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Parameterized Model Order Reduction via Quasi-Convex Optimization

Parameterized Model Order Reduction via Quasi-Convex Optimization. Kin Cheong Sou with Luca Daniel and Alexandre Megretski. Digital. Analog RF. RF Inductors. Mixed Signal. I. DSP. ADC. MEM resonators. LNA. ADC. Q. LO. Systems on Chip or Package. Interconnect & Substrate.

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Parameterized Model Order Reduction via Quasi-Convex Optimization

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  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

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