Yiyu Shi, Lei He Electrical Engineering Department, UCLA

1. Yiyu Shi, Lei He Electrical Engineering Department, UCLA http//:eda.ee.ucla.edu EMPIRE: An Efficient and Compact Multiple-Parameterized Model Order Reduction Method for Physical Optimization This work is partially supported by NSF Career Award and a UC Micro grant sponsored by Analog Devices, Intel and Mindspeed.

2. Outline • Background and motivation • EMPIRE algorithm • Experimental results • Conclusions

3. Parameterized MOR • Most physical design and optimization problems involve nonlinear optimization • Decap allocation, shields insertion, thermal via planning, and structured P/G clock network sizing • Sensitivities are needed to linearize the nonlinear objection function • Parameterized model order reduction can generate macromodels with all the parameters preserved • The moments of the parameters of the design (POD) are exactly the sensitvities • Moment matching sensivity matching • Previous works • [Daniel:TCAD’04] extends PRIMA to handle parameterized systems • But can handle only a small number of parameters and match moments up to a low order. • CORE [Li:ICCAD’05] uses explicit-and-implicit moment matching for parameterized interconnect model reduction • Still cannot match the moments of a huge number of parameters to a very high order • Cannot match the moments of different parameters with different accuracy

4. Importance of High Order POD Moments • Output integral w.r.t. a randomly selected parameter (pitch width) for a P/G mesh • The reduced model by CORE cannot match the original well when the reduced order is less than 70.

5. Major Contribution of EMPIRE • It is an efficient yet accurate model order reduction method for physical design with multiple parameters: • It uses implicit moment matching to match high order POD moments, • more accurate than the explicit moment matching used in CORE; • It can match the moments of different PODs with different accuracy according to their influence on the objective. • Experimental results show that compared with CORE and [Daniel:TCAD’04], EMPIRE reduces error by 47.8X at a similar runtime.

6. Outline • Background • EMPIRE algorithm • Experimental results • Conclusions

7. Framework of EMPIRE

8. Parameter Number Reduction • Canonical form of a general parameterized system (E0 + E1s1 + E2s2 + … + Etst) x = Bu y=LTx, • Define the significance of a parameter si as Any value in the range of si • Theorem 1: SIG(si) is the perturbation magnitude of si to the output. • Therefore, we can neglect the parameters that have relative small SIG values and thus reduce the total number of parameters.

9. Verification • Normalized output perturbation w.r.t. the 2-norm of the coefficient matrix • With the increase of the norm, the perturbation increases.

10. Framework of EMPIRE

11. Projection Space Collapse • Find the original projection matrix by the traditional algorithms • Calculate a new projection matrix V so that the weighted distance between colspan(V) and colspan( ) is minimized, i.e., Directly solve optimization problem • Three different methods • Nonlinear Programming (NP) • Sequential Least Square (sLS) • Sequential Barycenter Allocation (sBA) Iteratively solve the optimization problem sLS sBA NP Use quadratic approximation runtime accuracy

12. Three different methods • Find a new projection matrix V so that the weighted distance between colspan(V) and colspan( ) is minimized, i.e., • Nonlinear Programming (NP) • Directly solve the optimization problem • Expensive but provides optimal solution • Can be used for small scale problem • Sequential Least Square (sLS) • Solve the optimization problem incrementally • Each time find one column vector in that has the smallest distance to the vectors in V orthogonalized by the ones already found. • Sequential Barycenter Allocation (sBA) • Use the barycenter to approximate the optimal solution of sLS.

13. Verification • The weighted distance between two subspaces w.r.t the reduced order • With the increase of the reduced order, the distance decreases to zero. • sBA converges to zero the slowest. • NP converges to zero the fastest

14. Framework of EMPIRE

15. Frequency Domain Moment Expansion & Projection • Frequency domain moments are critical to the waveform accuracy matching more moments! • Let and , then Coefficient matrix corresponding to frequency variable Match up to q-th order of frequency domain moments • Projection

16. Outline • Background • EMPIRE algorithm • Experimental results • Conclusions

17. Experimental Settings • We use different sizes of extracted RLC meshes from industrial applications. • All the algorithms are implemented in MATLAB • We use a Linux workstation (P4 2.66G CPU and 2G RAM) • We compare the runtime, time/frequency domain accuracy and scalability of our hybrid algorithm with [Daniel:TCAD’04] and CORE.

18. Waveform Comparison (a) (b) • P/G RC meshes with 10000 nodes and 5000 parameters (pitch width) • EMPIRE is identical to the original in both time domain (a) and frequency domain (b) , more accurate compared with CORE and [Daniel:TCAD’04].

19. Waveform Comparison • Output integral w.r.t. a randomly selected parameter (pitch width) • EMPIRE is close to the original, more accurate than CORE and [Daniel:TCAD’04]

20. Scalability Comparison • Time domain waveform relative error w.r.t reduction size • EMPIRE has less error and coverges faster than CORE.

21. Runtime for EMPIRE • Runtime for EMPIRE w.r.t different original circuit size • A: NP (small scale problems) • B: sLS (medium scale problems) • C: SBA (large scale problems)

22. Runtime Comparison • Runtime comparison between the three methods on RC meshes of different scales. • EMPIRE has a similar runtime compared with CORE, and is 18.3X faster than [Daniel:TCAD’04] for model reduction time and 61.2X faster for simulation time. • In addition, [Daniel:TCAD’04] cannot finish large examples.

23. Conclusions and future work • We have developed an efficient yet accurate model order reduction method EMPIRE for physical design with multiple parameters: • Compared with CORE, with a small reduction size, it uses implicit moment matching to match high order POD moments, • more accurate than the explicit moment matching used in CORE; • It can match the moments of different PODs with different accuracy according to their influence on the objective. • Experimental results show that compared with CORE and [Daniel:TCAD’04], EMPIRE results in 47.8X improved accuracy at a similar runtime. • We will extend the EMPIRE algorithm and apply it in real physical design problems.

24. Thank You!