Model Order Reduction

1. Model Order Reduction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Joel Phillips, Cadence Berkeley Labs Jacob White, Massachusetts Instit. of Technology

2. Funct. Spec RTL Behav. Simul. Stat. Wire Model Logic Synth. Gate-level Net. Front-end Gate-Lev. Sim. Floorplanning Back-end Parasitic Extrac. Place & Route Layout Conventional Design Flow

3. Layout parasitics • Wires are not ideal. Parasitics: • Resistance • Capacitance • Inductance • Why do we care? • Impact on delay • noise • energy consumption • power distribution Picture from “Digital Integrated Circuits”, Rabaey, Chandrakasan, Nikolic

4. Parasitic Extraction thousands of wires e.g. critical path e.g. gnd/vdd grid Parasitic Extraction • identify some ports • produce equivalent circuit that models response of wires at those ports tens of circuit elements for gate level spice simulation

5. Electromagnetic Analysis (Tuesday) small surface panels with constant charge thin volume filaments with constant current million of elements Model Order Reduction (Today) tens of elements Parasitic Extraction (the two steps)

6. Why build reduced models? • Compression for Efficiency • It is possible to represent the system under study “precisely” with millions of elements • But the simulation is too slow with the complicated representation • Abstraction • I do not care at all about the precise representation • In fact I would rather those details were not even there. I may not be able to create or manipulate the precise representation at all.

7. Challenges for reduction algorithms • Accuracy • Must be controllable and predictable • Efficiency • Algorithms should be scalable to handle large systems • Numerical robustness • Algorithms should work reliably for all reasonable inputs & accuracy requests • Models must work in simulation • Composability : Combination of two good models is a good model

8. Overview • Introduction and Motivations • State-space models • Reduction via eigenmode analysis • Reduction via rational function fitting (point matching) • Reduction via moment matching (Pade, AWE) • Reduction via moment matching: (Projection Framework) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • Importance of preserving passivity • PRIMA

9. State-Space Models • Linear system of ordinary differential equations (ABCD form) Input State Output

10. State-Space Model Example:Interconnect Segment • Step 1: Identify internal state variables • Example : MNA uses node voltages & inductor current

11. + + - - State-Space Model Example:Interconnect Segment • Step 2: Identify inputs & outputs • Example : For Z-parameter representation, choose port currents inputs and port voltage outputs

12. + + - - State-Space Model Example:Interconnect Segment • Step 3: Write state-space & I/O equations • Example : KCL + inductor equation

13. State-Space Model Example:Interconnect Segment • Step 4: Identify state variables & matrices

14. LARGE! A linear circuit can be expressed as a state space model • So in general….

15. A canonical form for model order reduction Assuming A is non-singular we can cast the dynamical linear system into one canonical form for model order reduction Note: not necessarily always the best, but the simplest for educational purposes

16. Construct a linear system model with: • smaller complexity • same fidelity • small reduction cost 10 x 10 Our goal: smaller model, still accurate • Given a large linear system model: 500,000 x 500,000

17. Frequency Domain Representation Bilateral Laplace Transform: Key Transform Property:

18. Transfer Function: System Transfer Function Express y(s) as a function of u(s)

19. Connection Between the Transfer FunctionTime Domain Impulse Response Frequency domain representation u(s) y(s) = H(s) u(s) H(s) Linear system Time domain representation u(t) h(t) Linear system The transfer function H(s) is the Laplace Transform of the impulse response h(t)

20. Overview • Introduction and Motivations • State-space models • Reduction via eigenmode analysis • Reduction via rational function fitting (point matching) • Reduction via moment matching (Pade, AWE) • Reduction via moment matching: (Projection Framework) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • Importance of preserving passivity • PRIMA

21. Model Order Reduction via Eigenmode Analysis Pole-Residue Form Pole-Zero Form (SISO) • Ideas for reducing order: • Drop terms with small residues • Drop terms with large (“fast” modes) • Remove pole/zero near-cancellations • Cluster poles that are “together” • How to compute poles and residues?

22. Diagonalize E Computing Poles & Residues • Poles are eigenvalues of E-1 residues poles

23. Eigenvalue Based Reduction • Advantages • Conceptually familiar • Simple physical interpretation : retains dominant system modes/poles • Drawbacks • Relatively expensive : have to find the poles first • Relatively inefficient. For a given model size, many other approaches can provide better accuracy • Rule of thumb • Anything that can be done by manipulating pole/eigenvalues/eigenvectors can probably be done better with more sophisticated analysis, at the same or smaller cost.

