1. Reduced Order Modeling of Parameterized and Distributed Systems Luca Daniel, M.I.T.

2. 2 Parameterized Model Order Reduction Problem Classification Reducing Linear Systems  Moment Matching with Linear parameters  Moment Matching with NON-linear parameters  Quasi Convex Optimization approach Reducing NON-linear systems  Moment Matching with NON-linear parameters

3. 10 6 7 8 9 0 1 2 3 4 frequency |Z(f)| Field solvers can produce instance impedance vs. frequency curves. Motivation. Example: Analysis of RF micro-inductor How are the substrate eddy currents affecting the quality factor of the inductor? How are the displacement currents affecting the resonance of the inductor? Need to capture all 2 nd order effects

4. “Model Order Reduction” can help verify how the inductor performance (Q, resonance position, etc) affect the transceiver performance (distortion, interference rejection etc.) Modeling Requirements:  as accurate as a field solver  automatic and robust  model compatible with circuit simulators Motivation Example: MOR of RF micro-inductor LNA ADC I Q LO

5. Motivation Example: Parameterized MOR of RF micro-inductor LNA ADC I Q LO d W “Model Order Reduction” can help verify how the resonator or inductor performance (noise, quality factor, etc) affect the transceiver performance (distortion, interference rejection) “Parameterized Model Order Reduction” can help verify how the transceiver performance changes when I change wire widths and wire separation

6. Luca Daniel15 July, 2003 6 http://www.rle.mit.edu/cpg frequency [Hz] Motivation Example. Analysis of Integrated Power Electronics How is the magnetic fringing field from the core effecting eddy current losses? Analysis tools can produce for instance resistance vs. frequency curves. Spattered laminated NiFe core, electroplated windings [Daniel96] R ac

7. Luca Daniel15 July, 2003 7 http://www.rle.mit.edu/cpg Micro-inductor in a DC/DC power converter “Model Order Reduction” can help verifying how the 2 nd order effects of the power inductor (e.g. power loss vs. freq. curve) influences the power converter functionality (dynamics, overall power efficiency)“Model Order Reduction” can help verifying how the 2 nd order effects of the power inductor (e.g. power loss vs. freq. curve) influences the power converter functionality (dynamics, overall power efficiency) “Parameterized Model Order Reduction” can help verify how the functionality and the efficiency of the overall power converter changes when I change wire widths and wire separations?“Parameterized Model Order Reduction” can help verify how the functionality and the efficiency of the overall power converter changes when I change wire widths and wire separations? V in (t) V out (t)

8. 8 d W From Field Solvers to Parameterized Model Order Reduction (PMOR). 1M equations 10 equations PMOR Field Solvers discretize geometry and produce large systems PMOR produces a dynamical model: – automatically – match port impedance – small (10-15 ODEs)

9. Parameterized model order reduction. Problem classification linearity matrix size # parameters

10. 10 Parameterized Model Order Reduction Problem Classification matrix size # parameters linearity Non-Linear Systems Linear Time Invariant linearly parameterized non-linearly parameterized

11. 11 Parameterized Model Order Reduction. Applications interconnect RF inductors matrix size # parameters linearity linearly parameterized non-linearly parameterized Linear Time Invariant Non-Linear Systems LO LNA ADC MEMS Packages

12. 12 Parameterized Model Order Reduction. Previous work interconnect matrix size # parameters LO LNA ADC linearity Linear Time Invariant linearly parameterized non-linearly parameterized Non-Linear Systems MEMS Packages RF inductors statistical data mining Liu DAC99 Heydari ICCAD01 CMU Rutenbar02 Moment Matching: Pullela97, Weile99, Gunupudi00, Prud’homme02, Daniel02, Li05

13. 13 Parameterized Model Order Reduction Problem Classification Reducing Linear Systems  Moment Matching with Linear parameters  Moment Matching with NON-linear parameters  Quasi Convex Optimization approach Reducing NON-linear systems  Moment Matching with NON-linear parameters

14. DAC02 modeling tutorial 14 Motivation for geometrically parameterized modeling of interconnect In interconnect design often would like:In interconnect design often would like: –reliable functionality: minimize capacitive cross-talk,minimize capacitive cross-talk, minimize inductive cross-talk,minimize inductive cross-talk, minimize electromagnetic interferenceminimize electromagnetic interference –high speed: minimize resistanceminimize resistance minimize capacitanceminimize capacitance –low cost: minimize areaminimize area Need to explore tradeoff space and find optimal design!Need to explore tradeoff space and find optimal design!

