Fast Frequency Response Analysis using Model Order Reduction

Transcription

1 Fast Frequency Response Analysis using Model Order Reduction Peter Benner EU MORNET Exploratory Workshop Applications of Model Order Reduction Methods in Industrial Research and Development Luxembourg, November 6, 2015

2 Outline 1. Introduction 2. Model Reduction for Dynamical Systems 3. Balanced Truncation for Linear Systems 4. Interpolatory Model Reduction 5. Parametric Model Order Reduction (PMOR) 6. Conclusions c P. Benner Fast Frequency Response Analysis via MOR 2/24

3 Introduction Frequency Response Analysis Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system c P. Benner Fast Frequency Response Analysis via MOR 3/24

4 Introduction Frequency Response Analysis Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. 1 It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input c P. Benner Fast Frequency Response Analysis via MOR 3/24

5 Introduction Frequency Response Analysis Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. 1 It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. 1 Main tool in engineering to study the dynamic behavior of a system. 1 c P. Benner Fast Frequency Response Analysis via MOR 3/24

6 Introduction Frequency Response Analysis Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. 1 It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. 1 Main tool in engineering to study the dynamic behavior of a system. It is based on the Laplace/Fourier transforms, mapping a time-domain signal to frequency domain. 1 c P. Benner Fast Frequency Response Analysis via MOR 3/24

7 Introduction Frequency Response Analysis Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. 1 It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. 1 Main tool in engineering to study the dynamic behavior of a system. It is based on the Laplace/Fourier transforms, mapping a time-domain signal to frequency domain. { ẋ(t) = Ax(t) + Bu(t), For linear time-invariant (LTI) system Σ : y(t) = Cx(t) + Du(t), this requires the solution of a sequence of linear systems H(:, :, k) = C(jω k I n A) 1 B + D, k = 1,... K, where {ω 1,..., ω K } defines a frequency grid (in [rad/s]). 1 c P. Benner Fast Frequency Response Analysis via MOR 3/24

8 Introduction Frequency Response Analysis Linear time-invariant (LTI) system Σ : { ẋ(t) = Ax(t) + Bu(t), A R n n, B R n m Laplace (Fourier) transform y(t) = Cx(t) + Du(t), C R q n, D R q m Y (s) = ( C(sI n A) 1 B + D ) U(s) =: G(s)U(s), s C, where G R(s) q m is the (rational) transfer function of Σ. c P. Benner Fast Frequency Response Analysis via MOR 4/24

9 Introduction Frequency Response Analysis Linear time-invariant (LTI) system Σ : { ẋ(t) = Ax(t) + Bu(t), A R n n, B R n m Laplace (Fourier) transform y(t) = Cx(t) + Du(t), C R q n, D R q m Y (s) = ( C(sI n A) 1 B + D ) U(s) =: G(s)U(s), s C, where G R(s) q m is the (rational) transfer function of Σ. Visualization often by Bode (amplitude and phase), Bode magnitude (amplitude only), or sigma (only σ max (G(jω))) plots. c P. Benner Fast Frequency Response Analysis via MOR 4/24

10 Introduction Frequency Response Analysis Linear time-invariant (LTI) system Σ : { ẋ(t) = Ax(t) + Bu(t), A R n n, B R n m Laplace (Fourier) transform y(t) = Cx(t) + Du(t), C R q n, D R q m Y (s) = ( C(sI n A) 1 B + D ) U(s) =: G(s)U(s), s C, where G R(s) q m is the (rational) transfer function of Σ. Visualization often by Bode (amplitude and phase), Bode magnitude (amplitude only), or sigma (only σ max (G(jω))) plots. Requires K evaluations of the transfer function G(jω k ), k = 1,..., K, i.e., solution of K linear systems of equations with min{q, m} right-hand sides. c P. Benner Fast Frequency Response Analysis via MOR 4/24

11 Frequency Response Analysis Example: Beam Clamped beam (discretized elasticity equation). n = 348, m = q = 1, MATLAB R automatically chooses K = 389. Bode plot bode(sys) Source: The SLICOT Benchmark Collection for Model Reduction, c P. Benner Fast Frequency Response Analysis via MOR 5/24

12 Frequency Response Analysis Example: Beam Clamped beam (discretized elasticity equation). n = 348, m = q = 1, MATLAB R automatically chooses K = 389. Bode plot bodemag(sys) Source: The SLICOT Benchmark Collection for Model Reduction, c P. Benner Fast Frequency Response Analysis via MOR 5/24

13 Frequency Response Analysis Example: Beam Clamped beam (discretized elasticity equation). n = 348, m = q = 1, MATLAB R automatically chooses K = 389. Sigma plot sigma(sys) Source: The SLICOT Benchmark Collection for Model Reduction, c P. Benner Fast Frequency Response Analysis via MOR 5/24

