Parameterized Partial Differential Equations and the Proper Orthogonal D

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1 Parameterized Partial Differential Equations and the Proper Orthogonal Decomposition Stanford University February 04, 2014

2 Outline Parameterized PDEs The steady case Dimensionality reduction Proper orthogonal decomposition Projection-based model reduction Snapshot selection The unsteady case

3 Parameterized PDE Parametrized partial differential equation (PDE) L(W, x, t; µ) = 0 Associated boundary conditions B(W, x BC, t; µ) = 0 Initial condition W 0 (x) = W IC (x, µ) W = W(x, t) R q : state variable x Ω R d, d 3: space variable t 0: time variable µ D R p : parameter vector

4 Model parameterized PDE Advection-diffusion-reaction equation: W = W(x, t; µ) solution of W t + U W κ W = f R(W, t, µ R ) for x Ω with appropriate boundary and initial conditions Parameters of interest W(x, t; µ) = W D (x, t; µ D ) for x Γ D W(x, t; µ) n(x) = 0 for x Γ N W(x, 0; µ) = W 0 (x; µ IC ) for x Ω µ = [U 1,, U d, κ, µ R, µ D, µ IC ]

5 Semi-discretized problem The PDE is then discretized in space by one of the following methods Finite Differences approximation Finite Element method Finite Volumes method Discontinuous Galerkin method Spectral method... This leads to a system of N w = q N space ordinary differential equations (ODEs) dw = f(w, t; µ) dt in terms of the discretized state variable with initial condition w(0; µ) = w(µ) w = w(t; µ) R Nw This is the high-dimensional model (HDM)

6 Parameterized solutions Example: two dimensional advection-diffusion equation W + U W κ W = 0 for x Ω t with boundary and initial conditions W(x, t; µ) = W D (x, t; µ D ) for x Γ D W(x, t; µ) n(x) = 0 for x Γ N W(x, 0; µ) = W 0 (x) for x Ω Γ N Γ D U Ω Γ N Γ N

7 Parameterized solutions Example: two dimensional advection-diffusion equation W t with boundary and initial conditions 4 parameters of interest w R Nw with N w = 2, U W κ W = 0 for x Ω W(x, t; µ) = W D (x, t; µ D ) for x Γ D W(x, t; µ) n(x) = 0 for x Γ N W(x, 0; µ) = W 0 (x) for x Ω µ = [U 1, U 2, κ, µ D ] R 4

8 Parameterized solutions

9 Steady parameterized PDE Steady parameterized HDM f(w; µ) = 0 Linear case A(µ)w = b(µ) Example: steady advection-diffusion equation

10 Dimensionality reduction Consider the manifold of solutions M = {w(µ) s.t. µ D} R Nw Often dim(m) N w Therefore, M could be described in terms of a much smaller set of variables, rather than {e 1,, e Nw } Hence dimensionality reduction

11 Dimensionality reduction First idea: use solutions of the equation to describe M Consider pre-computed solution {w(µ 1 ),, w(µ m )} where {µ 1,, µ m } D Let µ D. Then approximate w(µ) as w(µ) α 1 (µ)w(µ 1 ) + + α m (µ)w(µ m ) where {α 1 (µ),, α m (µ)} are coefficients to be determined

12 Reduced-order basis There may be redundancies in the solutions {w(µ 1 ),, w(µ m )}. Better approach: remove the redundancies by considering an equivalent independent set {v 1,, v k } with k m such that span {v 1,, v k } = span {w(µ 1 ),, w(µ m )} V = [v 1,, v k ] R Nw k is a reduced-order basis with k N w

13 Basis construction Lagrange basis Hermite basis span {v 1,, v k } = span {w(µ 1 ),, w(µ m )} span (v 1,, v k )} = span { w(µ 1 ), w (µ 1 ),, w } (µ 1 ), w(µ 2 ), µ 1 µ p

14 Data compression It is possible to remove more information from the snapshots {w(µ 1 ),, w(µ m )} Consider the snapshot matrix W = [w(µ 1 ),, w(µ m )] Can we quantify the main information contained in W and discard the rest (noise)? This amount to data compression Here orthogonal projection will be used to compress the data

15 Orthogonal projection Let V R Nw k be an orthogonal matrix (V T V = I k ) which columns span S, a subspace of dimension k Let x R Nw. The orthogonal projection of x onto the subspace S is VV T x Projection matrix Π V,V = V(V T V) 1 V T = VV T special case 1: if x belongs to S Π V,V x = VV T x = x special case 2: if x is orthogonal to S Π V,V x = VV T x = 0

