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

Systems on Chip or Package Interconnect & Substrate Courtesy of Harris semiconductor RF Inductors MEM resonators 210/22/2010 DSP Digital LNA LO Analog RF ADC Mixed Signal I Q

From 3D Geometry to Circuit Model Fig. thanks to Coventor Need accurate mathematical models of components Describe components using Maxwell equations, Navier- Stokes equations, etc 310/22/2010

From 3D Geometry to Circuit Model Z(f)Z(f)Z(f)Z(f) Model generated by available field solver Field solver models usually high order 410/22/2010

RF Inductor Model Reduction Spiral radio frequency (RF) inductor Impedance State space model has 1576 states Reduced model has 8 states RL ff x full 1576 states - reduced 8 states x full 1576 states - reduced 8 states 510/22/2010

Parameter dependent RF inductor Two design parameters: -Wire width w -Wire separation d d w ff RL d = 1um d = 3um d = 5um d = 1um d = 3um d = 5um 610/22/2010 RF Inductor Parameter Dependency

Parameterized Reduced Modeling Parameterized reduced model G r (d,w) One reduced model with explicit dependency on parameters Fast generation of reduced model for all parameter values 710/22/2010 d w DSP LNA LO ADC I Q reduced model

Parameterized Reduced Model Example Parameter dependent complex system Parameterized reduced order model Coefficients depend explicitly on d Low order, inexpensive to simulate 810/22/2010

Continuous/Discrete-time Setups 10/22/20109 Continuous-timeDiscrete-time left-half plane & imaginary axis unit disk & unit circle

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 1010/22/2010 Parameterized Model Reduction Methods

Moment Matching Method 10/22/ Projection with UV = I Full model Reduced model with the moment matching properties 8 th order full 4 th order MM moments matched user specified

Rational Transfer Function Fitting 10/22/ input output Idea from system ID – reduced model matching I/O data Data in time domain or frequency domain Data from state space model or experiment measurement

Explanations in Two Steps 10/22/ Will present a rational transfer function fitting method First describe basic non-parameterized reduction Then extend basic method to parameterized setup

Non-Parameterized Model Order Reduction

Non-parameterized Problem Statement Given transfer function G(z) Find parameterized reduced model of order r subject to 1510/22/2010 Reduced model found as the solution dec. vars. roots inside unit circle H  norm error Can obtain state space realization from p(z) and q(z)

Difficulty with H  Norm Reduction 10/22/ Difficulty #2: abs. value on the “wrong” side iff Difficulty #1: stability constraint not convex if r >2 but branching solutions convex combo. of stable poly. not necessarily stable

Idea From Optimal Hankel Reduction 10/22/ s.t. anti-stable relaxation redefine dec. var. Solve AAK problem efficiently (Glover) suboptimal solution

Anti-Stable Relaxation in Rational Fit 10/22/ Similar to Hankel reduction, add anti-stable term subject to added DOF In Hankel MR, entire anti-stable term is decision variable Here, only numerator f is decision variable flip poles of q(z)

Combine Stable and Anti-stable Terms 10/22/ Combine stable and anti-stable terms in reduced model New decision variables are trigonometric polynomials

Stability and Positivity 10/22/ Can show Overcome Difficulty #1, trigonometric positivity  convex constraint a1a1 a2a2 Overcome Difficulty #2, the trouble making abs. value is gone!

Quasi-Convex Relaxation 10/22/ Quasi-convex relaxation subject to Original optimal H  norm model reduction problem subject to Quasi-convex problem, easy to solve

From Relaxation Back to H  Reduction 10/22/ 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

Quality of Suboptimal Reduced Model 10/22/ Minimizing upper bound of Hankel norm error H  norm error upper bound

Back to Big Picture – Model Reduction 10/22/ s.t. optimal a(z), b(z), c(z) suboptimal p(z), q(z) discussed How to solve it?

