{"id":3959,"date":"2018-01-02T10:00:12","date_gmt":"2018-01-02T15:00:12","guid":{"rendered":"http:\/\/mseas.mit.edu\/?p=3959"},"modified":"2021-07-06T12:51:36","modified_gmt":"2021-07-06T16:51:36","slug":"a-geometric-approach-to-dynamical-model-order-reduction","status":"publish","type":"post","link":"https:\/\/mseas.mit.edu\/?p=3959","title":{"rendered":"A Geometric Approach to Dynamical Model\u2013Order Reduction"},"content":{"rendered":"

Any model order reduced dynamical system that evolves a modal\u00a0<\/span>decomposition to approximate the discretized solution of a stochastic PDE can be<\/span>\u00a0related to a vector field tangent to the manifold of fixed rank matrices. The\u00a0<\/span>Dynamically Orthogonal (DO) approximation is the canonical reduced order model\u00a0<\/span>for which the corresponding vector field is the orthogonal projection of the\u00a0<\/span>original system dynamics onto the tangent spaces of this manifold. The embedded\u00a0<\/span>geometry of the fixed rank matrix manifold is thoroughly analyzed.\u00a0 The curvature\u00a0<\/span>of the manifold is characterized and related to the smallest singular value\u00a0<\/span>through the study of the Weingarten map.\u00a0 Differentiability results for the\u00a0<\/span>orthogonal projection onto embedded manifolds are reviewed and used to derive an\u00a0<\/span>explicit dynamical system for tracking the truncated Singular Value Decomposition\u00a0<\/span>(SVD)\u00a0 of a time-dependent matrix. It is demonstrated that the error made by the\u00a0<\/span>DO approximation remains controlled under the minimal condition that the original\u00a0<\/span>solution stays close to the low rank manifold, which translates into an explicit\u00a0<\/span>dependence of this error on the gap between singular values.\u00a0 The DO\u00a0<\/span>approximation is also justified as the dynamical system that applies\u00a0<\/span>instantaneously the SVD truncation to optimally constrain the rank of the reduced\u00a0<\/span>solution.\u00a0 Riemannian matrix optimization is investigated in this extrinsic\u00a0<\/span>framework to provide algorithms that adaptively update the best low rank\u00a0<\/span>approximation of a smoothly varying matrix.\u00a0 The related gradient flow provides a\u00a0<\/span>dynamical system that converges to the truncated SVD of an input matrix for\u00a0<\/span>almost every initial data.<\/span><\/p>","protected":false},"excerpt":{"rendered":"

Any model order reduced dynamical system that evolves a modal\u00a0decomposition to approximate the discretized solution of a stochastic PDE can be\u00a0related to a vector field tangent to the manifold of fixed rank matrices. The\u00a0Dynamically Orthogonal (DO) approximation is the canonical reduced order model\u00a0for which the corresponding vector field is the orthogonal projection of the\u00a0original system […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[31,33,5,64,192,184],"tags":[],"class_list":["post-3959","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","category-papers-in-refereed-journals-reduced-order-modeling","category-reduced-order-modeling"],"_links":{"self":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/3959","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=3959"}],"version-history":[{"count":3,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/3959\/revisions"}],"predecessor-version":[{"id":4224,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/3959\/revisions\/4224"}],"wp:attachment":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3959"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3959"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3959"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}