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17 Proper Orthogonal Decomposition POD seeks the subspace S of a given dimension k minimizing the projection error of the snapshots {w(µ 1 ),, w(µ m )} Mathematical formulation S = range(v) where V = argmin Y m w(µ i ) Π Y,Y w(µ i ) 2 2 i=1 s.t. Y T Y = I k

18 POD by eigenvalue decomposition Minimization problem V = argmin Y T Y=I k m w(µ i ) Π Y,Y w(µ i ) 2 2 i=1 Equivalent maximization problem V = argmax Y T Y=I k m YY T w(µ i ) 2 2 i=1 = argmax Y T Y=I k Y T W 2 F = argmax Y T Y=I k trace ( Y T WW T Y ) Solution: V is the matrix of eigenvectors {φ 1,, φ k } associated with the k largest eigenvalues of K = WW T

19 The method of snapshots POD: V is the matrix of eigenvectors {φ 1,, φ k } associated with the k largest eigenvalues of K = WW T K R Nw Nw is a large, dense matrix Its rank is at most m N w In 1987, Sirovich developed the method of snapshots by noticing that R = W T W R m m has the same non-zero eigenvalues {λ i } r i=1 as K r = rank(r) m N w and Rψ i = λψ i, i = 1,, r Exercise: relationship between {φ 1,, φ r } and {ψ 1,, ψ r }?

20 The method of snapshots Step 1: compute the eigenpairs {λ i, ψ i } r i=1 associated with R Rψ i = λψ i, i = 1,, r Step 2: compute φ i = 1 λi Wψ i, i = 1,, r In matrix form Φ = WΨΛ 1 2 POD reduced basis of dimension k r: V = [φ 1,, φ k ]

21 POD by singular value decomposition The POD basis V can also be computed by singular value decomposition (SVD) SVD of W: W = U r Σ r Z T r U r = [u 1,, u r] R Nw r : left singular vectors (U T r U r = I r) Σ r = diag(σ 1,, σ r) R r r : singular values Z r = [z 1,, z r] R m r : right singular vectors (Z T r Z r = I r) POD reduced basis of dimension k r V = [u 1,, u k ]

22 POD basis size selection The POD basis V can also be computed by singular value decomposition (SVD) SVD of W: W = U r Σ r Z T r POD reduced basis of dimension k r Relative projection error V k = [u 1,, u k ] e(k) = W V kv T k W F W F = Typically k is chosen so that e(k) < 0.1 r i=k+1 σ2 i r i=1 σ2 i

23 Projection-based model reduction High-dimensional model (HDM) A(µ)w = b(µ) Reduced-order modeling assumption using a reduced basis V w(µ) Vq(µ) q(µ): reduced (generalized) coordinates Inserting in the HDM equation AVq b N w equations in terms of k unknowns q Associated residual r(q) = A(µ)Vq b(µ)

24 Galerkin projection Residual equation r(q) = A(µ)Vq b(µ) N w equations with k unknowns Galerkin projection enforces the orthogonality of r(q) to range(v): V T r(q) = 0 Reduced equations: V T A(µ)Vq = V T b(µ) k equations in terms of k unknowns A k (µ)q = b k (µ)

25 Application to the steady advection diffusion equation m = 5 snapshots {µ i } 5 i=1

26 Application to the steady advection diffusion equation m = 5 snapshots {µ i } 5 i=1 POD basis of dimension k = 5

27 Application to the steady advection diffusion equation Projection error e(k) k

28 Application to the steady advection diffusion equation µ 5 = (U 1, U 2, κ, µ D ) = (1, 10, , 850)

29 Exercise: Petrov-Galerkin projection Other approach to get a unique solution to Least-squares approach A(µ)Vq b(µ) q = argmin A(µ)Vy b(µ) 2 y Exercise: give the equivalent set of equations satisfied by y Solution: V T A(µ) T A(µ)Vq = V T A(µ) T b(µ)