15. DAC02 modeling tutorial 15 The traditional design methodology The traditional design flow:The traditional design flow: –REPEAT design all interconnect wiresdesign all interconnect wires extract accurately parasitics all at onceextract accurately parasitics all at once –UNTIL noise and timing are within specs such procedure is not ideal for optimization!such procedure is not ideal for optimization! each iteration is very time consumingeach iteration is very time consuming

16. DAC02 modeling tutorial 16 Alternative design methodologies 1.Pre-characterize standard interconnect structures (e.g. busses): –using parasitic extraction and table lookup –or building parameterized and accurate low order models 2.However... if the model construction is fast enough can also: –build the interconnect structure model "on the fly" during layout –accounting for any topology in surrounding topologies already committed to layout –then use optimizer to choose the best parameter for optimal tradeoff design.

17. We construct a multi-parameter model of the bus parameterized in wire width W and separation d Example: an interconnect bus ………….. Wd accounting for surrounding topologyaccounting for surrounding topology

18. Parasitic extraction produces large state space models E.g. subdividing wires in short sections and using for instance Nodal Analysis ………….. Conductance matrix Large linear dynamical system

19. Our goal Given a large parameterized linear system:Given a large parameterized linear system: construct a reduced order system with similar frequency responseconstruct a reduced order system with similar frequency response

20. Background Non-parameterized Model order reduction 500,000 x 500,000 Given a large linear system model:Given a large linear system model: 20 x 20 Construct a linear system model with:Construct a linear system model with: -smaller complexity -same fidelity -small reduction cost

21. Taylor series expansion: U U Background Model order reduction (cont.) change basis and use only the first few vectors of the Taylor series expansion: equivalent to match first derivatives around expansion point

22. 22 Reducing matrices’ size: Congruence Transformation [PRIMA TCAD98] nxn qxn nxq nxq qxq

23. Discretizing wires and using Nodal Analysis ………….. Parameterized Model Order Reduction. Example: interconnect bus

24. 24 More in general... [D. TCAD04] [D. PhD04] UqUq It is a p-variables Taylor series expansion Once again change basis:  use first few vectors of the Taylor expansion,  matching first few derivatives with respect to each parameter

25. 25 nxn qxq nxq qxn Congruence transformations on each of the matrices Parameterized moment matching (cont.)

26. Example: model step responses for different W and d. ………….. N=16 wires h=1.2um L=1mm d=2um W=0.25um W=0.2um W=4um W=8um W 0 =1um d 0 =1um W d d=0.25um W=0.25um W=0.2um W=4um W=8um W 0 =1um d 0 =1um

27. d=2um W=0.25um W=0.2um W=4um W=8um W 0 =1um d 0 =1um Example: model crosstalk responses for different W and d. ………….. N=16 wires h=1.2um L=1mm d=0.25um W=0.25um W=0.2um W=4um W=8um W 0 =1um d 0 =1um W d

28. 28 Parameterized Model Order Reduction Problem Classification Reducing Linear Systems  Moment Matching with Linear parameters  Moment Matching with NON-linear parameters  Quasi Convex Optimization approach Reducing NON-linear systems  Moment Matching with NON-linear parameters

29. linearly parameterized Example: RF inductor Reduction of Non-linearly parameterized linear systems interconnectMEMS RF inductors matrix size # parameters LO LNA ADC linearity Non-Linear Systems non-linearly parameterized Linear Time Invariant CMU Rutenbar02 statistical data mining Liu DAC99 Heydari ICCAD01 Moment Matching Pullela97, Weile99, Prud’homme 02

