Model Reduction for Variable-Fidelity Optimization Frameworks

Transcription

1 Model Reduction for Variable-Fidelity Optimization Frameworks Karen Willcox Aerospace Computational Design Laboratory Department of Aeronautics and Astronautics Massachusetts Institute of Technology Workshop on Model Order Reduction, Coupled Problems and Optimization Sept , 2005 Leiden University

2 Model Reduction for Large-Scale Systems Many model reduction methods for largescale systems Krylov-based, proper orthogonal decomposition (POD), balanced truncation, modal analysis, Fourier model reduction, et al. Methodology mature for linear timeinvariant systems with few inputs/few outputs Open challenges: nonlinear, parametrically varying systems Model reduction for optimization versus just simulation

3 Model Reduction in Design and Optimization: Key Challenges Optimization-ready reduced-order models What is a good reduced model for design? Metrics beyond the transfer function? High dimensionality of parametric input space Reduction methods rely on small number of inputs (curse of dimensionality) Fidelity management Can the results of the reduced model be trusted? When is it time to update the model?

4 Outline Projection framework for model reduction An optimization framework for model reduction Joint work with O. Ghattas (UTexas), B. van Bloemen Waanders & B. Bader (Sandia National Laboratories) Variable-fidelity frameworks in design optimization Design variable mapping PhD research of T. Robinson (MIT), joint work with M. Eldred (Sandia National Laboratories) and R. Haimes (MIT)

5 Dynamical Systems

6 Reduced-Order Projection x = V x r

7 Reduced-Order Dynamical Systems x = V x r

8 Reduced-Order Basis Determine the projection where V contains m basis vectors so that m n and system dynamics are captured accurately: Most, but not all, reduction techniques use projection framework (all reduced-order models can be represented in projection framework)

9 Proper Orthogonal Decomposition Consider M snapshots (instantaneous state solutions) Construct kernel n n eigenvalue problem are eigenvectors of K with λ 1 > λ 2 > λ 3 > L If V m contains the first m eigenvectors, then Q m = V m V mt is the optimal projection in a least squares sense: Note: optimality and error bound applies to the reconstruction of sampled data, not to the ROM.

10 Method of Snapshots Sirovich: the eigenvectors of the kernel are linear combinations of the snapshots: The eigenvalue problem becomes: M M eigenvalue problem where R is the M M correlation matrix:

11 Reduction via POD 1. Simulate the high-order system to get M snapshots x j, j=1, 2,, M (could be for different parameters) 2. Construct the correlation matrix 3. Calculate the eigenvectors α i and eigenvalues λ i of R 4. Construct the basis functions 5. Select the most energetic basis functions using λ i 6. Project the governing equations onto the reduced basis Reduction via POD offers no guarantees of ROM quality (accuracy/stability). POD is a data-driven approach. Basis and ROM do not respect the governing equations.

12 Outline Projection framework for model reduction An optimization framework for model reduction Joint work with O. Ghattas (UTexas), B. van Bloemen Waanders & B. Bader (Sandia National Laboratories) Variable-fidelity frameworks in design optimization Design variable mapping PhD research of T. Robinson (MIT), joint work with M. Eldred (Sandia National Laboratories) and R. Haimes (MIT)

13 Reduction via Optimization

14 Optimization Framework: Goal-Oriented Error in reduced-order outputs for sample set

15 Optimization Framework Regularization term to yield basis vectors of unit length

16 Optimization Framework: Model-Based Reduced output predictions from solution of governing equations

17 PDE-Constrained Optimization Earthquake inverse modeling (CMU Quake Project, 2003) Optimal flow control (Biros & Ghattas, 2005) Accelerator shape optimization (SLAC)

18 Basis Computation Time integrals in objective function are replaced by a summation over a finite number of discrete time instants Method requires a priori computation of a snapshot set over S parameter instances and T time instants Assume the basis vectors are linear combinations of snapshots Reduces number of unknowns from mn to mst Typically ST n Use POD basis as initial guess for optimizer

