Online adaptive discrete empirical interpolation for nonlinear model reduction Benjamin Peherstorfer (Courant Institute, NYU) and Karen Willcox (MIT) 1 / 22
Introduction: Nonlinear PDEs with parameters Discretized equations stemming from (steady-state) nonlinear PDE A y ( µ ) + f ( y ( µ )) = 0 ���� � �� � N × N f : R N → R N ◮ Spatial domain Ω ⊂ R { 1 , 2 , 3 } ◮ Linear term A ∈ R N × N ◮ State variable y ∈ R N ◮ Parameter µ ∈ D ⊂ R d ◮ Nonlinear term f ( y ( µ )) ∈ R N 2 / 22
Introduction: Discrete Empirical Interpolation Method Derive POD-Galerkin reduced system ◮ Collect snapshots y ( µ 1 ) , . . . , y ( µ M ) ∈ R N ◮ Construct POD basis V ∈ R N × n , n ≪ N of reduced space ◮ Project onto reduced space V T AV y ( µ ) + V T ˜ f ( V y ( µ )) = 0 ˜ � �� � ���� ���� n × n n × N N × n DEIM interpolates f as linear combination of basis U ◮ Compute nonlinear snapshots f ( y ( µ 1 )) , . . . , f ( y ( µ M )) ∈ R N ◮ Compute DEIM basis U ∈ R N × m of nonlinear snapshots ◮ Select interpolation points in P ∈ R N × m at which to sample f V T AV y ( µ ) + V T U ( P T U ) − 1 P T f ( V ˜ ˜ y ( µ )) = 0 � �� � � �� � n × n n × m [Barrault et al., 2004], [Grepl et al., 2007], [Astrid et al., 2008], [Chaturantabut et al., 2010], [Carlberg et al., 2011], [Drohmann et al., 2012], [Drmac, Gugercin, 2016] 3 / 22
Nonlinear approximation Have t ∈ [ 1 , 3 ] , grid x = [ x 1 , . . . , x 64 ] T in [ 0 , 2 π ] time t = 1 5 function f solution f ( t ) = t ( sin ( x t ) + sin ( x t π/ 2 ) + sin ( x t π )) 0 Dimension of manifold induced by f is 1 −5 0 2 4 6 M = { f ( t ) | t ∈ [ 1 , 3 ] } ⊂ R 64 spatial domain 5 DEIM basis vector Classical reduced models approximate M with space mode 0 U = span { u 1 , . . . , u m } −5 0 2 4 6 Here, one-dimensional space spatial domain approximation U = span { u 1 } 5 DEIM approx 0 fails to approximate M −5 ⇒ Localized and adaptive reduced spaces 0 2 4 6 spatial domain 4 / 22
Nonlinear approximation Have t ∈ [ 1 , 3 ] , grid x = [ x 1 , . . . , x 64 ] T in [ 0 , 2 π ] time t = 1 5 function f solution f ( t ) = t ( sin ( x t ) + sin ( x t π/ 2 ) + sin ( x t π )) 0 Dimension of manifold induced by f is 1 −5 0 2 4 6 M = { f ( t ) | t ∈ [ 1 , 3 ] } ⊂ R 64 spatial domain 5 DEIM basis vector Classical reduced models approximate M with space mode 0 U = span { u 1 , . . . , u m } −5 0 2 4 6 Here, one-dimensional space spatial domain approximation U = span { u 1 } 5 DEIM approx 0 fails to approximate M −5 ⇒ Localized and adaptive reduced spaces 0 2 4 6 spatial domain 4 / 22
Nonlinear approximation Have t ∈ [ 1 , 3 ] , grid x = [ x 1 , . . . , x 64 ] T in [ 0 , 2 π ] time t = 2.98 5 function f solution f ( t ) = t ( sin ( x t ) + sin ( x t π/ 2 ) + sin ( x t π )) 0 Dimension of manifold induced by f is 1 −5 0 2 4 6 M = { f ( t ) | t ∈ [ 1 , 3 ] } ⊂ R 64 spatial domain 5 DEIM basis vector Classical reduced models approximate M with space mode 0 U = span { u 1 , . . . , u m } −5 0 2 4 6 Here, one-dimensional space spatial domain approximation U = span { u 1 } 5 DEIM approx 0 fails to approximate M −5 ⇒ Localized and adaptive reduced spaces 0 2 4 6 spatial domain 4 / 22
from global reduced spaces to localized subspaces U ⇒ U 1 , . . . , U k ◮ Eftang, Patera, Ronquist; 2010 ◮ Dihlmann, Drohmann, Haasdonk; 2011 ◮ Amsallem, Zahr, Farhat; 2012 ◮ Eftang, Stamm; 2012 ◮ Eftang, Patera; 2013 ◮ Maday, Stamm; 2013 ◮ P., Butnaru, Willcox, Bungartz; 2014 ◮ Zimmermann; 2014 ◮ Albrecht, Haasdonk, Kaulmann, Ohlberger; 2015 ◮ Schindler, Ohlberger; 2015, 2017 ◮ Buhr, Engwer, Ohlberger, Rave; 2017a,2017b ◮ ... 5 / 22
reduced models with online adaptive (DEIM) subspaces U ⇒ U 1 , . . . , U k ⇒ U ( t ) ◮ Amsallem, Zahr, Washabaugh; 2015 ◮ Carlberg; 2015 ◮ Kramer, P., Willcox; 2017 ◮ Lass; 2014 ◮ P., Willcox; 2015a, 2015b, 2016 ◮ Schindler, Ohlberger; 2015, 2017 ◮ Zahr, Farhat; 2015 ◮ ... 6 / 22
reduced models with online adaptive (DEIM) subspaces U ⇒ U 1 , . . . , U k ⇒ U ( t ) ◮ Amsallem, Zahr, Washabaugh; 2015 ◮ Carlberg; 2015 ◮ Kramer, P., Willcox; 2017 ◮ Lass; 2014 ◮ P., Willcox; 2015a, 2015b, 2016 ◮ Schindler, Ohlberger; 2015, 2017 ◮ Zahr, Farhat; 2015 ◮ ... P., Willcox: Online Adaptive Model Reduction for Nonlinear Systems via Low-Rank Updates . SIAM Journal on Scientific Computing, 37(4):A2123-A2150, SIAM, 2015. ◮ Sparse residual from full model ⇒ avoid pre-computed quantities ◮ Low-rank updates ⇒ online efficiency ◮ Additive updates ⇒ arbitrary changes to reduced basis 6 / 22
