PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment PDE-Constrained Optimization Using Hyper-Reduced Models Matthew J. Zahr and Charbel Farhat Institute for Computational and Mathematical Engineering Farhat Research Group Stanford University SIAM Conference on Optimization (CP13) May 19 - 22, 2014 San Diego, CA Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment 1 PDE-Constrained Optimization 2 HROM-Constrained Optimization 3 Numerical Experiment Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Problem Formulation Goal: Rapidly solve PDE-constrained optimization problems of the form minimize f ( w , µ ) w ∈ R N , µ ∈ R p (1) subject to R ( w , µ ) = 0 where R : R N × R p → R N is the discretized (nonlinear) PDE, w is the PDE state vector, µ is the vector of parameters, and N is assumed to be very large. Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Reduced-Order Model Model Order Reduction (MOR) assumption: state vector lies in low-dimensional affine subspace w = ¯ w + Φy where y ∈ R n are the reduced coordinates of w in the basis Φ ∈ R N × n and n ≪ N Substitute assumption into High-Dimensional Model (HDM), R ( w , µ ) = 0 R ( ¯ w + Φy , µ ) ≈ 0 Require projection of residual in low-dimensional left subspace , with basis Ψ ∈ R N × n to be zero R r ( y , µ ) = Ψ T R ( ¯ w + Φy , µ ) = 0 Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Bottleneck R r ( y , µ ) = Ψ T R ( ¯ w + Φy , µ ) = 0 y Φ Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Bottleneck R r ( y , µ ) = Ψ T R ( ¯ w + Φy , µ ) = 0 y w + ¯ Φ Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Bottleneck R r ( y , µ ) = Ψ T R ( ¯ w + Φy , µ ) = 0 y R ( w + ¯ ) Φ Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Bottleneck R r ( y , µ ) = Ψ T R ( ¯ w + Φy , µ ) = 0 Ψ T R Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Bottleneck R r ( y , µ ) = Ψ T R ( ¯ w + Φy , µ ) = 0 R r = y Ψ T R ( w + ¯ ) Φ Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Bottleneck ∂ R r ∂ y ( y , µ ) = Ψ T ∂ R ∂ y ( ¯ w + Φy , µ ) Φ y Φ Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Bottleneck ∂ R r ∂ y ( y , µ ) = Ψ T ∂ R ∂ y ( ¯ w + Φy , µ ) Φ y w + ¯ Φ Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Bottleneck ∂ R r ∂ y ( y , µ ) = Ψ T ∂ R ∂ y ( ¯ w + Φy , µ ) Φ y ∂ R ∂ w ( w + ¯ ) Φ Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Bottleneck ∂ R r ∂ y ( y , µ ) = Ψ T ∂ R ∂ y ( ¯ w + Φy , µ ) Φ Ψ T ∂ R Φ ∂ w Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Bottleneck ∂ R r ∂ y ( y , µ ) = Ψ T ∂ R ∂ y ( ¯ w + Φy , µ ) Φ ∂ R r y = Ψ T ∂ y ∂ R ∂ w ( w + ¯ ) Φ Φ Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Solution: Gappy POD Approximation Assume nonlinear terms (residual/Jacobian) lie in low-dimensional subspace R ( w , µ ) ≈ Φ R r ( w , µ ) where Φ ∈ R N × n R and r : R N × R p → R n R are the reduced coordinates; n R ≪ N Determine R by solving gappy least-squares problem a ∈ R nr || Z T Φ R a − Z T R ( w , µ ) || r ( w , µ ) = arg min where Z is a restriction operator Analytical solution Z T Φ R � † � Z T R ( w , µ ) � � r ( w , µ ) = Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Gappy POD in Practice (a) 253 sample nodes (b) 378 sample nodes (c) 505 sample nodes Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Hyper-Reduced Model Using the Gappy POD approximation, the hyper-reduced governing equations are R h ( y , µ ) = Ψ T Φ R Z T Φ R � † � Z T R ( ¯ � � w + Φy , µ ) = 0 where E = Ψ T Φ R Z T Φ R � † � is known offline and can be precomputed Z T R R r = E Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Hyper-Reduced Optimization Using the hyper-reduced model as a surrogate for the HDM in the PDE-constrained optimization, we have the hyper-reduced optimization problem ˜ minimize f ( y , µ ) y ∈ R n , µ ∈ R p subject to R h ( y , µ ) = 0 where R h : R k × R p → R k is the hyper-reduced PDE and y ∈ R k are the reduced coordinates, where k ≪ N . Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Hyper-Reduced Optimization Procedure HDM Update RB Optimizer HDM Compress RB, Φ HDM ROM R R R R R R R R R HDM HDM HDM O O O O O O O O O M M M M M M M M M Zahr and Farhat Hyper-Reduced Optimization w µ ∗ ∗ � Φ y w µ ∗ Φ � ������� � ����� Φ ������������ � �������� ������� � �������� ��� � ������� y Φ ������� � ����� ������������ � �������� ������� � �������� ��� � �������
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Hyper-Reduced Optimization Schematic r Φ ������� � ����� ������������ � �������� ������� � �������� ��� � ������� Zahr and Farhat Hyper-Reduced Optimization w µ ∗ ∗ � Φ y Φ ������� � ����� ������������ � �������� ������� � �������� ��� � �������
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Quasi-1D Euler Flow Quasi-1D Euler equations: ∂ U ∂t + 1 ∂ ( A F ) = Q A ∂x where 0 ρ ρu ρu 2 + p , , p ∂A U = ρu F = Q = A ∂x ( e + p ) u e 0 Semi-discretization = ⇒ finite volumes with Roe flux and entropy corrections Full discretization = ⇒ Backward Euler → steady state Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Nozzle Parametrization Nozzle parametrized with cubic splines using 13 control points and constraints requiring A ′′ ( x ) ≥ 0 convexity bounds on A ( x ) A l ( x ) ≤ A ( x ) ≤ A u ( x ) bounds on A ′ ( x ) at inlet/outlet A ′ ( x l ) ≤ 0, A ′ ( x r ) ≥ 0 Nozzle Parametrization 0.07 A l ( x ) 0.06 A u ( x ) A ( x ) Spline Points 0.05 Nozzle Height 0.04 0.03 0.02 0.01 0 0 0.05 0.1 0.15 0.2 0.25 x Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Parameter Estimation/Inverse Design For this problem, the goal is to determine the parameter µ ∗ such that the flow achieves some optimal or desired state w ∗ || w ( µ ) − w ∗ || minimize w ∈ R N , µ ∈ R p (2) subject to R ( w , µ ) = 0 c ( w , µ ) ≤ 0 where c are the nozzle constraints. This problem is solved using the HDM as the governing equation HDM-based optimization the HROM as the governing equation HROM-based optimization Zahr and Farhat Hyper-Reduced Optimization
PDE-Constrained Optimization HROM-Constrained Optimization Numerical Experiment Objective Function Convergence (a) Convergence (# HDM Evals) (b) Convergence (CPU Time) 6 10 6 10 HDM - based opt H DM - B ased Opt HROM - based opt H ROM - B ased Opt 5 5 10 10 Ob j ective Function 4 Ob j ective Function 4 10 10 3 3 10 10 2 2 10 10 1 1 10 10 0 0 10 10 0 5 10 15 20 25 30 0 500 1000 1500 2000 2500 3000 3500 C PU T im e (sec) # H DM Evaluations Zahr and Farhat Hyper-Reduced Optimization
Recommend
More recommend
