Parameterized Model Order Reduction via Quasi-Convex Optimization

1. Parameterized Model Order Reduction via Quasi-Convex Optimization Kin Cheong Sou with Luca Daniel and Alexandre Megretski

2. Digital Analog RF RF Inductors Mixed Signal I DSP ADC MEM resonators LNA ADC Q LO Systems on Chip or Package Interconnect & Substrate Courtesy of Harris semiconductor

3. Fig. thanks to Coventor From 3D Geometry to Circuit Model • Need accurate mathematical models of components • Describe components using Maxwell equations, Navier-Stokes equations, etc

4. Z(f) Z(f) Z(f) Z(f) From 3D Geometry to Circuit Model • Model generated by available field solver • Field solver models usually high order

5. RF Inductor Model Reduction • Spiral radio frequency (RF) inductor • Impedance • State space model has 1576 states • Reduced model has 8 states R L x full 1576 states - reduced 8 states x full 1576 states - reduced 8 states f f

6. d w RF Inductor Parameter Dependency • Parameter dependent RF inductor • Two design parameters: • Wire width w • Wire separation d d = 1um d = 3um d = 5um R L d = 1um d = 3um d = 5um f f

7. d I DSP ADC w LNA ADC Q LO Parameterized Reduced Modeling • Onereduced model with explicit dependency on parameters • Fast generation of reduced model for all parameter values Parameterized reduced model Gr(d,w) reduced model

8. Parameterized Reduced Model Example • Parameter dependent complex system • Parameterized reduced order model • Coefficients depend explicitly on d • Low order, inexpensive to simulate

9. Continuous/Discrete-time Setups Continuous-time Discrete-time left-half plane & imaginary axis unit disk & unit circle

10. Parameterized Model Reduction Methods • Parameterized moment matching methods • References: • [Grimme et al. AML 99] • [Daniel et al. TCAD 04] • [Pileggi et al. ICCAD 05] • [Bai et al. ICCAD 07] • Reduced model order increases with number of parameters rapidly • Require knowledge of state space model • Rational transfer function fitting methods • Does not require state space model • Reduced model order does not increase with number of parameters • More expensive than moment matching in general

11. Moment Matching Method Reduced model Full model Projection with UV = I moments matched with the moment matching properties 8th order full 4th order MM user specified

12. Rational Transfer Function Fitting • Idea from system ID – reduced model matching I/O data input output • Data in time domain or frequency domain • Data from state space model or experiment measurement

13. Explanations in Two Steps • Will present a rational transfer function fitting method • First describe basic non-parameterized reduction • Then extend basic method to parameterized setup

14. Non-Parameterized Model Order Reduction

15. Non-parameterized Problem Statement • Given transfer function G(z) • Find parameterized reduced model of order r dec. vars. • Reduced model found as the solution H norm error subject to roots inside unit circle • Can obtain state space realization from p(z) and q(z)

16. Difficulty with H Norm Reduction • Difficulty #2: abs. value on the “wrong” side iff convex combo. of stable poly. not necessarily stable branching solutions • Difficulty #1: stability constraint not convex if r >2 but

17. Idea From Optimal Hankel Reduction anti-stable relaxation s.t. s.t. suboptimal solution redefine dec. var. Solve AAK problem efficiently (Glover) s.t.

18. Anti-Stable Relaxation in Rational Fit • Similar to Hankel reduction, add anti-stable term added DOF flip poles of q(z) subject to • In Hankel MR, entire anti-stable term is decision variable • Here, only numerator f is decision variable

19. Combine Stable and Anti-stable Terms • Combine stable and anti-stable terms in reduced model • New decision variables are trigonometric polynomials

20. Stability and Positivity • Can show • Overcome Difficulty #1, trigonometric positivity  convex constraint • Overcome Difficulty #2, the trouble making abs. value is gone! a2 a1

21. Quasi-Convex Relaxation • Original optimal H norm model reduction problem subject to • Quasi-convex relaxation Quasi-convex problem, easy to solve subject to

22. From Relaxation Back to H Reduction • Obtain (a,b,c) by solving quasi-convex relaxation • Spectral factorize a to obtain stable denominator q* for some K • With q* found, search for numerator p* by solving convex problem

23. Quality of Suboptimal Reduced Model • Minimizing upper bound of Hankel norm error • H norm error upper bound

24. Back to Big Picture – Model Reduction discussed s.t. s.t. How to solve it? discussed discussed suboptimal p(z), q(z) optimal a(z), b(z), c(z)

25. Quasi-Convex Optimization

26. Quasi-Convex Optimization • Quasi-convex function is “almost convex” J(x) Function not necessarily convex All sub-level sets are convex sets x Local (also global) minima Local (but not global) minima • [Outer loop] Bisection search for objective value • [Inner loop] Convex feasibility problem (e.g. LP, SDP) • Convex problem algorithms: 1) interior-point method • 2) cutting plane method

27. iterate 2 iterate 1 Cutting Plane Method optimal point covering set call oracle call oracle Oracle return cut return cut kept removed kept removed • Optimization problem data described by oracle • What is the oracle in our model reduction problem?

28. Model Reduction Oracles • Given candidate a(z), b(z), c(z), check two conditions Oracle #1 (objective value): Discretize frequency  finite number of linear inequalities, “easy” Oracle #2 (positive denominator): Cannot discretize frequency!