30. Example: RF inductor d W n= 3 Design parameters:  wire dimensions and separations,  number of turns  type of substrate and distance from wires Performance parameters:  inductance  quality factor Q  resonance frequency  power  area Effects captured:  displacement currents affect resonance  skin effect in wires affect Q  proximity effect in wiresaffect Q  dielectricsaffect Q and resonance  substrate eddy currents affect Q and resonance  interference with other devices

31. 31 Example: PEEC Mixed Potential Integral Equation [Ruehli MTT74] current and charge conservation resistive effect magnetic coupling charge-voltagerelation

32. 32 PEEC Discretization Basis Functions [Ruehli MTT74, FastHenry94] PEEC subdivides the volumes into small thin filaments conductor k Enforcing this equation in each filament: produces branch equations conductor h

33. 33 PEEC Discretization Basis Functions [Ruehli MTT74, FastCap91] panel j Enforcing this equation in each panel: produces branch equations PEEC subdivides the surface into small panels panel i

34. 34 PEEC Discretization Basis Functions [Ruehli MTT74, MIT course 6.336J and 16.920J] PEEC discretizes volumes in short thin filaments, small surface panels thin volume filaments with constant current small surface panels with constant charge PEEC discretization gives branch equations:PEEC discretization gives branch equations:

35. 35 PEEC Discretization Example: On-Chip RF Inductor [D. BMAS03] x100um wire separation d = 1um-5um wire width W = 1um-5um overall dimensions = 600um x 600um wire thickness 1um picture not to scale W d

36. 36 Mesh (Loop) Analysis [FastHenry94, Kamon Trans Packaging98] Imposing current conservation with mesh (loop) analysis (KVL)

37. 37 Example of Field Solver output: current distributions on a package power grid input terminals

38. 38 Example of a Field Solver output: package powergrid admittance amplitude * 3 proximity templates per cross-section - 20 non-uniform thin filaments per cross-section

39. 39 From Field Solvers to a Dynamical Linear System Model Multiply out and introduce state Imposing current conservation with mesh (loop) analysis (KVL)

40. 40 Discretization produces a HUGE “nonlinearly parameterized” dynamical linear system [D. BMAS03] thin volume filaments with constant current small surface panels with constant charge E A Case 2 Laplace parameter (frequency) Case 1 geometrical parameters (W,d,s)

41. A polynomial interpolation approach [D. et al BMAS03] 1. Fit a low order polynomial (e.g. quadratic) to the evaluated matrices R and L A (W,d) ≈ A 0,0 + W A 1,0 + d A 0,1 + W 2 A 2,0 + Wd A 1,1 + d 2 A 0,2 E (W,d) ≈ E 0,0 + W E 1,0 + d E 0,1 + W 2 E 2,0 + Wd E 1,1 + d 2 E 0,2 [ A 0,0 + W A 1,0 + d A 0,1 + W 2 A 2,0 + Wd A 1,1 + d 2 A 0,2 +... s E 0,0 + sW E 1,0 + sd E 0,1 + sW 2 E 2,0 + sWd E 1,1 + sd 2 E 0,2 ] x= b u [ s E (W,d)- A (W,d) ] x= b u

42. A polynomial interpolation approach [D. et al BMAS03] 2. Selected a grid of 9 evaluation points for different combination of parameters (W,d) = (1um,1um), (1um,3um), (1um,5um), (3um,1um), (3um,3um), (3um,5um), (5um,1um), (5um,3um), (5um,5um) 3. Used the Volume Integral Equation code to generate system matrices E k = E (W k,d k ) and A k = A (W k,d k ) for each combination of parameters EkEk AkAk

43. A polynomial interpolation approach [D. et al BMAS03] 4. Calculating Interpolation coefficients Need to calculate 6 polynomial coefficients Hence need at least 6 equations imposing fit in 6 evaluation points However in general it is better to use more evaluation points than the minimum. For instance here we used the 9 evaluation points above and solved with a least square method

