CME 345: MODEL REDUCTION - Projection-Based Model Order Reduction

Transcription

1 CME 345: MODEL REDUCTION - Projection-Based Model Order Reduction Projection-Based Model Order Reduction Charbel Farhat and David Amsallem Stanford University cfarhat@stanford.edu 1 / 38

2 Outline 1 Solution Approximation 2 Orthogonal and Oblique Projections 3 Galerkin and Petrov-Galerkin Projections 4 Equivalent High-Dimensional Model 5 Error Analysis 2 / 38

3 Solution Approximation High-dimensional Model Ordinary Differential Equation (ODE) dw (t) = f(w(t), t) (1) dt w R N : State variable Initial condition: w() = w Output equation There is no parameter dependence for now y(t) = g(w(t), t) (2) 3 / 38

4 Solution Approximation Low Dimensionality of Trajectories In many cases, the trajectories of the solutions computed using High-Dimensional Models (HDMs) are contained in low-dimensional subspaces k S = dim (S) The state can be written exactly as a linear combination of vectors spanning S w(t) = q 1 (t)v q ks (t)v ks V S = [v 1,, v ks ] R N k S is a basis for S (q 1(t),, q ks (t)) are the generalized coordinates for w(t) in S q(t) = [q 1(t),, q ks (t)] T R k S is the reduced-order state vector In matrix form, the above expansion can be written as w(t) = V S q(t) 4 / 38

5 Solution Approximation Low Dimensionality of Trajectories Often, the exact basis V S is unknown but can be estimated empirically by a trial basis V R N k, k < N k and k S may be different The following ansatz is considered w(t) Vq(t) Substituting the above subspace approximation in Eq. (1) and in Eq. (2) leads to d Vq(t) dt = f(vq(t), t) + r(t) y(t) = g(vq(t), t) where r(t) is the residual due to the subspace approximation 5 / 38

6 Solution Approximation Low Dimensionality of Trajectories The residual r(t) R N accounts for the fact that Vq(t) is not in general an exact solution of problem (1) Since the basis V is assumed to be time-invariant and therefore d dq (Vq) = V dt dt V d q(t) dt = f(vq(t), t) + r(t) y(t) = g(vq(t), t) Set of N differential equations in terms of k unknowns Over-determined system (k < N) q 1 (t),, q k (t) 6 / 38

7 Orthogonal and Oblique Projections Orthogonality Let w and y be two vectors in R N w and y are orthogonal to each other with respect to the canonical inner product in R N if and only if w T y = w and y are orthonormal to each other with respect to the canonical inner product in R N if and only if w T y =, and w T w = 1, and y T y = 1 Let V be a matrix in R N k V is an orthogonal (orthonormal) matrix if and only if V T V = I k 7 / 38

8 Orthogonal and Oblique Projections Projections Definition A matrix Π R N N is a projection matrix if Π 2 = Π Some direct consequences range(π) is invariant under the action of Π and 1 are the only possible eigenvalues of Π let k be the rank of Π: Then, there exists a basis X such that [ ] Ik Π = X X 1 N k 8 / 38

9 Orthogonal and Oblique Projections Projections Consider [ Ik Π = X N k ] X 1 decompose X as X = [ X 1 X 2 ], where X1 R N k and X 2 R N (N k) then, w R N Πw range(x 1 ) = range(π) = S 1 w Πw range(x 2 ) = Ker(Π) = S 2 Π defines the projection onto S 1 parallel to S 2 R N = S 1 S 2 9 / 38

10 Orthogonal and Oblique Projections Orthogonal Projections Consider the case where S 2 = S 1 Let V R N k be an orthogonal matrix whose columns span S 1, and let w R N : The orthogonal projection of w onto the subspace S 1 is VV T w the equivalent projection matrix is Π V,V = VV T special case #1: If w belongs to S 1 Π V,V w = VV T w = w special case #2: If w is orthogonal to S 1 Π V,V w = VV T w = 1 / 38

11 Orthogonal and Oblique Projections Orthogonal Projections Π V,V w = VV T w 11 / 38

12 Orthogonal and Oblique Projections Orthogonal Projections Example: Helix in 3D (N = 3) let w(t) R 3 define a curve parameterized by t [, 6π] as follows w 1(t) cos(t) w(t) = w 2(t) w 3(t) = sin(t) t w (t) 15 w w 2.5 w / 38

13 Orthogonal and Oblique Projections Orthogonal Projections w (t) Π V, V w (t) Orthogonal projection onto range(v) = span(e 1, e 2) Π V,V w(t) = cos(t) sin(t) w w2.5 w / 38

14 Orthogonal and Oblique Projections Orthogonal Projections w (t) Π V, V w (t) Orthogonal projection onto range(v) = span(e 2, e 3) Π V,V w(t) = sin(t) t w w2.5 w / 38