14 Frequency Response Analysis Example: CD Player Modal model of a rotating swing arm in a CD player. n = 120, m = q = 2, MATLAB automatically chooses K = 445. Bode plot bode(sys) Source: The SLICOT Benchmark Collection for Model Reduction, c P. Benner Fast Frequency Response Analysis via MOR 6/24

15 Frequency Response Analysis Example: CD Player Modal model of a rotating swing arm in a CD player. n = 120, m = q = 2, MATLAB automatically chooses K = 445. Bode plot bodemag(sys) Source: The SLICOT Benchmark Collection for Model Reduction, c P. Benner Fast Frequency Response Analysis via MOR 6/24

16 Frequency Response Analysis Example: CD Player Modal model of a rotating swing arm in a CD player. n = 120, m = q = 2, MATLAB automatically chooses K = 445. Sigma plot sigma(sys) Source: The SLICOT Benchmark Collection for Model Reduction, c P. Benner Fast Frequency Response Analysis via MOR 6/24

17 Frequency Response Analysis Accelerating Frequency response calculations Use your Numerical Analysis... c P. Benner Fast Frequency Response Analysis via MOR 7/24

18 Frequency Response Analysis Accelerating Frequency response calculations Use your Numerical Analysis... Example (ISS Module 12A, structure model) n = 1412, m = 2, q = 3 MATLAB built-in command freqresp: Use sparse arithmetic: >> tic, h=freqresp(sys,[100]); toc Elapsed time is seconds. >> tic, hs=c*((100*i*speye(1412)-as)\b); toc Elapsed time is seconds. Note: solve (jωi A)X = B, rather than (jωi A) T Y T = C T as m < q. c P. Benner Fast Frequency Response Analysis via MOR 7/24

19 Frequency Response Analysis Accelerating Frequency response calculations Use your Numerical Analysis... Intelligent use of iterative methods, e.g., block-krylov methods, recycling Krylov subspaces, shift-invariance of Krylov subspaces,... [Frommer, de Sturler, Meerbergen, Morgan, Nabben Simoncini, Szyld, Vuik,... ] c P. Benner Fast Frequency Response Analysis via MOR 7/24

20 Frequency Response Analysis Accelerating Frequency response calculations Use your Numerical Analysis... Intelligent use of iterative methods, e.g., block-krylov methods, recycling Krylov subspaces, shift-invariance of Krylov subspaces,... [Frommer, de Sturler, Meerbergen, Morgan, Nabben Simoncini, Szyld, Vuik,... ] Here: employ model (order) reduction techniques! Replace A, B, C, D by (Â, ˆB, Ĉ, ˆD) R r r R r m R q r R q m with so that is small in desired frequency range! r n G(jω)U(jω) Ĝ(jω)U(jω) c P. Benner Fast Frequency Response Analysis via MOR 7/24

21 Model Reduction for Dynamical Systems Dynamical Systems { ẋ(t) = f (t, x(t), u(t)), x(t0 ) = x Σ : 0, y(t) = g(t, x(t), u(t)) with states x(t) R n, inputs u(t) R m, outputs y(t) R q. c P. Benner Fast Frequency Response Analysis via MOR 8/24

22 Model Reduction for Dynamical Systems Original System { ẋ(t) = f (t, x(t), u(t)), Σ : y(t) = g(t, x(t), u(t)). states x(t) R n, inputs u(t) R m, outputs y(t) R q. Goal: y ŷ < tolerance u for all admissible input signals. c P. Benner Fast Frequency Response Analysis via MOR 9/24

23 Model Reduction for Dynamical Systems Original System { ẋ(t) = f (t, x(t), u(t)), Σ : y(t) = g(t, x(t), u(t)). states x(t) R n, inputs u(t) R m, outputs y(t) R q. Reduced-Order Model ROM Σ : { ˆx(t) = f (t, ˆx(t), u(t)), ŷ(t) = ĝ(t, ˆx(t), u(t)). states ˆx(t) R r, r n inputs u(t) R m, outputs ŷ(t) R q. Goal: y ŷ < tolerance u for all admissible input signals. c P. Benner Fast Frequency Response Analysis via MOR 9/24

24 Model Reduction for Dynamical Systems Original System { ẋ(t) = f (t, x(t), u(t)), Σ : y(t) = g(t, x(t), u(t)). states x(t) R n, inputs u(t) R m, outputs y(t) R q. Reduced-Order Model ROM Σ : { ˆx(t) = f (t, ˆx(t), u(t)), ŷ(t) = ĝ(t, ˆx(t), u(t)). states ˆx(t) R r, r n inputs u(t) R m, outputs ŷ(t) R q. Goal: y ŷ < tolerance u for all admissible input signals. c P. Benner Fast Frequency Response Analysis via MOR 9/24