Nonlinear approximation through online adaptivity ... reduced state vector ˜ y ( µ M + i ) sample f samples of non- linear function outer loop process DEIM interpolant approximate nonlinear function approximation . ... 7 / 22
Nonlinear approximation through online adaptivity ... outer loop iteration reduced state vector ˜ y ( µ M +1 ) sample f samples of non- linear function process DEIM interpolant approximate nonlinear function approximation . ... 7 / 22
Nonlinear approximation through online adaptivity ... outer loop iteration reduced state reduced state vector ˜ y ( µ M +1 ) vector ˜ y ( µ M +2 ) sample f sample f samples of non- samples of non- linear function linear function process process DEIM interpolant approximate approximate nonlinear function nonlinear function approximation approximation . ... 7 / 22
Nonlinear approximation through online adaptivity ... outer loop iteration reduced state reduced state vector ˜ y ( µ M +1 ) vector ˜ y ( µ M +2 ) sample f sample f samples of non- samples of non- linear function linear function process process adapted DEIM DEIM interpolant adapt interpolant approximate approximate nonlinear function nonlinear function approximation approximation . ... 7 / 22
Nonlinear approximation through online adaptivity ... outer loop iteration reduced state reduced state reduced state vector ˜ y ( µ M +1 ) vector ˜ y ( µ M +2 ) vector ˜ y ( µ M +3 ) sample f sample f sample f samples of non- samples of non- samples of non- linear function linear function linear function process process process adapted DEIM adapted DEIM ... DEIM interpolant adapt adapt interpolant interpolant approximate approximate approximate nonlinear function nonlinear function nonlinear function approximation approximation approximation . ... 7 / 22
Nonlinear approximation through online adaptivity ... outer loop iteration reduced state reduced state reduced state vector ˜ y ( µ M +1 ) vector ˜ y ( µ M +2 ) vector ˜ y ( µ M +3 ) sample f sample f sample f samples of non- samples of non- samples of non- linear function linear function linear function process process process adapted DEIM adapted DEIM ... DEIM interpolant adapt adapt interpolant interpolant approximate approximate approximate nonlinear function nonlinear function nonlinear function approximation approximation approximation . ... Key ingredients of online adaptation ◮ Sparse residual from full model ⇒ avoid pre-computed quantities ◮ Low-rank updates ⇒ online efficiency ◮ Additive updates ⇒ arbitrary changes to DEIM space 7 / 22
Adaptive DEIM: Problem setup Outer loop ◮ Online phase consists of k = 1 , . . . , M ′ outer loop iterations ◮ Reduced solution ˜ y ( µ M + k ) requested at each iteration ◮ DEIM interpolant used to approximate f ( V ˜ y ( µ M + k )) Adaptive interpolant ◮ DEIM interpolant ( U 0 , P 0 ) built offline from snapshots ◮ Adaptation initialized with ( U 0 , P 0 ) ◮ In each step k = 1 , . . . , M ′ ◮ Adapt DEIM basis U k − 1 to obtain U k ◮ Adapt DEIM interpolation points matrix P k − 1 to obtain P k ◮ Derive adapted interpolant ( U k , P k ) ◮ Use ( U k , P k ) to approximate nonlinear term 8 / 22
Adaptive DEIM: Problem setup Outer loop ◮ Online phase consists of k = 1 , . . . , M ′ outer loop iterations ◮ Reduced solution ˜ y ( µ M + k ) requested at each iteration ◮ DEIM interpolant used to approximate f ( V ˜ y ( µ M + k )) Adaptive interpolant ◮ DEIM interpolant ( U 0 , P 0 ) built offline from snapshots ◮ Adaptation initialized with ( U 0 , P 0 ) ◮ In each step k = 1 , . . . , M ′ ◮ Adapt DEIM basis U k − 1 to obtain U k ◮ Adapt DEIM interpolation points matrix P k − 1 to obtain P k ◮ Derive adapted interpolant ( U k , P k ) ◮ Use ( U k , P k ) to approximate nonlinear term 8 / 22
Adaptive DEIM: Oversampling DEIM interpolates f at interpolation points { p 1 , . . . , p m } , i.e., � �� � P T � � U ( P T U ) − 1 P T f ( y ( µ )) − f ( y ( µ )) � 2 = 0 � Oversample nonlinear function ◮ Draw m s ∈ N points uniformly from { 1 , . . . , N } \ { p 1 , . . . , p m } ◮ First m points equal DEIM interpolation points ◮ Create sampling points matrix S ∈ R N × m + m s ◮ Solve regression problem to obtain coefficient c ( y ( µ )) = ( S T U ) + S T f ( y ( µ )) ◮ Residual r ( y ( µ )) = Uc ( y ( µ )) − f ( y ( µ )) In general, non-zero residual at sampling points, � S T r ( y ( µ )) � 2 > 0 9 / 22
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