A Geometric Approach to Dynamical Model Order Reduction
Any model order reduced dynamical system that evolves a modal decomposition to approximate the discretized solution of a stochastic PDE can be related to a vector field tangent to the manifold of fixed rank matrices. The dynamically orthogonal (DO) approximation is the canonical reduced-order model...
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| Format: | Article |
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Society for Industrial & Applied Mathematics (SIAM)
2019
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| Online Access: | http://hdl.handle.net/1721.1/120002 https://orcid.org/0000-0003-0122-5220 https://orcid.org/0000-0002-1869-3883 |
| _version_ | 1826188001125859328 |
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| author | Feppon, Florian Jeremy Lermusiaux, Pierre |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Feppon, Florian Jeremy Lermusiaux, Pierre |
| author_sort | Feppon, Florian Jeremy |
| collection | MIT |
| description | Any model order reduced dynamical system that evolves a modal decomposition to approximate the discretized solution of a stochastic PDE can be related to a vector field tangent to the manifold of fixed rank matrices. The dynamically orthogonal (DO) approximation is the canonical reduced-order model for which the corresponding vector field is the orthogonal projection of the original system dynamics onto the tangent spaces of this manifold. The embedded geometry of the fixed rank matrix manifold is thoroughly analyzed. The curvature of the manifold is characterized and related to the smallest singular value through the study of the Weingarten map. Differentiability results for the orthogonal projection onto embedded manifolds are reviewed and used to derive an explicit dynamical system for tracking the truncated singular value decomposition (SVD) of a time-dependent matrix. It is demonstrated that the error made by the DO approximation remains controlled under the minimal condition that the original solution stays close to the low rank manifold, which translates into an explicit dependence of this error on the gap between singular values. The DO approximation is also justified as the dynamical system that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. Riemannian matrix optimization is investigated in this extrinsic framework to provide algorithms that adaptively update the best low rank approximation of a smoothly varying matrix. The related gradient flow provides a dynamical system that converges to the truncated SVD of an input matrix for almost every initial datum. Key words. model order reduction, fixed rank matrix manifold, low rank approximation, singular value decomposition, orthogonal projection, curvature, Weingarten map, dynamically orthogonal approximation, Riemannian matrix optimization |
| first_indexed | 2024-09-23T07:53:04Z |
| format | Article |
| id | mit-1721.1/120002 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T07:53:04Z |
| publishDate | 2019 |
| publisher | Society for Industrial & Applied Mathematics (SIAM) |
| record_format | dspace |
| spelling | mit-1721.1/1200022022-09-30T00:47:57Z A Geometric Approach to Dynamical Model Order Reduction Feppon, Florian Jeremy Lermusiaux, Pierre Massachusetts Institute of Technology. Department of Mechanical Engineering Massachusetts Institute of Technology. Computation for Design and Optimization Program Feppon, Florian Jeremy Lermusiaux, Pierre Any model order reduced dynamical system that evolves a modal decomposition to approximate the discretized solution of a stochastic PDE can be related to a vector field tangent to the manifold of fixed rank matrices. The dynamically orthogonal (DO) approximation is the canonical reduced-order model for which the corresponding vector field is the orthogonal projection of the original system dynamics onto the tangent spaces of this manifold. The embedded geometry of the fixed rank matrix manifold is thoroughly analyzed. The curvature of the manifold is characterized and related to the smallest singular value through the study of the Weingarten map. Differentiability results for the orthogonal projection onto embedded manifolds are reviewed and used to derive an explicit dynamical system for tracking the truncated singular value decomposition (SVD) of a time-dependent matrix. It is demonstrated that the error made by the DO approximation remains controlled under the minimal condition that the original solution stays close to the low rank manifold, which translates into an explicit dependence of this error on the gap between singular values. The DO approximation is also justified as the dynamical system that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. Riemannian matrix optimization is investigated in this extrinsic framework to provide algorithms that adaptively update the best low rank approximation of a smoothly varying matrix. The related gradient flow provides a dynamical system that converges to the truncated SVD of an input matrix for almost every initial datum. Key words. model order reduction, fixed rank matrix manifold, low rank approximation, singular value decomposition, orthogonal projection, curvature, Weingarten map, dynamically orthogonal approximation, Riemannian matrix optimization United States. Office of Naval Research (Grant N00014-14-1-0725) United States. Office of Naval Research (Grant N00014-14-1-0476) 2019-01-11T19:13:59Z 2019-01-11T19:13:59Z 2018-01 2018-12-12T17:15:10Z Article http://purl.org/eprint/type/JournalArticle 0895-4798 1095-7162 http://hdl.handle.net/1721.1/120002 Feppon, Florian, and Pierre F. J. Lermusiaux. “A Geometric Approach to Dynamical Model Order Reduction.” SIAM Journal on Matrix Analysis and Applications 39, no. 1 (January 2018): 510–538. © 2018 Society for Industrial and Applied Mathematics. https://orcid.org/0000-0003-0122-5220 https://orcid.org/0000-0002-1869-3883 http://dx.doi.org/10.1137/16M1095202 SIAM Journal on Matrix Analysis and Applications Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Society for Industrial & Applied Mathematics (SIAM) SIAM |
| spellingShingle | Feppon, Florian Jeremy Lermusiaux, Pierre A Geometric Approach to Dynamical Model Order Reduction |
| title | A Geometric Approach to Dynamical Model Order Reduction |
| title_full | A Geometric Approach to Dynamical Model Order Reduction |
| title_fullStr | A Geometric Approach to Dynamical Model Order Reduction |
| title_full_unstemmed | A Geometric Approach to Dynamical Model Order Reduction |
| title_short | A Geometric Approach to Dynamical Model Order Reduction |
| title_sort | geometric approach to dynamical model order reduction |
| url | http://hdl.handle.net/1721.1/120002 https://orcid.org/0000-0003-0122-5220 https://orcid.org/0000-0002-1869-3883 |
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