44. A polynomial interpolation approach [D. et al BMAS03] (cont.) 6. Used the previously developed model reduction for linearly parameterized systems 5. Transformed the polynomial parameterized system into a linearly parameterized system introducing new parameters E0E0 E1E1 E2E2 E3E3 E4E4 E5E5 E 11 E 10 E9E9 E8E8 E7E7 E6E6 s1s1 s2s2 s3s3 s4s4 s5s5 s 11 s 10 s9s9 s8s8 s7s7 s6s6 [ A 0,0 + W A 1,0 + d A 0,1 + W 2 A 2,0 + Wd A 1,1 + d 2 A 0,2 +... s E 0,0 + sW E 1,0 + sd E 0,1 + sW 2 E 2,0 + sWd E 1,1 + sd 2 E 0,2 ] x= b u

45. 45 More in general... [D. TCAD04] [D. PhD04] UqUq It is a p-variables Taylor series expansion Once again change basis:  use first few vectors of the Taylor expansion,  matching first few derivatives with respect to each parameter

46. 46 Computational Complexity (time and memory) Computational Complexity Hankel reduction or Balance Realizations O(N 3 ) e.g. 10months for N=100,000 Moment matching O(q N 2 ) e.g. 8days for N=100,000 q=10 Moment Matching + pFFT O(q N log N) e.g. 8hours for N=100,000 q=10 Moment matching bottleneck: getting Eb, E 2 b,...

47. 47 pFFT: an O(NlogN) Matrix-Vector product for the Mixed Potential Integral equations [Phillips97] pFFT: an O(NlogN) Matrix-Vector product for the Mixed Potential Integral equations [Phillips97] Matrix-vector product Eb physical interpretation: (1) (2) (3) Picture by J. Phillips (1) Project charges into a 3D grid. Complexity O(N) (2) Calculate grid potentials, it is a convolution use FFT. Complexity O(n log n). (3) Interpolate potentials from grid. Complexity O(N)  given N charges on N panels  calculate the resulting N potentials grid points n ≈ N panels

48. 48 PEEC Discretization Example: On-Chip RF Inductor [D. BMAS03] x100um wire separation d = 1um-5um wire width W = 1um-5um overall dimensions = 600um x 600um wire thickness 1um picture not to scale W d

49. Results: Inductance vs. frequency frequency [ x10GHz ] Wire width = 5um separation = 1um, 2um, 3um, 4um, 5um frequency [ x10GHz ] Wire width = 1um separation = 1um, 2um, 3um, 4um, 5um L L __ original system (order 420) --- reduced model (order 12) __ original system (order 420) --- reduced model (order 12) Worst case error in resonance position = 3%

50. Results: Quality factor (Q=wL/R) vs. frequency Wire width = 1um separation = 1um, 2um, 3um, 4um, 5um Wire width = 5um separation = 1um, 2um, 3um, 4um, 5um frequency [ x10GHz ] Worst case error in amplitude = 4% __ original system (order 420) --- reduced model (order 12) __ original system (order 420) --- reduced model (order 12) Q Q

51. 51 Open issues in the PMOR Matrix Reduction step Model order grows as O(p m ) where p = # parameters and m = # derivatives matched for each parameter  however model order is linear in # of parameters when matching only one derivative per parameter (m = 1) and still produces good accuracy in our experiments.  furthermore, for higher accuracy instead of increasing # of matched derivatives, can instead match multiple points (or combine the two approaches) [Li, Liu, Nassif, Pileggi DATE05] W or d

52. Limitations and future work use better interpolation! Wire width = 1um separation = 1um, 2um, 3um, 4um, 5um frequency [ x10GHz ] __ original system (order 420) --- reduced model (order 12) Q very little error in the points used for fitting 1um, 3um, 5um Worst case errors far from fitting points 2um, 4um (3% in resonance position) so the critical step is the fitting!!