30 Offline/online decomposition for parametric systems Offline phase: computation of V from snapshots {w(µ 1 ),, w(µ m )} Online phase: construction and solution of V T A(µ)Vq = V T b(µ) Issue: constructing A k (µ) = V T A(µ)V and b k (µ) = V T b(µ) is expensive Exception in the case of affine parameter dependence: q A N w and q b N w Then A k (µ) = A(µ) = q A i=1 q A i=1 f (i) A (µ)a(i), b(µ) = f (i) A (µ)vt A (i) V, b(µ) = q b i=1 q b i=1 f (i) b (µ)b (i) f (i) b (µ)v T b (i) The following small dimensional matrices can be computed offline A (i) k b (i) k = VT A (i) V R k k, i = 1,, q A = VT b (i) R k, i = 1,, q b

31 Snapshot selection For a given number of snapshots m, what are the best snapshot parameter locations µ 1,, µ m D? The snapshots {w(µ 1 ),, w(µ m )} should be optimally placed so that they capture the physics over the parameter space D Difficult problem use a heuristic approach (Greedy algorithm)

32 Greedy approach Start by randomly selecting a parameter value µ 1 and compute w(µ 1 ) For i = 1, m, find the parameter µ i which presents the highest error between the ROM solution Vq(µ) and the HDM solution w(µ) This however requires knowing the HDM solution (unknown) Instead, for i = 1, m, find the parameter µ i for which the residual r(q(µ)) = AVq(µ) b(µ) is the highest where V T A(µ)Vq(µ) = V T b(µ) µ i = argmin A(µ)Vq(µ) b(µ) 2 µ D The parameter domain µ D can be in practice replaced by a search over a finite set µ {µ (1),, µ (N) } D

33 Application to the steady advection diffusion equation Parameter domain (U 1, U 2 ) [0, 10] [0, 10] U 2 6 initial U 1

34 Application to the steady advection diffusion equation

35 Application to the steady advection diffusion equation Maximum Residual Maximum Error Normalized quantity Iteration 6 7 8

36 Application to the steady advection diffusion equation Normalized quantity Maximum Residual Maximum Error Iteration

37 POD in time Consider an unsteady linear parametric problem dw (t) = A(µ)w(t) b(µ)u(t) dt where u(t) is given for a time interval t [t 0, t Nt ] For a given parameter µ, POD can also provide an optimal reduced basis associated with the minimization problem min V T V=I k tnt t 0 w(t, µ) VV T w(t, µ) 2 2 dt Solution by the method of snapshots: consider the solutions in time w(t 0, µ),, w(t Nt, µ). An approximation of the minimization problem is min V T V=I k N t i=0 δ i w(ti, µ) VV T w(t i, µ) 2 2 where δ 0,, δ Nt are appropriate quadrature weights

38 POD in time (continued) Equivalent maximization problem max V T V=I k N t i=0 δ i V T w(t i, µ) 2 2 Solution by POD by the method of snapshots with the associated snapshot matrix W = [ δ 0 w(t 0, µ),, δ Nt w(t Nt, µ)]

39 POD: Application to linearized aeroelasticity M = 0.99 N w = 2, 188, 394, m = 99 snapshots, k = 60 retained POD vectors Lift (lb)! 6000! 5000! 4000! 3000! 2000! 1000! 0! -1000! -2000! 0.00! 0.10! 0.20! 0.30! 0.40! 0.50! Time (s)! HDM (2,188,394)! ROM (69)!

40 Global vs. local strategies Global strategy build a ROB V that captures the behavior of unsteady systems for all µ D POD based on snapshots {w(t 0, µ 1),, w(t Nt, µ 1), w(t 0, µ 2),, w(t Nt, µ m)} not optimal for a given µ i Local strategy build a separate ROB V(µ) for every value of µ D database approach: build offline a set of ROBs {V(µ i)} m i=1 and use them online to build V(µ) for µ D each ROB V(µ i) is optimal at µ = µ i requires an adaptation approach online (see Lecture 3)

41 References Lecture notes on projection-based model reduction: amsallem/ca-cme345-ch3.pdf Lecture notes on POD: amsallem/ca-cme345-ch4.pdf Holmes, P., Lumley, J., and Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge Univ. Press, Cambridge, England, U.K., Sirovich, L.: Turbulence and the dynamics of coherent structures. Part I: coherent structures. Quarterly of applied mathematics 45(3), (1987). Patera A. and Rozza G., Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations, Version 1.0, Copyright MIT 2006, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering. Haasdonk, B., Ohlberger, M.: Efficient reduced models and a-posteriori error estimation for parametrized dynamical systems by offline/online decomposition. Stuttgart Research Centre for Simulation Technology (SRC SimTech) Preprint Series 23 (2009)