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Introduction Local Reduced-Order Models Application Conclusion Efficient, Parametrically-Robust Nonlinear Model Reduction using Local Reduced-Order Bases Matthew J. Zahr and Charbel Farhat Farhat Research Group Stanford University SIAM Computational Science and Engineering Conference February 25 - March 1, 2013 Zahr and Farhat

Introduction Local Reduced-Order Models Application Conclusion 1 Introduction 2 Local Reduced-Order Models Offline Phase Online Phase Fast, Reduced Basis Updating Hyperreduction 3 Application Burger’s Equation (Non-predictive) Potential Nozzle (Predictive) 4 Conclusion Zahr and Farhat

Introduction Local Reduced-Order Models Application Conclusion Motivation Complex, time-dependent problems Real-time analyses Model Predictive Control Many-query analyses Optimization Uncertainty-Quantification Zahr and Farhat

Introduction Local Reduced-Order Models Application Conclusion Model Order Reduction Framework Data collection Compression I. Full-order model Approximation 1: Projection II. Reduced-order model Data collection Compression Approximation 2: System approximation Reduced-order model + system approximation III. [Carlberg et. al. 2011] Zahr and Farhat

Introduction Local Reduced-Order Models Application Conclusion High-Dimensional Model Consider the nonlinear system of Ordinary Differential Equations (ODE), usually arising from the semi-discretization of Partial Differential Equation, d w dt = F ( w , t, µ ) where w ∈ R N state vector µ ∈ R d parameter vector F : R N × R × R d → R N nonlinearity of ODE This is the High-Dimensional Model (HDM). Zahr and Farhat

Introduction Local Reduced-Order Models Application Conclusion Fully Discretization of HDM Our approach to Model Order Reduction leverages dimensionality reduction at the fully discrete level Full, implicit (single-step) discretization of the governing equation yields a sequence of nonlinear systems of equations: R ( w ( n ) , t n , µ ; w ( n − 1) ) = 0 , n ∈ { 1 , 2 , . . . , N s } where w ( n ) = w ( t n ) R : R N × R × R d → R N From this point, we drop the dependence of R on the previous time step w ( n − 1) . Zahr and Farhat