53. Case #2. The parameter is frequency. Already discussed: Distributed Linear Systems Examples:  full-wave MPIE  MPIE using layered-media Green functions (e.g. for handling substrate or dielectrics)  frequency-dependent basis functions  frequency dependent discretizations

54. Polinomial interpolation for frequency [Phillips96] Polynomial approximation e.g. Taylor expansion, or a polynomial interpolation for E(s) Performance: Fast and accurate in the frequency band of interest Problem: Can not be used in a time domain circuit simulator because does not guarantee stability and passivity

55. Positive real transfer function in the complex plane for different frequencies Passive region Active region original system

56. Why does polynomial interpolation fail when applied to the Laplace parameter ‘s’? original system Passive region Active region Although accurate in the frequency band of interest Polynomial interpolation is unlikely to preserve GLOBAL properties such as positive realness because it is GLOBALLY not well-behaved

57. Using global uniformly convergent interpolants [Daniel02] If E(s) is strictly positive real, a GLOBALLY and UNIFORMLY convergent interpolant will eventually get close enough (for a large enough order M of the interpolant) and be positive-real as well. original system reduced system Proof: just choose accuracy of interpolation smaller than minimum distance from imaginary axis Passive region Active region

58. Parameterized reduction can enable automatic design of circuit components that can only be described by field solvers All methods presented are based on Krylov subspace moment matching projection Key observation: all these methods are compatible with O(NlogN) field solver based matrix-vector But of course all these methods also have all the arguable issues associated with moment matching Linearly parameterized (use p-variable Taylor series expansion  e.g. interconnect bus Non-linearly parameterized (use polynomial interpolation)  e.g. RF inductor (no substrate) Distributed systems i.e. non-linear in s (use globally convergent interpolant implemented with FFT)  e.g. substrate layered green functions or high order basis functions Conclusions

59. 59 Parameterized Model Order Reduction Problem Classification Reducing Linear Systems  Moment Matching with Linear parameters  Moment Matching with NON-linear parameters Example utilization of PMOR  Quasi Convex Optimization approach Reducing NON-linear systems  Moment Matching with NON-linear parameters

60. Accelerated Optical Topography Using Parameterized Model Order Reduction Junghoon Lee, Dimitry Vasilyev, Anne Vithayathil, Luca Daniel, and Jacob White Research Laboratory of Electronics Massachusetts Institute of Technology http://www.rle.mit.edu/cpg TH3D-3

61. Non-destructive Inspection of Fabricated Structures Spectroscopic ellipsometry –Shine light of =200~800nm –Measure the scattered light –Estimate the geometric parameters, e.g. w and h plasma-processing.com/insitu.htmwww.sopra-sa.com etched structurescattered field measurements

62. Model Based Approach Determine parameters p by solving Solve nonlinear least squares problem –Algorithms are iterative in nature –Scattering model evaluated for many p’s Need an efficient model of parameters p –Up to a dozen parameters –width (w), height (h), left/right sidewall angle, top curvature, etc.

63. Model Possibilities Tabulated library of scattered fields –Table grows exponentially as the number of parameters grow Full EM simulation for scattered field –Too computationally expensive (too slow) New approach: automatic extraction of parameterized low-order model –Automatically extract parameterized low-order model from pre-determined test structures –Use model in optimization during in-line process diagnosis

64. Surface Integral Formulation Discretized surface integral formulation –PMCHW formulation on dielectric interfaces –Discretization and RWG basis functions –Method of moments Resulting system of equations scattered field integral of incident wave weights for RWG basis discretized integral operators

65. Polynomial fitting to find explicit dependency –y(p; ): rough function of p and –A(p; ), C(p; ): smooth function of p and Polynomial fit of A and C Accurate since they are smooth Polynomial Fitting coefficient matrices

66. Parameterized Moment Matching Projection representation: approximate x f as Many different ways to choose V –Moment matching, balanced realization, POD, etc. –We used moment matching: match Taylor coefficients of y f (p; ) and y i (p; )

67. Parameterized Moment Matching Fitting with moment matching: for the constant term Overall reduced system

68. Accelerated Optimization with Reduced Model error radius original: 576 reduced: 30 (about x8000 faster) error width True: 97 nm Predicted: 97.3 nm original: 540 reduced: 50 (about x1000 faster) True: 105 nm Predicted: 104.9 nm

69. Conclusions Optical Inspection Problem: –Parameterized reduced model using polynomial fitting plus projection –Accelerates optimization –The method achieves good accuracy in practice Acknowledgements MARCO designated Interconnect Focus Centers, SRC, Singapore-MIT Alliance, and NSF