15 Orthogonal and Oblique Projections Orthogonal Projections w (t) Π V, V w (t) Orthogonal projection onto range(v) = span(e 1, e 3) Π V,V w(t) = cos(t) t w w2.5 w / 38

16 Orthogonal and Oblique Projections Oblique Projections This is the general case where S 2 may be distinct from S 1 Let V R N k and W R N k be two full-column rank matrices whose columns span respectively S 1 and S 2 The projection of w R N onto the subspace S 1 parallel to S 2 is V(W T V) 1 W T w the equivalent projection matrix is Π V,W = V(W T V) 1 W T special case #1: If w belongs to S 1 then w = Vq where q R k and Π V,W w = V(W T V) 1 W T Vq = Vq special case #2: If w is orthogonal to S2, i.e. w belongs to S 2 Π V,W w = V(W T V) 1 W T w = 16 / 38

17 Orthogonal and Oblique Projections Oblique Projections Π V,W w = V(W T V) 1 W T w 17 / 38

18 Orthogonal and Oblique Projections Oblique Projections Example: Helix in 3D bases V = [e 1, e 2], W = [e 1 + e 3, e 2 + e 3] projection matrix Π V,W = V(W T V) 1 W T = projected helix equation 1 1 Π V,W w(t) = 1 1 cos(t) sin(t) t = cos(t) + t sin(t) + t 18 / 38

19 Orthogonal and Oblique Projections Oblique Projections w (t) Π V, W w (t) 15 w w w / 38

20 Galerkin and Petrov-Galerkin Projections Model Order Reduction Start with an HDM d w(t) dt = f(w(t), t) y(t) = g(w(t), t) w() = w w R N : Vector of state variables y R q : Vector of output variables (typically q N) f(, ) R N : Completes the specification of the ODE of interest 2 / 38

21 Galerkin and Petrov-Galerkin Projections Model Order Reduction The goal is to construct a Reduced-Order Model (ROM) d dt q(t) = f r (q(t), t) y(t) = g r (q(t), t) where q R k : Vector of reduced-order state variables, k N y R q : Vector of output variables f r (, ) R k : Completes the definition of the reduced-order equations 21 / 38

22 Galerkin and Petrov-Galerkin Projections Model reduction requirements The Model Order Reduction (MOR) method should be computationally tractable be applicable to a large class of dynamical systems minimize a certain error criterion between the solution computed using the HDM and that computed the ROM preserve as many properties of the underlying HDM as possible 22 / 38

23 Galerkin and Petrov-Galerkin Projections Petrov-Galerkin Projection Recall the residual introduced by approximating w(t) as Vq(t) V d q(t) = f(vq(t), t) + r(t) dt Constrain this residual to be orthogonal to a subspace W defined by a test basis W R N k that is, define q(t) such that W T r(t) = Left-multiply the above equations by W T : This leads to the Petrov-Galerkin projection-based equations W T V d dt q(t) = WT f(vq(t), t) + 23 / 38

24 Galerkin and Petrov-Galerkin Projections Petrov-Galerkin Projection Assume that W T V is non-singular: In this case, the reduced-order equations are d dt q(t) = (WT V) 1 W T f(vq(t), t) y(t) = g(vq(t), t) After the above reduced-order equations have been solved, the subspace approximation of the high-dimensional state vector can be reconstructed, if needed, as follows w(t) = Vq(t) 24 / 38

25 Galerkin and Petrov-Galerkin Projections Galerkin Projection If W = V, the projection method is called a Galerkin projection If in addition V is orthogonal, the reduced-order equations become d dt q(t) = VT f(vq(t), t) y(t) = g(vq(t), t) 25 / 38

26 Galerkin and Petrov-Galerkin Projections Linear Time-Invariant Systems Special case: Linear Time-Invariant (LTI) systems f(w(t), t) = Aw(t) + Bu(t) g(w(t), t) = Cw(t) + Du(t) u R p : Vector of input variables corresponding ROM d dt q(t) = (WT V) 1 W T (AVq(t) + Bu(t)) y(t) = CVq(t) + Du(t) reduced-order LTI operators A r = (W T V) 1 W T AV R k k, k N B r = (W T V) 1 W T B R k p C r = CV R q k D r = D R q p 26 / 38

27 Galerkin and Petrov-Galerkin Projections Initial Condition High-dimensional initial condition w() = w R N Reduced-order initial condition q() = (W T V) 1 W T w R k in the high-dimensional state space, this gives Vq() = V(W T V) 1 W T w R k this is an oblique projection of w onto range(v) parallel to range(w) Error in the subspace approximation of the initial condition w() Vq() = (I N V(W T V) 1 W T )w Alternative: use an affine approximation w(t) = w() + Vq(t) (see Homework 1) 27 / 38