19 Optimization Framework vs. POD OPT: min error in computed data POD: min error in projected data

20 Outline Projection framework for model reduction An optimization framework for model reduction Joint work with O. Ghattas (UTexas), B. van Bloemen Waanders & B. Bader (Sandia National Laboratories) Variable-fidelity frameworks in design optimization Design variable mapping PhD research of T. Robinson (MIT), joint work with M. Eldred (Sandia National Laboratories) and R. Haimes (MIT)

21 Large-Scale Design Optimization Evaluation of f(u) and c(u) could be computationally expensive c(u) might comprise constraints from many different disciplines u may have large dimensionality For complex large-scale multidisciplinary systems, optimization may be computationally intractable with high-fidelity models.

22 Variable-Fidelity Models Reduced complexity of f( ) and c( ) Simplified physics Model order reduction Other surrogate models (data fit, multigrid, etc.) Reduced complexity of u May need mapping between u and

23 A Hierarchy of Models CFD: Euler/ Navier-Stokes Linearized Panel Code Classical Aerodynamics J.J. Alonso, I. Kroo, Stanford University Low-fidelity EM model: 30,000 DOF High-fidelity EM model: 800,000 DOF J. Castro, A. Giunta Sandia National Labs

24 Trust Region Model Management How to manage the fidelity of the reduced model? Can the results of the reduced model be trusted? When is it time to update the model? Alexandrov, Lewis et al., 1997

25 Corrections Guarantee convergence to optimum of high-fidelity model Additive Multiplicative surrogate model low-fidelity (reducedorder) model

26 Variable Design Descriptions How to create a reduced model for a large number of inputs (dimensionality of u high)? Solution: reduce the complexity of u (physically or numerically) Need mapping between u and Space mapping POD mapping: based on gappy POD theory

27 Space Mapping Developed by Bandler for electronics problems Particular form assumed for mapping between highand low-fidelity design vectors: Parameters, p, determined by solving an optimization problem: Need to choose sample points and form of the mapping Trust-region model management methods using space-mapping are not provably convergent

28 Corrected Space Mapping Trust region model management using space mapping can be made rigorous by correcting spacemapped model to at least first order Correcting the model after space mapping distorts global fit Better solution: Perform space mapping and correction simultaneously

29 Model Reduction in Design and Optimization: Key Components

30 2D Heat Conduction Example Objective function value Objective function value Optimized POD BT S=5 parameter instances T=20 time steps n=480 states q=47 outputs Number of modes Number of modes

31 Output Errors vs. State Errors S=5 parameter instances T=20 time steps n=480 states q=9 outputs Error Error in output Error Error Optimized basis POD basis Snapshot number Snapshot number 2-norm of 2 norm error of over over entire domain domain Optimized basis POD basis Snapshot number Snapshot number

32 Snapshot Reconstruction Errors Error in snapshot in for snapshot optimized basis for optimized basis x y Error in snapshot for Error in snapshot for initial basis POD basis y AEROSPACE COMPUTATIONAL x DESIGN LABORATORY

33 Extended Rosenbrock Problem Example: 3D extended Rosenbrock (N=3) x 2 =-2 x 2 =-1 x 2 =1 x 2 =2

34 Space Mapping Examples Low-fidelity: 2D quadratic High-fidelity: 2D Rosenbrock Problem Objective function value High fidelity function evaluations SQP method Multifidelity (1st order correction) Multifidelity (BFGS correction) Uncorrected space mapping 1st order corrected space mapping 2nd order corrected space mapping

35 Space Mapping Examples Low-fidelity: 2D Rosenbrock Problem High-fidelity: 3D Extended Rosenbrock Problem

36 Space Mapping Examples

37 Airfoil Design Problem Inverse design of an airfoil High-fidelity model: XFOIL potential flow solver Design variables are magnitudes of 36 bump functions Low-fidelity model: Joukowski transform Design variables are two Joukowski parameters

38 POD mapping example Airfoil design problem using POD mapping