25 Model Reduction for Dynamical Systems Original System { ẋ(t) = f (t, x(t), u(t)), Σ : y(t) = g(t, x(t), u(t)). states x(t) R n, inputs u(t) R m, outputs y(t) R q. Reduced-Order Model ROM Σ : { ˆx(t) = f (t, ˆx(t), u(t)), ŷ(t) = ĝ(t, ˆx(t), u(t)). states ˆx(t) R r, r n inputs u(t) R m, outputs ŷ(t) R q. Goal: y ŷ < tolerance u for all admissible input signals. Secondary goal: reconstruct approximation of x from ˆx. c P. Benner Fast Frequency Response Analysis via MOR 9/24

26 Introduction Linear Systems Linear, Time-Invariant (LTI) Systems ẋ = f (t, x, u) = Ax + Bu, A R n n, B R n m, y = g(t, x, u) = Cx + Du, C R q n, D R q m. c P. Benner Fast Frequency Response Analysis via MOR 10/24

27 Introduction Linear Systems Formulating model reduction in frequency domain Approximate the dynamical system ẋ = Ax + Bu, A R n n, B R n m, y = Cx + Du, C R q n, D R q m, by reduced-order system ˆx = ˆx + ˆBu,  R r r, ˆB R r m, ŷ = Ĉ ˆx + ˆDu, Ĉ R q r, ˆD R q m of order r n, such that y ŷ = Gu Ĝu G Ĝ u < tolerance u. c P. Benner Fast Frequency Response Analysis via MOR 10/24

28 Introduction Linear Systems Formulating model reduction in frequency domain Approximate the dynamical system ẋ = Ax + Bu, A R n n, B R n m, y = Cx + Du, C R q n, D R q m, by reduced-order system ˆx = ˆx + ˆBu,  R r r, ˆB R r m, ŷ = Ĉ ˆx + ˆDu, Ĉ R q r, ˆD R q m of order r n, such that y ŷ = Gu Ĝu G Ĝ u < tolerance u. = Approximation problem: min G Ĝ. order (Ĝ) r c P. Benner Fast Frequency Response Analysis via MOR 10/24

29 Application Areas structural mechanics / (elastic) multibody simulation systems and control theory micro-electronics / circuit simulation / VLSI design computational electromagnetics, design of MEMS/NEMS (micro/nano-electrical-mechanical systems), computational acoustics,... Peter Benner and Lihong Feng. Model Order Reduction for Coupled Problems Applied and Computational Mathematics: An International Journal, 14(1):3 22, Available from c P. Benner Fast Frequency Response Analysis via MOR 11/24

30 Application Areas structural mechanics / (elastic) multibody simulation systems and control theory micro-electronics / circuit simulation / VLSI design computational electromagnetics, design of MEMS/NEMS (micro/nano-electrical-mechanical systems), computational acoustics,... Current trend: more and more multi-physics problems, i.e., coupling of several field equations, e.g., electro-thermal (e.g., bondwire heating in chip design), fluid-structure-interaction,... Peter Benner and Lihong Feng. Model Order Reduction for Coupled Problems Applied and Computational Mathematics: An International Journal, 14(1):3 22, Available from c P. Benner Fast Frequency Response Analysis via MOR 11/24

31 Balanced Truncation for Linear Systems Basic idea { ẋ(t) = Ax(t) + Bu(t), Σ : with A stable, i.e., Λ (A) C, y(t) = Cx(t), is balanced, if system Gramians, i.e., solutions P, Q of the Lyapunov equations AP + PA T + BB T = 0, A T Q + QA + C T C = 0, satisfy: P = Q = diag(σ 1,..., σ n ) with σ 1 σ 2... σ n > 0. c P. Benner Fast Frequency Response Analysis via MOR 12/24

32 Balanced Truncation for Linear Systems Basic idea { ẋ(t) = Ax(t) + Bu(t), Σ : with A stable, i.e., Λ (A) C, y(t) = Cx(t), is balanced, if system Gramians, i.e., solutions P, Q of the Lyapunov equations AP + PA T + BB T = 0, A T Q + QA + C T C = 0, satisfy: P = Q = diag(σ 1,..., σ n ) with σ 1 σ 2... σ n > 0. {σ 1,..., σ n } are the Hankel singular values (HSVs) of Σ. c P. Benner Fast Frequency Response Analysis via MOR 12/24

