrate uncertainty in such systems with external stochastic forcing. <\/p>","protected":false},"excerpt":{"rendered":"<p>p> The science of uncertainty quantification has gained a lot of attention over recent years. This is because models of real processes always contain some elements of uncertainty, and also because real systems can be better described using stochastic components. Stochastic models can therefore be utilized to provide a most informative prediction of possible future [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[42,5,44],"tags":[],"class_list":["post-3992","post","type-post","status-publish","format-standard","hentry","category-meche-theses","category-publications","category-masters-theses"],"_links":{"self":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/3992","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\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3992"}],"version-history":[{"count":1,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/3992\/revisions"}],"predecessor-version":[{"id":3993,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/3992\/revisions\/3993"}],"wp:attachment":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3992"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3992"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3992"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}