(Randomized) Localized Model Order Reduction Kathrin Smetana - PowerPoint PPT Presentation

(Randomized) Localized Model Order Reduction Kathrin Smetana (University of Twente) March 24, 2020 ICERM Workshop “Algorithms for Dimension and Complexity Reduction” K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 1 / 49

Collaborators Andreas Buhr Anthony T Patera Julia Schleuß (formerly University of Münster) (MIT) (University of Münster) Lukas ter Maat Olivier Zahm (University of Twente) (INRIA) K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 2 / 49

Motivation § Model order reduction ... ... allows to perform computations for many different configurations (parameters, geometry,...) very fast ... without jeopardizing accuracy § Topic of this talk: Localization and randomization facilitate (nearly) real-time simulations of large-scale problems K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 3 / 49

Projection-based model order reduction Outline § Projection-based model order reduction in a nutshell Randomized error estimation § Localized Model Order Reduction Constructing optimal local approximation spaces (in space) Approximating optimal local approximation spaces via random sampling Generating quasi-optimal local approximation spaces in time by random sampling K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 3 / 49

Projection-based model order reduction Parametrized Partial Differential Equation § Parameter vector µ P P ; compact parameter set P Ă R P § Parametrized PDE: Given any µ P P , find u p µ q P X , s.th. in X 1 . A p µ q u p µ q “ f p µ q § Ω Ă R 3 : bounded domain with Lipschitz boundary B Ω 0 p Ω q d Ă X Ă H 1 p Ω q d ( d “ 1 , 2 , 3); X 1 : dual space § H 1 § A p µ q : X Ñ X 1 : inf-sup stable, continuous linear differential operator § f p µ q : X Ñ R : continuous linear form K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 4 / 49

Projection-based model order reduction Parametrized Partial Differential Equation § Parameter vector µ P P ; compact parameter set P Ă R P § Parametrized PDE: Given any µ P P , find u p µ q P X , s.th. in X 1 . A p µ q u p µ q “ f p µ q § High-dimensional discretization: § Introduce high-dimensional FE space X N Ă X with dim p X N q “ N (assume small discretization error) § High-dimensional approximation: Given any µ P P , find u N p µ q P X N , s.th. A p µ q u N p µ q “ f p µ q in X N 1 . § Issue: Require u N p µ q in real time and/or for many µ P P . K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 4 / 49

Projection-based model order reduction Parametrized Partial Differential Equation § Parameter vector µ P P ; compact parameter set P Ă R P § Parametrized PDE: Given any µ P P , find u p µ q P X , s.th. in X 1 . A p µ q u p µ q “ f p µ q § High-dimensional discretization: § Introduce high-dimensional FE space X N Ă X with dim p X N q “ N (assume small discretization error) § High-dimensional approximation: Given any µ P P , find u N p µ q P X N , s.th. A p µ q u N p µ q “ f p µ q A p µ q P R N ˆ N , f p µ q P R N . § Issue: Require u N p µ q in real time and/or for many µ P P . K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 4 / 49

Projection-based model order reduction Projection-based model order reduction: key concept § Exploit: u N p µ q belongs to “solution manifold” M N “ t u N p µ q | µ P P u Ă X N of typically very low dimension § Offline: Construct reduced space X N from solutions u N p ¯ µ i q , i “ 1 , ..., N (e.g. by a Greedy algorithm, Proper Orthogonal Decomposition,...) § Online: Galerkin projection on X N : Given any µ ˚ P P , find u N p µ ˚ q P X N , s.th. A p µ ˚ q u N p µ ˚ q “ f p µ ˚ q in p X N q 1 . K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 5 / 49

Projection-based model order reduction Construction of reduced basis B via randomization § First Goal: Given a matrix S P R m ˆ n and an integer k find an orthonormal matrix Q of rank k such that S « QQ ˚ S . § Approach: § Draw k random vectors r j P R n (say standard Gaussian) § Form sample vectors y j “ Sr j P R m j “ 1 , . . . , k . § Orthonormalize y j Ý Ñ q j , j “ 1 , . . . , k and define Q “ r q 1 , . . . , q k s § Result: If S has exactly rank k then q j , j “ 1 , . . . , k span the range of S at high probability. But also in the general case q j , j “ 1 , . . . , k often perform nearly as good as the k leading left singular vectors of S § Compute randomized SVD: § Form C “ Q ˚ S which yields S « QC U Σ V ˚ and set B “ Q r § Compute SVD of of the small matrix C “ r U map which is approximately low rank For a review see for instance [Halko, Martinsson, Tropp 2011] K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 6 / 49