33 Balanced Truncation for Linear Systems Basic idea { ẋ(t) = Ax(t) + Bu(t), Σ : with A stable, i.e., Λ (A) C, y(t) = Cx(t), is balanced, if system Gramians, i.e., solutions P, Q of the Lyapunov equations AP + PA T + BB T = 0, A T Q + QA + C T C = 0, satisfy: P = Q = diag(σ 1,..., σ n ) with σ 1 σ 2... σ n > 0. {σ 1,..., σ n } are the Hankel singular values (HSVs) of Σ. Compute balanced realization (needs P, Q!) of the system via state-space transformation T : (A, B, C) (TAT 1, TB, CT 1 ) ([ ] A11 A 12 =, A 21 A 22 [ B1 B 2 ], [ C 1 C 2 ] ). c P. Benner Fast Frequency Response Analysis via MOR 12/24

34 Balanced Truncation for Linear Systems Basic idea { ẋ(t) = Ax(t) + Bu(t), Σ : with A stable, i.e., Λ (A) C, y(t) = Cx(t), is balanced, if system Gramians, i.e., solutions P, Q of the Lyapunov equations AP + PA T + BB T = 0, A T Q + QA + C T C = 0, satisfy: P = Q = diag(σ 1,..., σ n ) with σ 1 σ 2... σ n > 0. {σ 1,..., σ n } are the Hankel singular values (HSVs) of Σ. Compute balanced realization (needs P, Q!) of the system via state-space transformation T : (A, B, C) (TAT 1, TB, CT 1 ) ([ ] A11 A 12 =, A 21 A 22 Truncation (Â, ˆB, Ĉ) = (A 11, B 1, C 1 ). [ B1 B 2 ], [ C 1 C 2 ] ). c P. Benner Fast Frequency Response Analysis via MOR 12/24

35 Balanced Truncation for Linear Systems Properties Reduced-order model is stable with HSVs σ 1,..., σˆn. c P. Benner Fast Frequency Response Analysis via MOR 12/24

36 Balanced Truncation for Linear Systems Properties Reduced-order model is stable with HSVs σ 1,..., σˆn. Adaptive choice of r via computable error bound: y ŷ 2 (2 n ) u 2. k=ˆn+1 σ k c P. Benner Fast Frequency Response Analysis via MOR 12/24

37 Balanced Truncation for Linear Systems Properties Reduced-order model is stable with HSVs σ 1,..., σˆn. Adaptive choice of r via computable error bound: y ŷ 2 (2 n ) u 2. k=ˆn+1 σ k Practical implementation Rather than solving Lyapunov equations for P, Q (n 2 unknowns!), find S, R R n s with s n such that P SS T, Q RR T. Reduced-order model directly obtained via small-scale (s s) SVD of R T S! No O(n 3 ) or O(n 2 ) computations necessary! c P. Benner Fast Frequency Response Analysis via MOR 12/24

38 Computational Examples Electro-Thermal Simulation of Integrated Circuit (IC) [Source: Evgenii Rudnyi, CADFEM GmbH] Simplorer R test circuit with 2 transistors. Conservative thermal sub-system in Simplorer: voltage temperature, current heat flow. Original model: n = , m = q = 2 Computing time (on Intel Xeon dualcore 3GHz, 1 Thread): Main computational cost for set-up data 22min. Computation of reduced models from set-up data: 44 49sec. (r = 20 70). Bode plot (MATLAB on Intel Core i7, 2,67GHz, 12GB): 7.5h for original system, < 1min for reduced system. c P. Benner Fast Frequency Response Analysis via MOR 13/24

39 Computational Examples Electro-Thermal Simulation of Integrated Circuit (IC) [Source: Evgenii Rudnyi, CADFEM GmbH] Original model: n = , m = q = 2 Computing time (on Intel Xeon dualcore 3GHz, 1 Thread): Bode plot (MATLAB on Intel Core i7, 2,67GHz, 12GB): 7.5h for original system, < 1min for reduced system. Bode magnitude plot Hankel Singular Values 10 2 Transfer functions of original and reduced systems 10 0 Computed Hankel singular values σ max (G(jω)) original ROM 20 ROM 30 ROM 40 ROM 50 ROM 60 ROM ω magnitude index c P. Benner Fast Frequency Response Analysis via MOR 13/24

40 Computational Examples Electro-Thermal Simulation of Integrated Circuit (IC) [Source: Evgenii Rudnyi, CADFEM GmbH] Original model: n = , m = q = 2 Computing time (on Intel Xeon dualcore 3GHz, 1 Thread): Bode plot (MATLAB on Intel Core i7, 2,67GHz, 12GB): 7.5h for original system, < 1min for reduced system. Absolute Error Relative Error σ max (G(jω) G r (jω)) absolute model reduction error ROM 20 ROM 30 ROM 40 ROM 50 ROM 60 ROM 70 σ max (G(jω) G r (jω)) / G relative model reduction error ROM 20 ROM 30 ROM 40 ROM 50 ROM 60 ROM ω ω c P. Benner Fast Frequency Response Analysis via MOR 13/24