29. Positivity Check • Check only finite number of stationary points r = 8 case stationary points  • Much harder to check in the parameterized case

30. Back to Big Picture – Model Reduction discussed s.t. s.t. Solved with cutting plane method discussed discussed suboptimal p(z), q(z) optimal a(z), b(z), c(z)

31. Parameterized Model Order Reduction

32. Problem Statement • Given parameter dependent transfer function G(z,d) • Find parameterized reduced model of order r • Reduced model found as the solution design parameter stable for all d subject to

33. Parameterized Reduced Model Example • Parameter dependent complex system • Parameterized reduced order model • Coefficients depend explicitly on d • Low order, inexpensive to simulate

34. Parameterized Decision Variables • Decision variables = parameterized trig. poly. • When evaluated on unit circle, i.e.

35. Parameterized Quasi-Convex Relaxation • Parameterized quasi-convex relaxation subject to • Solution technique similar to non-parameterized case • Some extension requires more care, e.g. Parameterized positivity check is hard!

36. Parameterized Positivity Check denominator … a simple parameter dependency denominator = multivariate trigonometric polynomial e.g. • Positivity check of multivariable trig. poly. is hard • Another variant is multivariable ordinary polynomial our focus

37. Positivity Check of Multivariate Polynomials

38. Checking Polynomial Positivity – Special Cases • Univariate case simple, check the roots of derivative Is it true for all x, • Multivariate quadratic form is easy but important polynomial nonnegative matrix positive semidefinite

39. Checking Polynomial Positivity – General Case • Positivity check of general multivariate polynomial is hard [from Parrilo & Lall] Question: • What if we still write out “quadratic form”? Monomials of relevant degrees = Q(Gram matrix)

40. Checking Polynomial Positivity – General Case • To find Q, equate coefficients of all monomials Generally, linear constraints on Q, i.e.L(Q) = 0 • Gram matrix Q is typically not unique. If we can find Q ≥ 0

41. Semidefinite Program/LMI Optimization • Standard form: Read Boyd and Vandenberghe’s SIAM review linear objective linear constraints pos. def. matrix variable • Efficiently solvable in theory and practice • Polynomial-time algorithm available • Efficient free solvers: SeDuMi, SDPT3, etc. • Lots of applications • KYP lemma, Lyapunov function search, filter design, circuit sizing, MAX-CUT, robust optimization …

42. Positivity Check is Sufficient Only Quadratic case General case spans R3 does not span R3 • Requiring Q ≥ 0 sufficient but not necessary! Positive? Can you find Q ≥ 0?

43. Sum of Squares (SOS) • Finding Q ≥ 0 equivalent to sum of squares decomposition • In our example, we can find sum of squares  positive semidefinite Q  nonnegativity

44. Wrap Up

45. d w xfull model -QCO PROM 4 Turn RF Inductor PMOR • 4 turn RF inductor with substrate • Circle: training data • Triangle: test data

46. Summary (1) • Motivation for model reduction in design automation • PDE  high order ODE  reduced ODE • Parameterized reduced modeling facilitates design • Model reduction based on rational transfer function fitting • H problem difficult, resort to anti-stable relaxation • Relaxation easy to solve, closely related to H problem • Quasi-convex optimization • Efficient algorithms exist (e.g. cutting plane method) • Cutting plane method in model reduction setting

47. Summary (2) • Parameterized model reduction • Reduced rational transfer function, coefficients are function of design parameters • Easily extended from non-parameterized case, except positivity check is difficult • Positivity check of multivariate polynomials • Univariate case easy, quadratic case easy • General case requires semidefinite programs, only sufficient • Related to sum of squares optimization

48. Some References (1) • Parameterized reduced modeling • Moment matching: Eric Grimme’s PhD thesis • Parameterized moment matching: • L. Daniel, O. Siong, C. L., K. Lee, and J. White, “A multiparameter moment matching model reduction approach for generating geometrically parameterized interconnect performance models,” IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, vol. 23, no. 5, pp. 678–693. • Parameterized rational fitting: • Kin Cheong Sou; Megretski, A.; Daniel, L.; , "A Quasi-Convex Optimization Approach to Parameterized Model Order Reduction," Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on , vol.27, no.3, pp.456-469, March 2008 • MIMO rational fitting/interpolation: • A. Sootla, G. Kotsalis, A. Rantzer, “Multivariable Optimization-Based Model Reduction”, IEEE Transactions on Automatic Control,54:10, pp. 2477-2480, October 2009 • Lefteriu, S. and Antoulas, A. C. 2010. A new approach to modeling multiport systems from frequency-domain data. Trans. Comp.-Aided Des. Integ. Cir. Sys. 29, 1 (Jan. 2010), 14-27

49. Some References (2) • Convex/quasi-convex optimization • Convex optimization: • S. Boyd and L. Vandenberghe, “Convex Optimization”, Cambridge University Press, 2004. • Ellipsoid Cutting plane method: • Bland, Robert G., Goldfarb, Donald, Todd, Michael J. Feature Article--The Ellipsoid Method: A SurveyOPERATIONS RESEARCH 1981 29: 1039-1091 • Multivariate polynomials and sum of squares • Ordinary polynomial case: PabloParrilo’s PhD thesis • Trigonometric polynomial case: • B. Dumitrescu, “Positive Trigonometric Polynomials and Signal Processing Applications”, Springer, 2007