24. Defining Accuracy • Time-domain response should be “close” • For which possible inputs? • Frequency response should match • At what frequencies?

25. Matching Frequency Response • Ensure accuracy for only some inputs? • Example: • low frequency inputs, • or some band, • or some points in the frequency response matching some part of the frequency response Original

26. Overview • Introduction and Motivations • State-space models • Reduction via eigenmode analysis • Reduction via rational function fitting (point matching) • Reduction via moment matching (Pade, AWE) • Reduction via moment matching: (Projection Framework) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • Importance of preserving passivity • PRIMA

27. Model Order Reduction via Rational Transfer Function Fitting Original System Transfer Function: rational function Model Reduction = Find a low order (q << N) rational function matching reduced order rational function

28. Rational Transfer Function Fitting: Degrees of Freedom Reduced Model Dynamical System coefficients Reduced Model Transfer Function coefficients

29. Rational Transfer Function Fitting: Degrees of Freedom (cont.) Reduced Model Transfer Function Apply any invertible change of variables to the state I I Many Dynamical Systems have the same transfer function!!

30. Can match 2q points • cross multiplying generates a linear system For i = 1 to 2q Rational Transfer Function Fitting: via Point Matching

31. Rational Transfer Function Fitting: Point Matching matrix can be ill-conditioned • Columns contain progressively higher powers of the test frequencies: problem is numerically ill-conditioned • also... missing data can cause severe accuracy problems

32. Overview • Introduction and Motivations • State-space models • Reduction via eigenmode analysis • Reduction via rational function fitting (point matching) • Reduction via moment matching (Pade, AWE) • Reduction via moment matching: (Projection Framework) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • Importance of preserving passivity • PRIMA

33. Point matching vs. Moment Matching Point matching: can be very inaccurate in between points Moment (derivatives) matching: accurate around expansion point, but inaccurate on wide frequency band

34. Frequency Domain "Moments" (or Taylor coefficients) of the transfer function Taylor Series Expansion of the original transfer function around s=0 The Taylor coef. = frequency domain moments = = derivatives of the transfer function (up to a constant)

35. Time domain moments of the impulse response Definition:

36. Compare: Hence the the Taylor coeff. are, up to a constant, the time-domain moments of the circuit response. Connection to the time-domain moments of the circuit response Time-domain moments

37. Rational function fitting via moment matching: Pade Approximation (AWE)

38. Rational function fitting via moment matching: Pade Approximation (AWE) • Step 1: calculate the first 2q moments of H(s) • Step 2: calculate the 2q coeff. of the Pade’ approx, matching the first 2q moments of H(s)

39. Step 2: Calculation of Pade’ coeff. (AWE) For coeff. a’s solve the following linear system: For coeff. b’s simply calculate:

40. Pade matrix can be very ill-conditioned • matrix powers converge to the eigenvector corresponding to the largest eigenvalue. Columns become linearly dependent for large q the problem is numerically very ill-conditioned!

41. Overview • Introduction and Motivations • State-space models • Reduction via eigenmode analysis • Reduction via rational function fitting (point matching) • Reduction via moment matching (Pade, AWE) • Reduction via moment matching: (Projection Framework) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • Importance of preserving passivity • PRIMA

42. Projection Framework:Change of variables reduced state Note: q << N original state

43. Projection Framework • Original System • Substitute • Note: now few variables (q<<N) in the state, but still thousands of equations (N)

44. Projection Framework (cont.) • Reduction of number of equations: test multiplying by VqT • If V and U biorthogonal

45. Projection Framework (cont.) qxn qxq nxn nxq

46. Projection Framework Change of variables Equation Testing

47. Approaches for picking V and U • Use Eigenvectors • Use Time Series Data • Compute • Use the SVD to pick q < k important vectors • Use Frequency Domain Data • Compute • Use the SVD to pick q < k important vectors • Use Singular Vectors of System Grammians? • Use Krylov Subspace Vectors?

48. Overview • Introduction and Motivations • State-space models • Reduction via eigenmode analysis • Reduction via rational function fitting (point matching) • Reduction via moment matching (Pade, AWE) • Reduction via moment matching: (Projection Framework) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • Importance of preserving passivity • PRIMA

49. Intuitive view of Krylov subspace choice for change of base projection matrix Taylor series expansion: • change base and use only the first few vectors of the Taylor series expansion: equivalent to match first derivatives around expansion point U

50. Combine point and moment matching: multipoint moment matching • Multipole expansion points give larger band • Moment (derivates) matching gives more • accurate behavior in between expansion points