Model reduction of large-scale systems Thanos Antoulas Rice - PowerPoint PPT Presentation

Model reduction of large-scale systems Thanos Antoulas Rice University and Jacobs University email: aca@rice.edu URL: www.ece.rice.edu/ ˜ aca DIMACS Workshop on Perspectives and Future Directions in Systems and Control Theory May 23 - May 27, 2011 Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 1 / 42

Motivating example I: pollution propagation (Fukushima) Velocity 0.4 0.3 0.2 0.1 ξ 2 0 − 0.1 − 0.2 − 0.3 − 0.4 0 0.2 0.4 0.6 0.8 1 1.2 ξ 1 Stokes equation and advection diffusion equation Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 2 / 42

Motivating example II: Viscous fingering in porous media (EOR) Enhanced oil recovery from underground reservoirs Darcy’s law and advection diffusion equation (twice) Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 3 / 42

Motivating example III: VLSI circuits CMOS Chip 3D Silicon Chip 10 8 components nanometer details several GHz speed several km interconnect ≈ 10 layers Interconnect analysis : signal distortions & delays ⇒ Maxwell’s equations Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 4 / 42

Motivating example IV: A steel cooling model 1.000 0.500 Advection diffusion equation Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 5 / 42

Motivating example V: Driven Cavity Flow A cavity is filled with viscoelastic material and is excited through shearing forces u ( t ) of the lid. We are interested in the displacement of the material, w (ˆ x , t ) , at the center. ✛ ✲ ∂ Ω 1 u ( t ) Ω ˆ x ✟ ✟ ✙ r w (ˆ x , t ) ∂ Ω 0 ⇒ wave equation with hereditary damping Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 6 / 42

Outline 1 Introduction Approximation: SVD based methods 2 POD Balanced reduction 3 Approximation: Krylov-based or interpolatory methods Choice of interpolation points: Passivity preserving reduction Choice of interpolation points: Optimal H 2 reduction Reduction of models in generalized form 4 Conclusions and References Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 7 / 42

Introduction Outline 1 Introduction Approximation: SVD based methods 2 POD Balanced reduction 3 Approximation: Krylov-based or interpolatory methods Choice of interpolation points: Passivity preserving reduction Choice of interpolation points: Optimal H 2 reduction Reduction of models in generalized form 4 Conclusions and References Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 8 / 42

Introduction The overall problem Physical system ⇓ PDEs ⇓ ODEs ⇓ Model reduction = DATA ⇐ (Reduced number of ODEs) ⇓ Simulation, Design, Control Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 9 / 42

Introduction Model reduction via projection f (˙ E ˙ x ( t ) , x ( t ) , u ( t )) = 0 x ( t ) = Ax ( t ) + Bu ( t ) Given is or . y ( t ) = h ( x ( t ) , u ( t )) y ( t ) = Cx ( t ) + Du ( t ) Common framework for (most) model reduction methods: Petrov-Galerkin projective approximation . k ) ⊂ C n . Choose k -dimensional subspaces, V k = Range ( V k ) , W k = Range ( W Find v ( t ) = V k x k ( t ) ∈ V k , x k ∈ C r , such that E ˙ v ( t ) − Av ( t ) − B u ( t ) ⊥ W r ⇒ W ∗ k ( EV k ˙ x k ( t ) − AV k x k ( t ) − B u ( t )) = 0 , y k ( t ) = CV k x k ( t ) + Du ( t ) , Reduced order system E k = W ∗ A k = W ∗ B k = W ∗ k EV k , k AV k , k B , C k = CV k . Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 10 / 42

Introduction The quality of the reduced system depends on the choice of V r and W r . n k Norms : • H ∞ -norm: n E , A B k E k , A k B k worst output error ⇒ � y ( t ) − ˆ y ( t ) � for � u ( t ) � = 1. • H 2 -norm: � h ( t ) − ˆ C D C k D k h ( t ) � Consider a system described by implicit nonlinear equations (DAEs) : f (˙ x ( t ) , x ( t ) , u ( t )) = 0 , y ( t ) = h ( x ( t ) , u ( t )) , with: u ( t ) ∈ R m , x ( t ) ∈ R n , y ( t ) ∈ R p . Approximate by means of a Petrov-Galerkin projection Π = V k W ∗ k : W ∗ k f ( V k ˙ x k ( t ) , V k x k ( t ) , u ( t )) = 0 , y k ( t ) = h ( V k x k ( t ) , u ( t )) where x k ∈ R k , k ≪ n . The approximation is ”good” if x − Π x is ”small”. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 11 / 42