70. 70 Parameterized Model Order Reduction Problem Classification Reducing Linear Systems  Moment Matching with Linear parameters  Moment Matching with NON-linear parameters  Quasi Convex Optimization approach Reducing NON-linear systems  Moment Matching with NON-linear parameters

71. A Quasi-Convex Optimization Approach to Parameterized Model Order Reduction Linear analog components e.g. interconnect, packages, RF inductors Field Solver Linear Models Parameterized Reduced Linear Models

72. Review: Quasi-Convex setup for NON-parameterized MOR convex set Stability: Passivity: quasi-convex set Standard problem. Use for example by the ellipsoid algorithm

73. Parameterized MOR setup Let design parameter p explicitly enter (a,b,c) More difficult to check stability Construct guaranteed stable parameterized reduced model

74. Parameterized stability Need to check a(z,p) > 0 for all z, for all p Cannot use the trick in the non-parameterized case Make a(z,p) a multivariate trigonometric polynomial Check positivity of multivariate trigonometric polynomial by using sum-of-squares argument trig polyordinary polytrig polytrig polys with

75. Positivity of multivariate trig. poly Given a(z,p), solving the semi-definite program Otherwise, claim a(z,p)  0 at some point If feasible and y* < 0, then

76. Example 7: Parameterized RF inductor Construct reduced order models in the design space Identify dominant poles z*(p) of each ROM Construct “non-dominant” systems such that where K(p), A(p) are scalars functions G 1 (z,p) is the “non-dominant” system Construct parameterized ROM of each “non-dominant” system Interpolate the scalars functions Combine both parameterized models to obtain the final model Circle: training points Triangle: test points

77. Example 6: Parameterized RF inductor Circle: training points Quality factor for W=16.5 µm, D = 1,5,18,20 µm Triangle: test points ----- full model ___ our QCO PMOR

78. Parameterized RF inductor (cont.) Quality factor for W=16.5 µm, D = 1,5,18,20 µm Frequency of the peak Q factor for W = 16.5 µm

79. Summary of Quasi-Convex Optimization based Parameterized Model Order Reduction QCO competes reasonably well with existing alg (e.g. PRIMA) for reducing large systems But in addition: QCO can reduce models with frequency dependent matrices QCO is very flexible in imposing constraints such as stability and passivity QCO can be extended to parameterized MOR problems

80. 80 Parameterized Model Order Reduction Problem Classification Reducing Linear Systems  Moment Matching with Linear parameters  Moment Matching with NON-linear parameters  Quasi Convex Optimization approach Reducing NON-linear systems  Moment Matching with NON-linear parameters

81. 81 “Field Solver Accurate” and Automatic Parameterized Reduced Order Modeling of NonLinear Analog and MEMs Components Luca Daniel, M.I.T. luca@mit.edu with contributions from Brad Bond www.rle.mit.edu

82. 82 Parameterized Model Order Reduction. Applications interconnect RF inductors matrix size # parameters linearity linearly parameterized non-linearly parameterized Linear Time Invariant Non-Linear Systems LO LNA ADC MEMS Packages

83. 83 Case 3. Reduction of NonLinear Systems Non-Linear analog components e.g. MEMs, VCO, LNA Parameterized NonLinear Reduction PDE Field Solvers or Circuit Simulators

84. Parameterized Model Order Reduction. Linear Systems Moment Matching approaches - (Pullela97, Gunupudi00, Prud’homme02, Daniel02, Li05) -Can handle nonlinear dependence on parameter -Can handle extremely large systems Optimization based approaches - (Sou05) -Good for fitting data from measurements -Cannot construct large order reduced models Statistical Data Mining - (Liu02) -Can handle nonlinear parameter dependence and nonlinear systems -Does not work well for extremely large systems Truncated Balance Realizations - (Heydari01, Phillips04) - Does not handle extremely large systems

85. Previous work on Non-Parameterized MOR for nonlinear systems Representation of F(x) using a polynomial (e.g. Taylor’s expansions, Volterra Series) [Phillips00]***** Representation of F(x) using a polynomial (e.g. Taylor’s expansions, Volterra Series) [Phillips00]***** Representation of F(x) using several linearizations (Trajectory Piece-Wise Linear TPWL) [Rewienski01] Representation of F(x) using several linearizations (Trajectory Piece-Wise Linear TPWL) [Rewienski01] Representation of F(x) with several polynomials (PWP PieceWise Polynomial) [Dong03] Representation of F(x) with several polynomials (PWP PieceWise Polynomial) [Dong03]

