SAMSON: A Generalized Second-order SAMSON: A Generalized Second-order Arnoldi Method for Reducing Multiple Source Arnoldi Method for Reducing Multiple Source Linear Network with Susceptance Linear Network with Susceptance Yiyu Shi, Hao Yu and Lei He EE Department, UCLA Partially supported by NSF and UC MICRO Analog Devices, Intel and d Mindspeed Mindspeed Partially supported by NSF and UC MICRO Analog Devices, Intel an Address comments to lhe@ee.ucla.edu Address comments to lhe@ee.ucla.edu
Outline Outline � Review and Motivation � SAMSON Algorithm � Experimental Results � Conclusions
Motivation for Model Order Reduction Motivation for Model Order Reduction � Deep submicron design needs to consider a large number of linear elements � Interconnect, Substrate, P/G grid, and Package � Accurate extraction leads to the explosion of data storage and runtime � Need efficient macro-model Nonlinear Elements Nonlinear Elements Reduced Model Linear Elements
Non-RHS MOR vs. RHS MOR Non-RHS MOR vs. RHS MOR � � RHS MOR reduces output vector y(s) Non-RHS MOR reduces transfer function H(s) � [Chiprout, ICCAD’04] � explicit moment matching is used First Order Method to handle L � (lack in numerical stability) � PRIMA [Pileggi et al, TCAD’98] Second Order Methods to handle � � EKS [J. M. Wang et al, DAC’00] S (susceptance) � Implicit moment matching based on � ENOR [Sheehan, DAC’99] Incremental orthonormalization � SMOR [Pileggi et al, ICCAD’02] � frequency domain shifting to deal with � SAPOR [Su et al, ICCAD’04] [Liu 1/s and 1/s 2 terms et al, ASPDAC’05] � IEKS [Y. Lee et al, TCAD’05] � Main Limitation � Based on the observation that there are no 1/s and 1/s 2 terms for PWL Can only match up to ceil(n/n p ) � sources in finite time moments Accuracy is significantly limited � when the port number n p is large
Problems Still Remain Unsolved Problems Still Remain Unsolved � None of the existing RHS MOR methods can deal with RCS circuits with susceptance elements � Both EKS and IEKS are first order methods � Directly applying them to RCS circuits cannot guarantee passivity � There is still much room to improve accuracy � Incremental Orthonormalization causes error to accumulate � When matching high order moments, it becomes inaccurate � None of the existing MOR methods can handle arbitrary independent inputs � Especially when they contain 1/s i terms (i>0) � frequency domain shifting (inaccurate) � In the existence of infinite PWL sources � EKS and IEKS cannot consider s=0 (cannot perform DC analysis)
Major Contributions of SAMSON Major Contributions of SAMSON � It is an RHS MOR method � Compare with SAPOR and other non-RHS methods, it is more accurate � Can handle a large number of ports � It can deal with all kinds of input sources accurately without frequency domain shifting or incremental orthonormalization � Numerically more stable, more efficient and more accurate in the whole frequency domain, especially at DC (s=0) � It is based upon generalized second order Arnoldi method � Can handle RCS circuits with passivity
Outline Outline � Review and Motivation � SAMSON Algorithm � Experimental Results � Conclusions
SAMSON Algorithm: Overview SAMSON Algorithm: Overview Second order state m atrices, incidence m atrices and arbitrary inputs Generalized Current Excitation Vector Generation Augm ented System Transform ation System Linearization Projection Projected output vector
Generalized Current Excitation Vector Generalized Current Excitation Vector Γ � Define the generalized current + + = ( G sC ) V ( s ) BJ excitation vector J ex as e s = J BJ ex e Original system � J ex can be divided into two categories equation � Rational + + + n a a s a s ... = 0 1 n J ( s ) + + + ex m b b s b s ... 0 1 m Superposed � Irrational SIMO � Can be expanded into Taylor series system and take dominant terms. Then it equation becomes a special case of the Rational category n ∑ = − + i m J s J s ( ) m n 1 ∑ ∑ ex i γ = Θ i i s V ( s ) s = i 0 i i � Multiply both sides by = = i 0 i 0 + + + m b b s b n s ... 0 1