Introduction Issues and requirements Issues with large-scale systems Storage – Computational speed – Accuracy 1 System theoretic properties 2 Requirements for model reduction Approximation error small 1 Structure preservation (e.g. stability/passivity) 2 Procedure computationally efficient and automatic 3 In addition: many ports, parameters, nonlinearities, ... . 4 Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 12 / 42

Approximation: SVD based methods Outline 1 Introduction Approximation: SVD based methods 2 POD Balanced reduction 3 Approximation: Krylov-based or interpolatory methods Choice of interpolation points: Passivity preserving reduction Choice of interpolation points: Optimal H 2 reduction Reduction of models in generalized form 4 Conclusions and References Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 13 / 42

Approximation: SVD based methods Approximation methods: Overview PPPPPPP ✟ ✟ ✟ ✟ ✟ ✙ P q Krylov SVD • Realization � ❅ • Interpolation � ❅ � ✠ ❅ ❘ • Lanczos • Arnoldi Nonlinear systems Linear systems • POD methods • Balanced truncation • Empirical Gramians • Hankel approximation Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 14 / 42

Approximation: SVD based methods SVD Prototype approximation problem : SVD (Singular Value Decomposition): A = U Σ V ∗ Singular values Σ provide trade-off between accuracy and complexity. original 599 × 726 k = 10 k = 50 Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 15 / 42

Approximation: SVD based methods POD POD POD (Proper Orthogonal Decomposition) : Consider: f (˙ x ( t ) , x ( t ) , u ( t )) = 0, y ( t ) = h ( x ( t ) , u ( t )) . Snapshots of the state: X = [ x ( t 1 ) x ( t 2 ) · · · x ( t N )] ∈ R n × N . SVD : X = U Σ V ∗ ≈ U k Σ k V ∗ k , k ≪ n . Approximation of the state: x k ( t ) = U ∗ x ( t ) ≈ U k x k ( t ) , x k ( t ) ∈ R k k x ( t ) ⇒ Project state and output equations. Reduced order system: U ∗ k f ( U k ˙ x k ( t ) , U k x k ( t ) , u ( t )) = 0 , y k ( t ) = h ( U k x k ( t ) , u ( t )) ⇒ x k ( t ) eolves in a low-dimensional space. Issues with POD : (a) Choice of snapshots, (b) singular values not I/O invariants, (c) computation of U ∗ k f costly. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 16 / 42

Approximation: SVD based methods POD Viscous fingering in porous media ∇ · u = 0 (incompressibility) ∇ π = − µ u (Darcy’s law) ∂ c ∂ t + u · ∇ c = α ∇ 2 c + β f ( c ) (convection, diffusion for c ) ∂ Θ ∂ t + u · ∇ Θ = γ ∇ 2 Θ + δ f ( c ) (convection, diffusion for Θ ) u : velocity, π : pressure, c : concentration, Θ : temperature, α, β, γ, δ constants, µ ( c , Θ) : viscosity of injected fluid, f ( c ) nonlinear function of c . Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 17 / 42

Approximation: SVD based methods Balanced reduction SVD methods: balanced truncation Given linear system ( E , A , B , C ) , det E � = 0 , ( A , E ) stable, use state and output . This implies the computation of the gramians which satisfy the generalized Lyapunov equations : APE ∗ + EPA ∗ + BB ∗ = 0 , P > 0 , A ∗ QE + E ∗ QA + C ∗ C = 0 , Q > 0 ⇒ � λ i ( PE ∗ QE ) : σ i = Hankel singular values : provide trade-off between accuracy and complexity. Properties Stability is preserved 1 Global error bound: σ k + 1 ≤� H ( s ) − ˆ H ( s ) � ∞ ≤ 2 ( σ k + 1 + · · · + σ n ) 2 Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 18 / 42

Approximation: SVD based methods Balanced reduction Iterative solution of Lyapunov equations Drawbacks Dense computations, matrix factorizations and inversions ⇒ may be 1 ill-conditioned; number of operations O ( n 3 ) Bottleneck: solution of Lyapunov equations: APE ∗ + EPA ∗ + BB ∗ = 0 . 2 For large A such equations cannot be solved exactly. Instead, since P > 0 ⇒ square root L exists : P = LL ∗ . Hence compute approximations V to L : ˆ P = VV ∗ : rank V = k ≪ n : V ∗ = = L ∗ ≈ ˆ P L V P Iterative solution: ADI, modified Smith (guaranteed convergence). Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 19 / 42