Projection-based model order reduction Construction of reduced basis B via randomization § First Goal: Given a matrix S P R m ˆ n and an integer k find an orthonormal matrix Q of rank k such that S « QQ ˚ S . § Approach: § Draw k random vectors r j P R n (say standard Gaussian) § Form sample vectors y j “ Sr j P R m j “ 1 , . . . , k . § Orthonormalize y j Ý Ñ q j , j “ 1 , . . . , k and define Q “ r q 1 , . . . , q k s § Result: If S has exactly rank k then q j , j “ 1 , . . . , k span the range of S at high probability. But also in the general case q j , j “ 1 , . . . , k often perform nearly as good as the k leading left singular vectors of S § Compute randomized SVD: § Form C “ Q ˚ S which yields S « QC U Σ V ˚ and set B “ Q r § Compute SVD of of the small matrix C “ r U Works also if S is not a data matrix but some linear map which is approximately low rank K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 6 / 49

Projection-based model order reduction References for randomized construction of reduced models § Hochman et al 2014 § Alla, Kutz 2015 § Zahm, Nouy 2016 § Balabanov, Nouy 2019, 2019 § Cohen, Dahmen, DeVore, Nichols 2020 § Saibaba 2020 K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 7 / 49

Projection-based model order reduction A posteriori error estimation § A posteriori error estimator is important both to construct reduced order models via the greedy algorithm to certify the approximation: how large is the error (in some QoI)? Proposition (A posteriori error bound) The error estimator r ∆ N p µ q “ β LB p µ q ´ 1 } f p µ q ´ A p µ q u N p µ q} X N 1 with β LB p µ q ď β N p µ q satisfies ∆ N p µ q ď γ N p µ q } u N p µ q ´ u N p µ q} X ď r β LB p µ q} u N p µ q ´ u N p µ q} X , x A p µ q v , w y x A p µ q v , w y where β N p µ q : “ inf } v } X } w } X and γ N p µ q “ sup } v } X } w } X . v P X N sup v P X N sup w P X N w P X N § Problem: Good estimate of stability constants often computationally infeasible; using simply the residual may perform very poorly, especially say for Helmholtz-type problems. K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 8 / 49

Projection-based model order reduction Outline § Projection-based model order reduction in a nutshell Randomized error estimation § Localized Model Order Reduction Constructing optimal local approximation spaces (in space) Approximating optimal local approximation spaces via random sampling Generating quasi-optimal local approximation spaces in time by random sampling References: § KS, Zahm, Patera, Randomized residual-based error estimators for parametrized equations. SIAM J. Sci. Comput., 2019. § KS, Zahm, Randomized residual-based error estimators for the proper generalized decomposition approximation of parametrized problems, Internat. J. Numer. Methods Engrg., to appear, 2020. K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 8 / 49

Randomized a posteriori error estimation References for randomization within error estimation § Cao, Petzold 2004, Homescu, Petzold, Serban 2005 § Drohmann, Carlberg 2015, Trehan, Carlberg, and Durlofsky 2017 § Manzoni, Pagani, Lassila 2016 § Janon, Nodet, Prieur 2016 § Zahm, Nouy 2016 § Buhr, KS 2018 § Balabanov, Nouy 2019 § Eigel, Schneider, Trunschke, Wolf 2020 K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 9 / 49

Randomized a posteriori error estimation Goal/Motivation Randomized a posteriori error estimation § Goal: Develop a posteriori error estimator for model order reduction that does not contain constants whose estimation is expensive (avoid estimating inf-sup constant and thus improve effectivity of estimator) § Setting: We query a finite number of parameters for which we want to estimate the approximation error; allows computing statistics in UQ § Approach: Exploit concentration inequalities: Proposition (Concentration inequality, Johnson-Lindenstrauss) Choose rows of matrix Φ P R K ˆ N say as K independent copies of standard ? K and let S Ă R N be a finite set. Gaussian random vectors scaled by 1 { Moreover, assume K ě p C p z q{ ε 2 q log p # S { δ q . Then we have � ( p 1 ´ ε q} x ´ y } 2 2 ď } Φ x ´ Φ y } 2 2 ď p 1 ` ε q} x ´ y } 2 @ x , y P S ě 1 ´ δ. P 2 see for instance [Boucheron, Lugosi, Massart 2012], [Vershynin 2018] K Smetana (k.smetana@utwente.nl) Localized Model Order Reduction March 24, 2020 10 / 49