28 Equivalent High-Dimensional Model Question: Which HDM would produce the same solution as that given by the ROM? recall the reduced-order equations d dt q(t) = (WT V) 1 W T f(vq(t), t) y(t) = g(vq(t), t) the corresponding reconstructed high-dimensional state solution is w(t) = Vq(t) pre-multiplying the above reduced-order system by V leads to d dt w(t) = V(WT V) 1 W T f( w(t), t) ỹ(t) = g( w(t), t) the associated initial condition is w() = Vq() = V(W T V) 1 W T w() 28 / 38

29 Equivalent High-Dimensional Model Recall the projector Π V,W Π V,W = V(W T V) 1 W T Definition Equivalent HDM with the initial condition d dt w(t) = Π V,Wf( w(t), t) ỹ(t) = g( w(t), t) w() = Π V,W w() The equivalent dynamical function is f(, ) = Π V,W f(, ) 29 / 38

30 Equivalent High-Dimensional Model Equivalence Between Two Reduced-Order Models Consider the Petrov-Galerkin-Based ROM d dt q(t) = (WT V) 1 W T f(vq(t), t) y(t) = g(vq(t), t) q() = (W T V) 1 W T w() Lemma Choosing two different bases V and W that respectively span the same subspaces V and W results in the same reconstructed solution w(t) In other words, subspaces are more important than bases... 3 / 38

31 Equivalent High-Dimensional Model Equivalence Between Two Reduced-Order Models Consequences given an HDM, a corresponding ROM is uniquely defined by its associated Petrov-Galerkin projector Π V,W this projector is itself uniquely defined by the two subspaces hence W = range(w) and V = range(v) ROM (W, V) W and V belong to the Grassmann manifold G(k, N), which is the set of all subspaces of dimension k in R N 31 / 38

32 Error Analysis Definition Question: Can we characterize the error of the solution computed using the ROM relative to the solution obtained using the HDM? E ROM (t) = w(t) w(t) = w(t) Vq(t) assume here a Galerkin projection and an associated orthogonal basis V T V = I k projector Π V,V = VV T the error vector can be decomposed into two orthogonal components E ROM (t) = w(t) Π V,V w(t) + Π V,V w(t) Vq(t) ( ) = (I N Π V,V ) w(t) + V V T w(t) q(t) = E V (t) + E V (t) 32 / 38

33 Error Analysis Orthogonal Components of the Error Vector Error component orthogonal to V E V (t) = (I N Π V,V ) w(t) Interpretation: The exact trajectory does not strictly belong to V = range(v) Error component parallel to V E V (t) = V ( V T w(t) q(t) ) Interpretation: An equivalent but different dynamical system is solved Note that E V (t) can be computed without executing the ROM and therefore can provide an a priori error estimate 33 / 38

34 Error Analysis Orthogonal Components of the Error Vector V? w(t) (I N V,V )w(t) w(t) Vq(t) V V(V T w(t) Vq(t) q(t)) 34 / 38

35 Error Analysis Orthogonal Components of the Error Vector Adapted from A New Look at Proper Orthogonal Decomposition, Rathiman and Petzold., SIAM Journal of Numerical Analysis, Vol. 41, No. 5, / 38

36 Error Analysis Orthogonal Components of the Error Vector Again, consider the case of an orthogonal Galerkin projection One can derive an ODE associated with the error component lying in V in terms of that lying in V de V dt (t) = Π V,V (f(w(t), t) f(w(t) E V (t) E V (t), t)) with the initial condition E V () = In the case of an autonomous linear system the ODE has the simple form dw (t) = Aw(t) dt de V dt (t) = Π V,VAE V (t) + Π V,V AE V (t) where E V (t) acts as a forcing term 36 / 38

37 Error Analysis Orthogonal Components of the Error Vector Theorem One can then derive the following error bound E ROM (t) ( F (T, V T AV) 2 V T AV ) E V (t) where denotes the L 2 ([, T ], R N ) function norm, T f 2 = f (τ) 2 2dτ, and F (T, M) denotes the linear operator defined by F (T, M) : L 2 ([, T ], R N ) L 2 ([, T ], R N ) ( t ) u t e M(t τ) u(τ)dτ Error bounds for the nonlinear case can be found in A New Look at Proper Orthogonal Decomposition, Rathiman and Petzold., SIAM Journal of Numerical Analysis, Vol. 41, No. 5, / 38

38 Error Analysis Preservation of Model Stability If A is symmetric and the projection is an orthogonal Galerkin projection, the stability of the HDM is preserved during the reduction process (Hint: Consider the equivalent HDM and analyze the sign of d dt ( wt w)) However, if A is not symmetric, the stability of the HDM is not preserved: For example, consider a linear HDM characterized by the following unsymmetric matrix A = [ the eigenvalues of A are {.1127,.8873} (stable model) consider next the reduced-order basis V [ ] 1 V = A r = [1], and therefore the ROM obtained using Galerkin projection, is not asymptotically stable ] 38 / 38