SAMSON Algorithm: Overview SAMSON Algorithm: Overview Second order state m atrices, incidence m atrices and arbitrary inputs Generalized Current Excitation Vector Generation Augm ented System Transform ation System Linearization Projection Projected output vector
Augmented System Transformation Augmented System Transformation � To linearize, we have to make + m 1 n ∑ ∑ γ = Θ i i LHS exactly one order higher s V ( s ) s i i than RHS by raising the order of = = i i 0 0 LHS by n-m . Superposed � Introduce auxiliary variables V 1 , V 2 , … , V n satisfying SIMO system Here we assume = = V sV , V sV , equation m<n. The 1 1 2 transformation for the n<m case can = Augmented ..., V sV be derived − − − n m 1 n m system similarly equation � Insert into superposed SIMO system equation => + m 1 ∑ ∑ ∑ ⎡ ⎤ + − n γ = Θ + n Θ i n m i n 1 i s V ( s ) s ∑ s Ψ = − i ⎢ ⎥ i n m = i = i s U i 0 i 0 = i i 0 ⎢ ⎥ ⎣ ⎦ 0 = i 0 = T U [ V , V , V ,..., V ] − 1 2 n m
SAMSON Algorithm: Overview SAMSON Algorithm: Overview Second order state m atrices, incidence m atrices and arbitrary inputs Generalized Current Excitation Vector Generation Augm ented System Transform ation System Linearization Projection Projected output vector
System Linearization System Linearization ∑ ⎡ ⎤ � Expand the augmented system + n Θ n 1 i ∑ s = + σ Ψ = i ⎢ ⎥ s s at a given frequency = i s U i 0 0 i ⎢ ⎥ + ⎣ ⎦ 0 n 1 n = ∑ ∑ i 0 ⇒ σ σ = σ i i A U ( ) R i i = T U [ V , V , V ,..., V ] = = i 0 i 0 − 1 2 n m � Introduce auxiliary variables Z 1 , Augmented Z 2 , … Z n , satisfying system σ + = equation A U Z R + n 1 n n − σ + = σ + = ( A U Z ) Z R e.g., A U Z R − − + n 1 n n n n n 1 n 1 + ⇒ σ + σ = σ n n n 1 A U Z R ... + n n n 1 Insert it into − σ + = ( A U Z ) Z R 0 1 1 0 σ + + = σ + n 1 n A U ... R ... + n 1 n and we can obtain and we get + σ − σ = A A U Z R ( ) − + σ + = σ + n n 1 A U Z R ( ) ... ... 0 1 1 0 − n n n 1
System Linearization System Linearization ∑ ⎡ ⎤ � Expand the augmented system + n Θ n 1 i ∑ s = + σ Ψ = i ⎢ ⎥ at a given frequency = i s s s U i 0 0 i ⎢ ⎥ ⎣ ⎦ + 0 n 1 n = ∑ ∑ i 0 ⇒ σ σ = σ i i A U ( ) R = i i T U [ V , V , V ,..., V ] = = i 0 i 0 − 1 2 n m � Introduce auxiliary variables Z 1 , Augmented Z 2 , … Z n , satisfying system σ + = equation A U Z R + n 1 n n − σ + = ( A U Z ) Z R − − n n n 1 n 1 Moment ... space equation − σ + = ( A U Z ) Z R 0 1 1 0 ⎡ ⎤ ⎡ ⎤ q V − σ = 0 and we can obtain ⎢ ⎥ ⎢ ⎥ ( I A ) ⎣ ⎦ ⎣ ⎦ p + σ − σ = D ( A A ) U Z R 0 0 1 1 0
SAMSON Algorithm: Overview SAMSON Algorithm: Overview Second order state m atrices, incidence m atrices and arbitrary inputs Generalized Current Excitation Vector Generation Augm ented System Transform ation System Linearization Projection Projected output vector
Projection Projection ⎡ ⎤ ⎡ ⎤ V q − σ = 0 ⎢ ⎥ ⎢ ⎥ ( I A ) � Find projection matrix using method ⎣ ⎦ ⎣ ⎦ D p similar to PRIMA 0 � But only take the first N rows (N is Moment the number of the original variables) space equation � The moment calculation is efficient because the linearized system is sparse Projection Q matrix Projected system equation Γ ) ) ˆ + + = ˆ ˆ � Directly project on J ex instead of B ( G s C ) V ( s ) J ex s
Outline Outline � Review and Motivation � SAMSON Algorithm � Experimental Results � Conclusions
Experimental Setting Experimental Setting � Experiments are run on a PC with Intel Pentium IV 2.66G CPU and 1G RAM. � All methods are implemented in MATLAB � Time domain responses are calculated by IFFT (Inverse Fast Fourier Transformation with 1024 sampling points) � The examples to be presented are from real industry applications (courtesy of Rio Design Automation) � Power planes and packages are modeled by RCS meshes � On chip power/ground grids are modeled by RC meshes � Vias and bumps are modeled by RC elements � PWL sources are generated from SPICE characterization of FPGA circuits
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