Quasi-Convex Optimization

10/22/ J(x)J(x) x All sub-level sets are convex sets Quasi-convex function is “almost convex” Local (also global) minimaLocal (but not global) minima Function not necessarily convex [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

Cutting Plane Method optimal point covering set iterate 1 iterate /22/2010 Optimization problem data described by oracle What is the oracle in our model reduction problem? Oracle call oracle return cut kept removed call oracle return cut kept removed

Model Reduction Oracles 10/22/ Oracle #1 (objective value): Oracle #2 (positive denominator): Discretize frequency  finite number of linear inequalities, “easy” Given candidate a(z), b(z), c(z), check two conditions Cannot discretize frequency!

Positivity Check 10/22/ stationary points r = 8 case Check only finite number of stationary points Much harder to check in the parameterized case 

Back to Big Picture – Model Reduction 10/22/ s.t. optimal a(z), b(z), c(z) suboptimal p(z), q(z) discussed Solved with cutting plane method

Parameterized Model Order Reduction

Problem Statement Given parameter dependent transfer function G(z,d) Find parameterized reduced model of order r subject to stable for all d 3210/22/2010 Reduced model found as the solution design parameter

Parameterized Reduced Model Example Parameter dependent complex system Parameterized reduced order model Coefficients depend explicitly on d Low order, inexpensive to simulate 3310/22/2010

3410/22/2010 Parameterized Decision Variables Decision variables = parameterized trig. poly. When evaluated on unit circle, i.e.

Parameterized Quasi-Convex Relaxation 10/22/ subject to Parameterized quasi-convex relaxation Solution technique similar to non-parameterized case Some extension requires more care, e.g. Parameterized positivity check is hard!

Parameterized Positivity Check 10/22/ 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 …

Positivity Check of Multivariate Polynomials

Checking Polynomial Positivity – Special Cases 10/22/ Univariate case simple, check the roots of derivative Is it true for all x, Multivariate quadratic form is easy but important polynomial nonnegativematrix positive semidefinite

Checking Polynomial Positivity – General Case 10/22/ Positivity check of general multivariate polynomial is hard Question: [from Parrilo & Lall] = Q (Gram matrix) Monomials of relevant degrees What if we still write out “quadratic form”?

Checking Polynomial Positivity – General Case 10/22/ To find Q, equate coefficients of all monomials Gram matrix Q is typically not unique. If we can find Q ≥ 0 Generally, linear constraints on Q, i.e. L(Q) = 0

Semidefinite Program/LMI Optimization 10/22/ linear objective linear constraints pos. def. matrix variable Standard form: 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 … Read Boyd and Vandenberghe’s SIAM review

Positivity Check is Sufficient Only 10/22/ spans R 3 does not span R 3 Quadratic case General case Requiring Q ≥ 0 sufficient but not necessary! Positive? Can you find Q ≥ 0 ?

Sum of Squares (SOS) 10/22/ Finding Q ≥ 0 equivalent to sum of squares decomposition In our example, we can find sum of squares  positive semidefinite Q  nonnegativity

Wrap Up

4 Turn RF Inductor PMOR d w x full model - QCO PROM 4510/22/ turn RF inductor with substrate Circle: training data Triangle: test data

Summary (1) 10/22/ 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

Summary (2) 10/22/ 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

Some References (1) 10/22/ 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 , 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 , October 2009 Lefteriu, S. and Antoulas, A. C A new approach to modeling multiport systems from frequency- domain data. Trans. Comp.-Aided Des. Integ. Cir. Sys. 29, 1 (Jan. 2010), 14-27

Some References (2) 10/22/ Convex/quasi-convex optimization Convex optimization: S. Boyd and L. Vandenberghe, “Convex Optimization”, Cambridge University Press, Ellipsoid Cutting plane method: Bland, Robert G., Goldfarb, Donald, Todd, Michael J. Feature Article--The Ellipsoid Method: A Survey OPERATIONS RESEARCH : Multivariate polynomials and sum of squares Ordinary polynomial case: Pablo Parrilo’s PhD thesis Trigonometric polynomial case: B. Dumitrescu, “Positive Trigonometric Polynomials and Signal Processing Applications”, Springer, 2007