41 Computational Examples Elastic Multi-Body Simulation (EMBS) FEM Resolving complex 3D geometries can involve millions of degrees of freedom. EMBS: ROM is used as surrogate in simulation runs with varying forcing terms. Source: ITM, U Stuttgart c P. Benner Fast Frequency Response Analysis via MOR 14/24

42 Computational Examples MOR in EMBS Christine Nowakowski, Patrick Kürschner, Peter Eberhard, Peter Benner. Model Reduction of an Elastic Crankshaft for Elastic Multibody Simulations ZAMM, 93(4): , c P. Benner Fast Frequency Response Analysis via MOR 15/24

43 Computational Examples EMBS in Tribological Study of Combustion Engine Consider coupling laws between elements of the combustion engine; tribological contacts describe the relative motion between solids separated by fluid film lubrication. Need to compute hydrodynamic pressure distribution. Crankshaft modeled as elastic body, all other parts rigid. LTI system with n = 84, 252, m = q = 35. Structure of crank drive: Crankshaft of a four-cylinder engine: c P. Benner Fast Frequency Response Analysis via MOR 16/24

44 Computational Examples EMBS in Tribological Study of Combustion Engine Crankshaft modeled as elastic body, all other parts rigid. LTI system with n = 84, 252, m = q = 35. ROM of order r = 70 computed by the different methods, including second-order variant of balanced truncation (< 2min to compute ROM): (Hankel) singular values ε [ ] Modal CraigBampton Krylov tang. Krylov BT pp singular value no. j f [Hz] Christine Nowakowski, Patrick Kürschner, Peter Eberhard, Peter Benner. Model Reduction of an Elastic Crankshaft for Elastic Multibody Simulations ZAMM, 93(4): , c P. Benner Fast Frequency Response Analysis via MOR 16/24

45 Computational Examples Mechatronics / Piezo-Actuated Spindle Head Used for localized actuation to superimpose micro motions of machine tool. Descriptor LTI system with n = 290, 137, m = q = 9. Piezo-actuated structure (CAD): FEM model: Source: Fraunhofer IWU Chemnitz/Dresden c P. Benner Fast Frequency Response Analysis via MOR 17/24

46 Computational Examples Mechatronics / Piezo-Actuated Spindle Head Used for localized actuation to superimpose micro motions of machine tool. Descriptor LTI system with n = 290, 137, m = q = 9. ROM of orders r = computed by variant of balanced truncation for descriptor systems, sigma plot (left) and relative errors (right): Mohammad Monir Uddin, Jens Saak, Burkhard Kranz, Peter Benner. Computation of a Compact State Space Model for an Adaptive Spindle Head Configuration with Piezo Actuators using Balanced Truncation. Production Engineering, 6(6): , c P. Benner Fast Frequency Response Analysis via MOR 17/24

47 Computational Examples Mechatronics / Piezo-Actuated Spindle Head Used for localized actuation to superimpose micro motions of machine tool. Descriptor LTI system with n = 290, 137, m = q = 9. ROM of orders r = computed by variant of balanced truncation for descriptor systems, Bode magnitude plots for 1 9 (left), 9 9 (right): Mohammad Monir Uddin, Jens Saak, Burkhard Kranz, Peter Benner. Computation of a Compact State Space Model for an Adaptive Spindle Head Configuration with Piezo Actuators using Balanced Truncation. Production Engineering, 6(6): , c P. Benner Fast Frequency Response Analysis via MOR 17/24

48 Computational Examples Mechatronics / Piezo-Actuated Spindle Head Used for localized actuation to superimpose micro motions of machine tool. Descriptor LTI system with n = 290, 137, m = q = 9. ROM of orders r = computed by variant of balanced truncation for descriptor systems, simulation times: system dimension execution time (sec) speedup 290, , , , , ,923 Mohammad Monir Uddin, Jens Saak, Burkhard Kranz, Peter Benner. Computation of a Compact State Space Model for an Adaptive Spindle Head Configuration with Piezo Actuators using Balanced Truncation. Production Engineering, 6(6): , c P. Benner Fast Frequency Response Analysis via MOR 17/24

49 Interpolatory Model Reduction Computation of reduced-order model by projection Given linear (descriptor) system Eẋ = Ax + Bu, y = Cx with transfer function G(s) = C(sE A) 1 B, a ROM is obtained using truncation matrices V, W R n r with W T V = I r ( (VW T ) 2 = VW T is projector) by computing Ê = W T EV, Â = W T AV, ˆB = W T B, Ĉ = CV. Petrov-Galerkin-type (two-sided) projection: W V, Galerkin-type (one-sided) projection: W = V. c P. Benner Fast Frequency Response Analysis via MOR 18/24