Introduction Local Reduced-Order Models Application Conclusion Model Order Reduction with Local Bases The goal of reducing the computational cost and resources required to solve a large-scale system of ODEs is attempted through dimensionality reduction Specifically, the (discrete) trajectory of the solution in state space is assumed to lie in a low-dimensional affine subspace w ( n ) ≈ w ( n − 1) + Φ( w ( n − 1) ) y ( n ) Φ( w ( n − 1) ) ∈ R N × k w ( w ( n − 1) ) Reduced Basis y ( n ) ∈ R k w ( w ( n − 1) ) Reduced Coordinates where k w ( w ( n − 1) ) ≪ N Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Overview In practice, N V bases are computed in an offline phase: Φ i ∈ R N × k i w Each basis, Φ i , is associated with a representative vector in state space, w i c Then, Φ( w ( n − 1) ) . = Φ i , where || w ( n − 1) − w i c || ≤ || w ( n − 1) − w j c || for all j ∈ { 1 , 2 , . . . , N V } . Contrived Example 1 � � � x ( t ) � d x ( t ) 2 + y ( t ) 2 = sin x ( t ) y ( t ) − dt x ( t ) 2 + y ( t ) 2 � x (0) � � − 1 � = y (0) 0 Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Data Collection HDM sampling (snapshot collection) Simulate HDM at one or more parameter configurations { µ 1 , . . . , µ n } and collect snapshots w ( j ) Combine in snapshot matrix W Figure : Contrived Example: HDM 0.5 0.4 0.3 0.2 0.1 y 0 − 0.1 − 0.2 − 0.3 − 0.4 − 1 − 0.5 0 0.5 1 1.5 x Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Data Organization Snapshot clustering Cluster snapshots using the k-means algorithm based on their relative distance in state space Store the center of each cluster, w i c W partitioned into cluster snapshot matrices W i Figure : Contrived Example: Snapshot Clustering 0.5 0.4 0.3 0.2 0.1 y 0 − 0.1 Cluster 1 − 0.2 Cluster 2 − 0.3 Cluster 3 − 0.4 − 1 − 0.5 0 0.5 1 1.5 x Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Data Compression Modify snapshot matrices W i by subtracting a reference w from each column ˆ we T vector, ¯ W i = W i − ¯ Apply POD method to each cluster: Φ i = POD( ˆ W i ) Figure : Contrived Example: Basis Construction 0.5 0.4 0.3 0.2 0.1 y 0 − 0.1 HDM − 0.2 Subspace 1 Subspace 2 − 0.3 Subspace 3 − 0.4 − 1 − 0.5 0 0.5 1 1.5 x Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Overview The MOR assumption is substituted into the HDM to obtain the over-determined nonlinear system of equations: R ( w ( n − 1) + Φ i y ( n ) , t n , µ ) = 0 Since the above system does not have a solution, in general, we seek the solution that minimizes the residual of the HDM in the chosen affine subspace: y ( n ) = arg min || R ( w ( n − 1) + Φ i y , t n , µ ) || 2 y ∈ R ki w This is the Reduced-Order Model (ROM) Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Inconsistency Recall the MOR assumption: w ( n ) − w ( n − 1) ≈ Φ i y ( n ) n w ( n ) − w ( switch ) ≈ Φ i � y ( k ) k = switch where w ( switch ) is the most recent state to initiate a switch between bases. Recall the reduced bases are constructed as Φ i = POD we T � � W i − ¯ Basis construction consistent with MOR assumption only if w = w ( switch ) ¯ Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Solution: Fast Basis Updating We seek a reduced basis of the form: Φ i = POD ( W i − w ( switch ) e T ) ˆ we T + ( ¯ w − w ( switch ) ) e T ) = POD ( W i − ¯ w − w ( switch ) ) e T ) = POD ( ˆ W i + ( ¯ ˆ Φ is the (truncated) left singular vectors of a matrix that is a rank-one update of a matrix, ˆ W i , whose (truncated) left singular vectors is readily available, Φ i . Fast updates available [Brand 2006]. Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Figure : Contrived Example: ROM Solution No Basis Updating Basis Updating 0.8 0.8 0.6 0.6 0.4 0.4 y 0.2 y 0.2 0 0 HDM HDM Subspace 1 Subspace 1 Subspace 2 Subspace 2 − 0.2 − 0.2 Subspace 3 Subspace 3 Local ROM Local ROM − 0.4 − 0.4 − 1 − 0.5 0 0.5 1 1.5 − 1 − 0.5 0 0.5 1 1.5 x x Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Figure : Contrived Example: ROM Solution No Basis Updating Basis Updating 0.8 0.8 0.6 0.6 0.4 0.4 y 0.2 y 0.2 0 0 HDM HDM Subspace 1 Subspace 1 Subspace 2 Subspace 2 − 0.2 − 0.2 Subspace 3 Subspace 3 Local ROM Local ROM − 0.4 − 0.4 − 1 − 0.5 0 0.5 1 1.5 − 1 − 0.5 0 0.5 1 1.5 x x Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Figure : Contrived Example: ROM Solution No Basis Updating Basis Updating 0.8 0.8 0.6 0.6 0.4 0.4 y 0.2 y 0.2 0 0 HDM HDM Subspace 1 Subspace 1 Subspace 2 Subspace 2 − 0.2 − 0.2 Subspace 3 Subspace 3 Local ROM Local ROM − 0.4 − 0.4 − 1 − 0.5 0 0.5 1 1.5 − 1 − 0.5 0 0.5 1 1.5 x x Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Figure : Contrived Example: ROM Solution No Basis Updating Basis Updating 0.8 0.8 0.6 0.6 0.4 0.4 y 0.2 y 0.2 0 0 HDM HDM Subspace 1 Subspace 1 Subspace 2 Subspace 2 − 0.2 − 0.2 Subspace 3 Subspace 3 Local ROM Local ROM − 0.4 − 0.4 − 1 − 0.5 0 0.5 1 1.5 − 1 − 0.5 0 0.5 1 1.5 x x Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Figure : Contrived Example: ROM Solution No Basis Updating Basis Updating 0.8 0.8 0.6 0.6 0.4 0.4 y 0.2 y 0.2 0 0 HDM HDM Subspace 1 Subspace 1 Subspace 2 Subspace 2 − 0.2 − 0.2 Subspace 3 Subspace 3 Local ROM Local ROM − 0.4 − 0.4 − 1 − 0.5 0 0.5 1 1.5 − 1 − 0.5 0 0.5 1 1.5 x x Zahr and Farhat

Introduction Offline Phase Local Reduced-Order Models Online Phase Application Hyperreduction Conclusion Figure : Contrived Example: ROM Solution No Basis Updating Basis Updating 0.8 0.8 0.6 0.6 0.4 0.4 y 0.2 y 0.2 0 0 HDM HDM Subspace 1 Subspace 1 Subspace 2 Subspace 2 − 0.2 − 0.2 Subspace 3 Subspace 3 Local ROM Local ROM − 0.4 − 0.4 − 1 − 0.5 0 0.5 1 1.5 − 1 − 0.5 0 0.5 1 1.5 x x Zahr and Farhat