86. x3x3 x1x1 x7x7 x6x6 x5x5 Use collection of linear models Background – TPWL [Reiwenski01] x1x1 x2x2 x 10 x8x8 x4x4 x2x2 x 14 x 15 xaxa xbxb x0x0 x 11 x9x9 x 13 #linearizations =#samples n n = 10 4 #samples = 100 100 10000 = LARGE Model i only valid near x i

87. Background – TPWL: Picking Linearization Points x1x1 x2x2 t y(t) Use training trajectories to pick linearization points Linearization at current state x i State Space Time Domain Simulation

88. Background – TPWL: Weighting / Simulation [Riewinski01,Tiwary05,Dong05] Use weighting functions to combine linear models during simulation x1x1 x2x2 Current state Linearization 3 Linearization 2 Linearization 1 C – poorly approximated also well approximated Well approximated

89. 89 Background – TPWL Reduction of the Linearized Systems Model from linearization 1 A1A1 A1A1 K1K1 K1K1 Model from linearization 2 A2A2 A2A2 K2K2 K2K2 Model from linearization k AkAk AkAk KkKk KkKk =

90. 90 Background – TPWL Constructing the projection matrix V Use moments from EACH linear model to construct V =

91. Outline Introduction Background NonLinear Parameterized Model Order Reduction (NLPMOR) –Constructing the system –How to pick linearization points –How to construct V Examples Results Conclusions Future Work

92. NLPMOR – Constructing the System Start with system possessing nonlinear dependence on state x and parameters s j Obtain Linear dependence on new parameters Linearize to get weighted sum of linear models By linear approximation, polynomial fitting to data points… Single linear model Weighting basis functions

93. Non-Linear Parameterized Reduction [Bond, Daniel ICCAD05] x1x1 x2x2 time y(t) Use training trajectories to pick linearization points s1s1 s2s2 Train again at different points in parameter space Parameter Space Repeat at different points in parameter space to populate state space State Space

94. Non-Linear Parameterized Reduction [Bond, Daniel ICCAD05] Populate relevant regions of state-space with linear models from training at different inputs or different points in parameter-space Parameter Space s2s2 s1s1 x1x1 x2x2 State Space

95. NLPMOR – Constructing V: 2 Options Single variable Taylor series expansion of x for each model, as in TPWL p-variable Taylor series expansion of x as in PMOR, but for each model OR “PMOR V” “MOR V” Option 1 Option 2

96. 96 Non-Linear Parameterized Reduction Constructing V For qth order model with p parameters and k linear models, total number of vectors for V is O(kp m ) Keeping all vectors would results in a huge “reduced” model Hence we perform an SVD on V and keep only the q most important vectors SVD NN O(kp m )q

97. NLPMOR – 4 Algorithms Train at single p-space point Train at multiple p-space points Linearize in parameters s j Original Model Train to pick linearization points x i MOR V PMOR V MOR V PMOR V Parameter Space MOR PMOR TPWL ALG1ALG2 ALG3 Single p- space point training Multiple p- space point training MOR V PMOR V TPWL ALG2 ALG1 ALG3

98. Outline Introduction Background NLPMOR Examples –Circuit –Beam Results Conclusions Future Work

99. Analog Circuit Example Nonlinear terms Analog circuit with nonlinear components distributed throughout Using constitutive relations for each element, apply KCL at each node to obtain state-space model Picture by Michał Rewienski

100. Analog Circuit Example – State Space System State-Space System Resistor values Turn on voltage Saturation current Capacitor values Possible parameters Single equation at node j

101. Analog Circuit Example – One possible parameterized model Selected parameter: α Obtained linear dependence on parameters Time (s) Voltage (V) Full Model ALG1 ROM Training point Evaluation point Single point training Multiple point training MOR V PMOR V ALG1