50 Interpolatory Model Reduction Computation of reduced-order model by projection Given linear (descriptor) system Eẋ = Ax + Bu, y = Cx with transfer function G(s) = C(sE A) 1 B, a ROM is obtained using truncation matrices V, W R n r with W T V = I r ( (VW T ) 2 = VW T is projector) by computing Ê = W T EV, Â = W T AV, ˆB = W T B, Ĉ = CV. Petrov-Galerkin-type (two-sided) projection: W V, Galerkin-type (one-sided) projection: W = V. Rational Interpolation/Moment-Matching Choose V, W such that and G(s j ) = Ĝ(s j ), j = 1,..., k, d i ds i G(s j) = d i ds i Ĝ(s j ), i = 1,..., K j, j = 1,..., k. c P. Benner Fast Frequency Response Analysis via MOR 18/24

51 Interpolatory Model Reduction Theorem (simplified) [Grimme 97, Villemagne/Skelton 87] If span { (s 1 E A) 1 B,..., (s k E A) 1 B } Ran(V ), span { (s 1 E A) T C T,..., (s k E A) T C T } Ran(W ), then G(s j ) = Ĝ(s j), d ds G(s j) = d ds Ĝ(s j), for j = 1,..., k. c P. Benner Fast Frequency Response Analysis via MOR 18/24

52 Interpolatory Model Reduction Theorem (simplified) [Grimme 97, Villemagne/Skelton 87] If span { (s 1 E A) 1 B,..., (s k E A) 1 B } Ran(V ), span { (s 1 E A) T C T,..., (s k E A) T C T } Ran(W ), then G(s j ) = Ĝ(s j), d ds G(s j) = d ds Ĝ(s j), for j = 1,..., k. Remarks: computation of V, W from rational Krylov subspaces, e.g., dual rational Arnoldi/Lanczos [Grimme 97], Iter. Rational Krylov-Alg. (IRKA) [Antoulas/Beattie/Gugercin 06/ 08]. c P. Benner Fast Frequency Response Analysis via MOR 18/24

53 Interpolatory Model Reduction Theorem (simplified) [Grimme 97, Villemagne/Skelton 87] If span { (s 1 E A) 1 B,..., (s k E A) 1 B } Ran(V ), span { (s 1 E A) T C T,..., (s k E A) T C T } Ran(W ), then G(s j ) = Ĝ(s j), d ds G(s j) = d ds Ĝ(s j), for j = 1,..., k. Remarks: using Galerkin/one-sided projection (W V ) yields G(s j ) = Ĝ(s j), but in general d ds G(s j) d ds Ĝ(s j). c P. Benner Fast Frequency Response Analysis via MOR 18/24

54 Interpolatory Model Reduction Theorem (simplified) [Grimme 97, Villemagne/Skelton 87] If span { (s 1 E A) 1 B,..., (s k E A) 1 B } Ran(V ), span { (s 1 E A) T C T,..., (s k E A) T C T } Ran(W ), then G(s j ) = Ĝ(s j), d ds G(s j) = d ds Ĝ(s j), for j = 1,..., k. Remarks: k = 1, standard Krylov subspace(s) of dimension K: range (V ) = K K ((s 1 I A) 1, (s 1 I A) 1 B). moment-matching methods/padé approximation [Freund/Feldmann 95], d i ds i G(s 1) = d i ds i Ĝ(s 1 ), i = 0,..., K 1(+K). c P. Benner Fast Frequency Response Analysis via MOR 18/24

55 Remarks: Interpolatory Model Reduction k = 1, standard Krylov subspace(s) of dimension K: range (V ) = K K ((s 1 I A) 1, (s 1 I A) 1 B). moment-matching methods/padé approximation [Freund/Feldmann 95], d i ds i G(s 1) = d i ds i Ĝ(s 1 ), i = 0,..., K 1(+K). News: Adaptive choice of interpolation points and number of moments to be matched based on dual-weighted residual based error estimate! Lihong Feng, Jan G. Korvink, Peter Benner. A Fully Adaptive Scheme for Model Order Reduction Based on Moment-Matching. IEEE Transactions on Components, Packaging, and Manufacturing Technology, in press. Lihong Feng, Athanasios C. Antoulas, Peter Benner Some a posteriori error bounds for reduced order modelling of (non-)parametrized linear systems. MPI Magdeburg Preprints MPIMD/15-17, October c P. Benner Fast Frequency Response Analysis via MOR 18/24

56 Interpolatory Model Reduction Micro-electronics: clock tree Each segment has 4 RL pairs in series, representing the wiring on a chip, with four capacitors to ground, representing the wire-substrate interaction, n = 6, 134, m = q = 1 (SISO). c P. Benner Fast Frequency Response Analysis via MOR 19/24

