Model Reduction for Dynamical Systems

Transcription

1 MAX PLANCK INSTITUTE Otto-von-Guericke Universitaet Magdeburg Faculty of Mathematics Summer term 2015 Model Reduction for Dynamical Systems Lecture 10 Peter Benner Lihong Feng Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Magdeburg, Germany FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 1/31

2 Contents 1 Linear parametric systems 2 PMOR based on Multi-moment matching 3 A Robust Algorithm 4 IRKA based PMOR 5 Steady systems 6 Extension to nonlinearities Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 2/31

3 Example 1 A microthruster Upper-left 1 : the structure of an array of pyrotechnical thrusters. Lower-right: the structure of a 2D-axisymmetric model. A model of the microthruster unit. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 3/31

4 Example 1 When the PolySilicon (green) in the middle is excited by a current, the fuel below is ignited and the explosion will occur through the nozzle. The thermal process can be modeled by a heat transfer partial differential equation, while the heat exchange through device interfaces is modeled by convection boundary conditions with different film coefficients h t, h s, h b. The film coefficients h t, h s, h b respectively describe the heat exchange on the top, side, and bottom of the microthruster with the outside surroundings. The values of the film coefficients can change from 1 to 10 9 Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 4/31

5 Example 1 After finite element discretization of the 2D-axisymmetric model, a parameterized system is derived, Eẋ = (A h t A t h s A s h b A b )x + B y = Cx. (1) Here, h t, h s, h b are the parameters and the dimension of the system is n = 4, 257. We observe the temperature at the center of the PolySilicon heater changing with time and the film coefficient, which defines the output of the system 2. 2 Detailed description of the parameterized system can be find at Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 5/31

6 Example 2 The second example is a butterfly gyroscope. The parameterized system is obtained by finite element discretization of the model for the gyroscope (The details of the model can be found in [Moosmann07]). Scheme of the butterfly gyroscope [Moosmann07]. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 6/31

7 Example 2 The paddles of the device are excited to a vibration z(t), where all paddles vibrate in phase. With the external rotation φ, the Coriolis force acts upon the paddles, which causes an out-of-phase movement measured as the z-displacement difference δz between the two red dotted nodes. The interesting output of the system is δz, the difference of the displacement z(t) between the two end nodes depicted as red dots on the same side of the bearing. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 7/31

8 Example 2 The system is of the following form: M(d)ẍ + D(θ, α, β, d)ẋ + T (d)x = Bu(t) y = Cx. (2) M(d) = (M 1 + dm 2 ), D(θ, α, β, d) = θ(d 1 + dd 2 ) + αm(d) + βt (d), and T (d) = (T d T 2 + dt 3 ). Parameters d, θ, α, β. d is the width of the bearing, and θ is the rotation velocity along the x axis. α, β are used to form the Rayleigh damping matrices αm(d), βt (d) in D(θ, α, β, d). The dimension of the system is n = Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 8/31

9 Example 3 The third example is a silicon-nitride membrane 3. This structure resembles a micro-hotplate similar to other micro-fabricated devices such as gas sensors [GrafBT04] and infrared sources [SpannSH05]. Temperature distribution over the silicon-nitride membrane. 3 Picture courtesy of T. Bechtold, IMTEK, University of Freiburg, Germany. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 9/31

10 Example 3 The model of the silicon-nitride membrane is a system with four parameters [BechtoldHRG10]. (E 0 + ρc p E 1 )ẋ + (K 0 + κk 1 + hk 2 )x = Bu(t) y = Cx. (3) The mass density ρ in kg/m 3, the specific heat capacity c p in J/kg/K, the thermal conductivity in W/m/K, and the heat transfer coefficient h in W/m 2 /K. The dimension of the system is n = Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 10/31

11 PMOR based on Multi-moment matching In frequency domain Using Laplace transform, the system in time domain is transformed into E(s 1,..., s p )x = Bu(s p ), y = L T x, (4) where the matrix E R n n is parametrized. The new parameter s p is in fact the frequency parameter s, which corresponds to time t. In case of a nonlinear and/or non-affine dependence of the matrix E on the parameters, the system in (4) is first transformed to an affine form (E 0 + s 1 E 1 + s 2 E s p E p )x = Bu(s p ), y = L T x. (5) Here the newly defined parameters s i, i = 1,..., p, might be some functions (rational, polynomial) of the original parameters s i in (4). Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 11/31

12 PMOR based on multi-moment matching To obtain the projection matrix V for the reduced model, the state x in (5) is expanded into a Taylor series at an expansion point s 0 = ( s 0 1,..., s0 p) T as below, x = [I (σ 1 M σ p M p )] 1 Ẽ 1 Bu(s p ) = [σ 1 M σ p M p ] m Ẽ 1 Bu(s p ) m=0 = m=0 m (k k p) k 2=0... m k p k p 1=0 F m k 2,...,k p (M 1,..., M p ) (6) where σ i = s i s 0 i, Ẽ = E 0 + s 0 1 E s 0 pe p, M i = Ẽ 1 E i, i = 1, 2,... p, and B M = Ẽ 1 B. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 12/31