102. Micromachined Switch Example Governing Equations Discretize Picture by Michał Rewienski

103. Micromachined Switch Example State Space System State-Space System Beam width Material Properties Beam height Possible Parameters

104. Micromachined Switch Example One possible parameterized model u(t) = 5.5 2, t > 0 Full Model ALG2 ROM Training point Evaluation point Single training Multiple training MOR V PMOR V ALG2 System parameterized in

105. Outline Introduction Background NLPMOR Examples Results –Algorithm Comparison –Algorithm Cost Conclusions Future Work

106. Analog Circuit Example – Algorithm 1 1 0.9 1.4 1e-101.5e-100.5e-10 Full Model ALG1 ROM - Training points - Evaluation points Single training Multiple training MOR V PMOR V ALG1

107. Results – Benefit of PMOR V I d0 Accuracy IdId TPWL – MOR V ALG1 – PMOR V Trained at I d0 Single p- space point training Multiple p- space point training MOR V PMOR V TPWL ALG1 Error TPWL ALG1 IdId I d0 Analog Circuit TPWL ROM ALG1 ROM

108. Analog Circuit Example 1 0.5 1.5 1e-101.5e-100.5e-10 This model is parameterized in I d and 1/R, and was created using ALG2 2 2e-10 - Training points - Evaluation points Full Model ALG2 ROM Single training Multiple training MOR V PMOR V ALG2

109. Results – Benefit of multiple training points in p-space Single p- space point training Multiple p- space point training MOR V PMOR V TPWL ALG2 I d0 Accuracy IdId TPWL – trained at I d0 ALG2 – trained at I d1, I d2 I d2 I d1 I d2 I d0 % Error TPWL ALG2 % Error time (s) MEMs Switch TPWL ROM ALG2 ROM Analog Circuit

110. Results – Algorithm Comparisons Each algorithm covers a different region of p-space Parameter Space Validity of Models S1S1 S2S2 s 2A s 1A s 1B s 2B All algorithms produce model with same reduced order and same number of linear pieces Single p- space point training Multiple p- space point training MOR V PMOR V TPWL ALG2ALG1 ALG3

111. Results – Algorithm Comparisons Each algorithm has a different cost Algorithm Cost of constructing V (in system solves) Training cost (trajectories made per input) TPWL ALG 1 ALG 2 ALG 3 O(km) O(kp m ) O(kr p m) O(kr p p m ) 1 rprp rprp 1 Majority of cost lies in training Majority of cost lies in making vectors for V Expensive in both training and creating V Cheap p - # of parameters k - # of linear models per trajectory r - # of parameter values used m - # of moments matched per parameter Single point training Multiple point training MOR V PMOR V TPWL ALG2 ALG1 ALG3

112. Conclusions NLPMOR –Captures nonlinear effects –Preserves parameter dependence Algorithm choice is situation/system dependent –TPWL, ALG1 can provide high accuracy locally in parameter space –ALG2, ALG3 can provide global accuracy in parameter space –ALG1, ALG3 : expensive to create V –ALG2, ALG3 : expensive to train Single point training Multiple point training MOR V PMOR V TPWL ALG2 ALG1 ALG3

113. Future Work in Parameterized Model Order Reduction of Non-Linear Systems Ensure stability and passivity of Non-Linear reduced order models Try other approaches for Projection Matrix –e.g. TBR, Quasi-Convex Optimization Try other approaches for capturing nonlinearity –e.g. Volterra series, or piece-wise polynomial

114. Summary of the course Model Order Reduction of Linear Systems  Krylov + TBR two step procedure 1 st step: Krylov Moment Matching Projection  PVL or PRIMA (for passive systems) 2 nd step Truncated Balance Realization (TBR) or Passive-TRB  Distributed Passive Systems: Quasi Convex Optimization Laguerre interpolation Model Order Reduction of NonLinear Systems  Weakly Nonlinear: use Volterra series + moment matching  Strongly Nonlinear: use TPWL + moment matching (and/or TBR) Parameterized Model Order Reduction  Linear: moment matching OR quasi-convex optimization  NonLinear: TWPL + moment matching