57 Interpolatory Model Reduction Micro-electronics: clock tree Each segment has 4 RL pairs in series, representing the wiring on a chip, with four capacitors to ground, representing the wire-substrate interaction, n = 6, 134, m = q = 1 (SISO). Sigma plot (r = 18): Relative error (r = 18): c P. Benner Fast Frequency Response Analysis via MOR 19/24

58 Interpolatory Model Reduction Micro-electronics: SPICE MNA model... Model of a CMOS-inverter driven two-bit bus determined, using modified modal analysis, by SPICE. n = 980, m = q = 4 (MIMO), ROM of order r = 48. Bode plot (magnitude, 1 4): Bode plot (phase, 1, 4 4): Source: The SLICOT Benchmark Collection for Model Reduction, c P. Benner Fast Frequency Response Analysis via MOR 20/24

59 Parametric Model Order Reduction (PMOR) The PMOR Problem Approximate the dynamical system E(p)ẋ = A(p)x + B(p)u, E(p), A(p) R n n, y = C(p)x, B(p) R n m, C(p) R q n, by reduced-order system Ê(p) ˆx = Â(p)ˆx + ˆB(p)u, Ê(p), Â(p) R r r, ŷ = Ĉ(p)ˆx, ˆB(p) R r m, Ĉ(p) R q r, of order r n, such that y ŷ = Gu Ĝu G Ĝ u < tolerance u p Ω R d. c P. Benner Fast Frequency Response Analysis via MOR 21/24

60 Parametric Model Order Reduction (PMOR) The PMOR Problem Approximate the dynamical system E(p)ẋ = A(p)x + B(p)u, E(p), A(p) R n n, y = C(p)x, B(p) R n m, C(p) R q n, by reduced-order system Ê(p) ˆx = Â(p)ˆx + ˆB(p)u, Ê(p), Â(p) R r r, ŷ = Ĉ(p)ˆx, ˆB(p) R r m, Ĉ(p) R q r, of order r n, such that y ŷ = Gu Ĝu G Ĝ u < tolerance u p Ω R d. = Approximation problem: min order (Ĝ) r G Ĝ. c P. Benner Fast Frequency Response Analysis via MOR 21/24

61 Parametric Model Order Reduction (PMOR) Example: Microsystems/MEMS Design (butterfly gyro) Applications: inertial navigation, electronic stability control (ESP). Voltage applied to electrodes induces vibration of wings, resulting rotation due to Coriolis force yields sensor data. FE model of second order: N = n = , m = 1, q = 12. Sensor for position control based on acceleration and rotation. Source: MOR Wiki c P. Benner Fast Frequency Response Analysis via MOR 22/24

62 Parametric Model Order Reduction (PMOR) Example: Microsystems/MEMS Design (butterfly gyro) Parametric FE model: M(d)ẍ(t) + D(θ, d, α, β)ẋ(t) + T (d)x(t) = Bu(t). c P. Benner Fast Frequency Response Analysis via MOR 22/24

63 Parametric Model Order Reduction (PMOR) Example: Microsystems/MEMS Design (butterfly gyro) Parametric FE model: M(d)ẍ(t) + D(θ, d, α, β)ẋ(t) + T (d)x(t) = Bu(t), where M(d) = M 1 + dm 2, D(θ, d, α, β) = θ(d 1 + dd 2) + αm(d) + βt (d), T (d) = T T2 + dt3, d with width of bearing: d, angular velocity: θ, Rayleigh damping parameters: α, β. c P. Benner Fast Frequency Response Analysis via MOR 22/24

64 Parametric Model Order Reduction (PMOR) Example: Microsystems/MEMS Design (butterfly gyro) Response surfaces: σ max (G(jω, p) vs. p at ω, original... and reduced-order model. Computation times: ca. 1 week ca. 1.5 hours c P. Benner Fast Frequency Response Analysis via MOR 22/24

65 Conclusions A huge savings in computational time and energy can be obtained using reduced-order models in frequency response analysis. c P. Benner Fast Frequency Response Analysis via MOR 23/24

66 Conclusions A huge savings in computational time and energy can be obtained using reduced-order models in frequency response analysis. ROMs can be computed by many methods, e.g., based on system-theoretic considerations, by c P. Benner Fast Frequency Response Analysis via MOR 23/24

67 Conclusions A huge savings in computational time and energy can be obtained using reduced-order models in frequency response analysis. ROMs can be computed by many methods, e.g., based on system-theoretic considerations, by novel implementation variants of balanced truncation using efficient numerical linear algebra, c P. Benner Fast Frequency Response Analysis via MOR 23/24