13 PMOR based on multi-moment-matching σ 0 : L T B M : the 0th order multi-moment; the columns in B M : the 0th order moment vectors. σ 1 : L T M i B M, i = 1, 2,..., p: the first order multi-moments; the columns in M i B M, i = 1, 2,..., p: the first order moment vectors. σ 2 :... ; the columns in M 2 i B M, i = 1, 2,..., p, (M 1 M i + M i M 1 )B M, i = 2,..., p, (M 2 M i + M i M 2 )B M, i = 3,..., p,..., (M p 1 M p + M p M p 1 )B M : the second order moment vectors..... Since the coefficients corresponding not only to s = s p, but also to those associated with the other parameters s i, i = 1,..., p 1 are, we call them as multi-moments of the transfer function. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 13/31

14 PMOR based on multi-moment-matching For the general case, the projection matrix V is constructed as range {V } = colspan{ m q m=0 m (k p+...+k 3) k 2=0... m k p m k p 1=0 k p=0 F m k 2,...,k p (M 1,..., M p )B M } = colspan{b M, M 1 B M, M 2 B M,..., M p B M, (M 1 ) 2 B M, (M 1 M 2 + M 2 M 1 )B M,..., (M 1 M p + M p M 1 )B M, (M 2 ) 2 B M, (M 2 M 3 + M 3 M 2 )B M,...}. (7) Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 14/31

15 A Robust Algorithm Taking a closer look at the power series expansion of x in (6), we get the following equivalent, but different formulation, x = [I (σ 1 M σ p M p )] 1 Ẽ 1 Bu = [σ 1 M σ p M p ] m B M u m=0 = B M u + [σ 1 M σ p M p ]B M u +[σ 1 M σ p M p ] 2 B M u [σ 1 M σ p M p ] j B M u +... (8) By defining x 0 = B M, x 1 = [σ 1 M σ p M p ]B M, x 2 = [σ 1 M σ p M p ] 2 B M,..., x j = [σ 1 M σ p M p ] j B M,..., we have x = (x 0 + x 1 + x x j + )u and obtain the recursive relations Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 15/31

16 A Robust Algorithm x 0 = B M, x 1 = [σ 1 M σ p M p ]x 0, x 2 = [σ 1 M σ p M p ]x 1,... x j = [σ 1 M σ p M p ]x j 1,.... If we define a vector sequence based on the coefficient matrices of x j, j = 0, 1,... as below, R 0 = B M, R 1 = [M 1 R 0, M 2 R 0,..., M p R 0 ], R 2 = [M 1 R 1, M 2 R 1,..., M p R 1 ],. R j = [M 1 R j 1, M 2 R j 1,..., M p R j 1 ], (9). Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 16/31

17 A Robust Algorithm and let R be the subspace spanned by the vectors in R j, j = 0, 1,, m: R = colspan{r 0,..., R j,..., R m }, then there exists z R q, such that x Vz. Here the columns in V R n q is a basis of R. We see that the terms in R j, j = 0, 1,..., m are the coefficients of the parameters in the series expansion (8). They are also the j-th order moment vectors. How to compute an orthonormal basis V? Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 17/31

18 Algorithm 1: Compute V = [v 1, v 2,..., v q ] [Benner, Feng 14] Initialize a 1 = 0, a 2 = 0, sum = 0. Compute R 0 = Ẽ 1 B. if (multiple input) then Orthogonalize the columns in R 0 using MGS: [v 1, v 2,..., v q1 ] = orth{r 0 } with respect to a user given tolerance ε > 0 specifying the deflation criterion for numerically linearly dependent vectors. sum = q 1 % q 1 is the number of columns remaining after deflation w.r.t. ε.) else Compute the first column in V: v 1 = R 0 / R 0 2 sum = 1 end if Compute the orthonormal columns in R 1, R 2,..., R m iteratively as below: Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 18/31

19 continued for i = 1, 2,..., m do a 2 = sum; for t = 1, 2,..., p do if a 1 = a 2 then stop else for j = a 1 + 1,... a 2 do w = Ẽ 1 E t v j ; col = sum + 1; for k = 1, 2,..., col 1 do h = vk T w; w = w hv k end for if w 2 > ε then v col = w w 2 ; sum = col; end if end for end if end for a 1 = a 2 ; end for Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 19/31

20 Adaptively select expansion points Let µ = ( s 1,..., s p ), (µ) is an error estimation, or error bound for ˆx/ŷ, the state/output of the system computed from ROM. Greedy algorithm: Adaptive selection of the expansion points µ i V = []; ɛ = 1; Initial expansion point: µ 0 ; i = 1; Ξ train : a large set of the samples of µ WHILE ɛ > ɛ tol i=i+1; µ i = ˆµ; Use Algorithm 1 to compute V i = span{r 0,..., R q } µ i ; V = [V, V i ]; ˆµ = arg max µ Ξ train (µ); ɛ = (ˆµ); END WHILE. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 20/31

21 Experimental results Example 1: A MEMS model with 4 parameters (benchmark available at M(d)ẍ + D(θ, α, β, d)ẋ + T (d)x = Bu(t), y = Cx. Here, M(d) = (M 1 + dm 2 ), T (d) = (T d T 2 + dt 3 ), D(θ, α, β, d) = θ(d 1 + dd 2 ) + αm(d) + βt (d) R n n, n=17,913. Parameters, d, θ, α, β. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 21/31

22 θ [10 7, 10 5 ], s 2π 1 [0.05, 0.25], d [1, 2]. Ξ train : 3 random θ, 10 random s, 5 random d, α = 0, β = 0 [Salimbahrami et al. 06]. Totally 150 samples of µ (µ i ) ε max true ε tol = (µ i ) ε max true ith iteration step ith iteration step V µ i = span{b M, R 1, R 2} µ i, i = 1,..., 33. ɛ tol = 10 7, ε max true = max H(µ) Ĥ(µ), ROM size=804. µ Ξ train Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 22/31

23 (µ i ) ε max true ε tol = (µ i ) ε max true ith iteration step ith iteration step V µ i = span{b M, R 1} µ i, i = 1,..., 36. ɛ tol = 10 7, ROM size=210. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 23/31

24 Example 2: a silicon nitride membrane (E 0 + ρc p E 1 )dx/dt + (K 0 + κk 1 + hk 2 )x = bu(t) y = Cx. Here, the parameters ρ [3000, 3200], c p [400, 750], κ [2.5, 4], h [10, 12], f [0, 25]Hz Ξ train : 2250 random samples have been taken for the four parameters and the frequency. ε re true = max H(µ) Ĥ(µ) / H(µ), ˆ re (µ) = ˆ (µ)/ Ĥ(µ) µ Ξ train V µ i =span{b M,R 1 }, ɛ re tol = 10 2, n = 60, 020, r = 8, iteration ε re true ˆ re (µ i ) Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 24/31

25 Ξtrain : 3 samples for κ, 10 samples for the frequency. Ξvar : 16 samples for κ, 51 samples for the frequency κ Frequency (Hz) Relative error of the final ROM over Ξvar. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 25/31

26 IRKA based PMOR Consider a linear parametric system C(p 1, p 2,, p l ) dx dt = G(p 1, p 2,, p l )x + B(p 1, p 2,, p l )u(t), y(t) = L(p 1, p 2,, p l ) T x, (10) where the system matrices C(p 1, p 2,, p l ), G(p 1, p 2,, p l ), B(p 1, p 2,, p l ), LT (p 1, p 2,, p l ), are (maybe, nonlinear, non-affine) functions of the parameters p 1, p 2, p l. A straight forward way is [Baur, et.al 11]: Set a group of samples of µ = (p 1,..., p l ): µ 0,..., µ l. For each sample µ i = (p1 i,..., pi l ), i = 1..., l, implement IRKA to get a projection matrix, W i, V i. The final projection matrix: range(v ) = orth(v 1,..., V l ), range(w ) = orth(w 1,..., W l ), W = W (V T W ) 1. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 26/31

27 IRKA based PMOR The reduced parametric model is: Parametric ROM W T C(p 1, p 2,, p l )V dx dt = W T G(p 1, p 2,, p l )Vx +W T B(p 1, p 2,, p l )u(t), y(t) = L(p 1, p 2,, p l ) T Vx, Question: How to select the samples of µ? Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 27/31

28 How to deal with nonaffine matrices? Nonafine matrices are those matrices that cannot be written as: E(p 1,..., p l ) = E 0 + p 1 E , p l E l. PMOR based on multi-moment-matching cannot directly deal with nonaffine case. We must first approximate with affine matrices. IRKA can deal with nonaffine matrices directly. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 28/31

29 Why and How MOR for Steady systems? Steady parametric systems E(p 1,..., p l )x = B(p 1,..., p l ) Solving steady systems for multi-query tasks is also time-consuming. Application of PMOR based on multi-moment-matching to steady systems is straight forward. IRKA cannot. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 29/31

30 Applicable to nonlinear parametric systems? Nonlinear parametric systems: or f (µ, x) = b(µ), E(µ) dx dt = A(µ)x + f (µ, x) = B(µ)u(t), y(t) = L(µ) T x, µ = (p 1,..., p m ), x = x(µ, t). PMOR could be extended to solve weakly nonlinear parametric systems. IRKA can only deal with linear parametric systems. Good candidates for MOR of general nonlinear parametric systems are POD and reduced basis methods. To be introduced: POD and reduced basis method for linear and nonlinear parametric systems. Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 30/31

31 References 1 Daniel, L., Siong, O., Chay, L., Lee, K., White, J.: A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 22(5), (2004). 2 Peter Benner and Lihong Feng: A robust algorithm for parametric model order reduction based on implicit moment-matching. In Reduced Order Methods for Modeling and Computational Reduction. A. Quarteroni, G. Rozza (editors), 9: , Series MS &A, 2014, Springer. 3 U. Baur and C. Beattie and P. Benner and S. Gugercin: Interpolatory Projection Methods for Parameterized Model Reduction. SIAM Journal on Scientific Computing, 33: , And many more... Max Planck Institute Magdeburg Peter Benner Lihong Feng, Model Reduction for Dynamical Systems 31/31