68 Conclusions A huge savings in computational time and energy can be obtained using reduced-order models in frequency response analysis. ROMs can be computed by many methods, e.g., based on system-theoretic considerations, by novel implementation variants of balanced truncation using efficient numerical linear algebra, variants of rational interpolation / moment-matching. c P. Benner Fast Frequency Response Analysis via MOR 23/24

69 Conclusions A huge savings in computational time and energy can be obtained using reduced-order models in frequency response analysis. ROMs can be computed by many methods, e.g., based on system-theoretic considerations, by novel implementation variants of balanced truncation using efficient numerical linear algebra, variants of rational interpolation / moment-matching. Savings increase by a considerable factor for parametric systems. c P. Benner Fast Frequency Response Analysis via MOR 23/24

70 Conclusions A huge savings in computational time and energy can be obtained using reduced-order models in frequency response analysis. ROMs can be computed by many methods, e.g., based on system-theoretic considerations, by novel implementation variants of balanced truncation using efficient numerical linear algebra, variants of rational interpolation / moment-matching. Savings increase by a considerable factor for parametric systems. Current and future work: c P. Benner Fast Frequency Response Analysis via MOR 23/24

71 Conclusions A huge savings in computational time and energy can be obtained using reduced-order models in frequency response analysis. ROMs can be computed by many methods, e.g., based on system-theoretic considerations, by novel implementation variants of balanced truncation using efficient numerical linear algebra, variants of rational interpolation / moment-matching. Savings increase by a considerable factor for parametric systems. Current and future work: combine MOR methods with reduction methods for parameter space (a lot of progress in this direction in recent years, mostly for instationary problems so far), c P. Benner Fast Frequency Response Analysis via MOR 23/24

72 Conclusions A huge savings in computational time and energy can be obtained using reduced-order models in frequency response analysis. ROMs can be computed by many methods, e.g., based on system-theoretic considerations, by novel implementation variants of balanced truncation using efficient numerical linear algebra, variants of rational interpolation / moment-matching. Savings increase by a considerable factor for parametric systems. Current and future work: combine MOR methods with reduction methods for parameter space (a lot of progress in this direction in recent years, mostly for instationary problems so far), extend balanced truncation and rational interpolation techniques in computationally feasible way to nonlinear systems, c P. Benner Fast Frequency Response Analysis via MOR 23/24

73 Conclusions A huge savings in computational time and energy can be obtained using reduced-order models in frequency response analysis. ROMs can be computed by many methods, e.g., based on system-theoretic considerations, by novel implementation variants of balanced truncation using efficient numerical linear algebra, variants of rational interpolation / moment-matching. Savings increase by a considerable factor for parametric systems. Current and future work: combine MOR methods with reduction methods for parameter space (a lot of progress in this direction in recent years, mostly for instationary problems so far), extend balanced truncation and rational interpolation techniques in computationally feasible way to nonlinear systems, merge the best features of so far competing methods. c P. Benner Fast Frequency Response Analysis via MOR 23/24

74 Conclusions A huge savings in computational time and energy can be obtained using reduced-order models in frequency response analysis. ROMs can be computed by many methods, e.g., based on system-theoretic considerations, by novel implementation variants of balanced truncation using efficient numerical linear algebra, variants of rational interpolation / moment-matching. Savings increase by a considerable factor for parametric systems. Current and future work: combine MOR methods with reduction methods for parameter space (a lot of progress in this direction in recent years, mostly for instationary problems so far), extend balanced truncation and rational interpolation techniques in computationally feasible way to nonlinear systems, merge the best features of so far competing methods. How do we get MOR into the pro software packages in CAE / CSE? c P. Benner Fast Frequency Response Analysis via MOR 23/24

75 Further Reading 1. U. Baur, P. Benner, and L. Feng. Model Order Reduction for Linear and Nonlinear Systems: a System-Theoretic Perspective Arch. Comp. Meth. Engrg., 21(4): , P. Benner. Solving large-scale control problems. IEEE Control Systems Magazine, 24(1):44 59, P. Benner and A. Bruns. Parametric model order reduction of thermal models using the bilinear interpolatory rational Krylov algorithm. Mathematical and Computer Modelling of Dynamical Systems, 21(2): , P. Benner, S. Gugercin, and K. Willcox. A Survey of Model Reduction Methods for Parametric Systems. SIAM Review, 57(4), 2015 (to appear). Available online as MPI Magdeburg Preprint MPIMD/13-14, August P. Benner and J. Saak. Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey. GAMM-Mitteilungen, 36(1):32-52, V. Simoncini. Computational methods for linear matrix equations (survey article). March 2013, c P. Benner Fast Frequency Response Analysis via MOR 24/24