Efficient Hermite-based Variability Analysis using

Approximate Decoupling Technique

Tuan-Anh Pham1, Emad Gad2, Michel Nakhla 1 and Ramachandra Achar 1

1Dept. of Electronics, Carleton University , Ottawa, ON, Canada K1S 5B6,

Email: tapham,msn,achar@doe.carleton.ca2School of Electrical Engineering and Computer Science University of Ottawa

Ottawa, ON, Canada K1N 6N5, Email: egad@eecs.uottawa.ca

Abstract—This paper presents a new approach aimed atlimiting the growth of the computational cost of variabilityanalysis, using the Hermite-based Polynomial Chaos (PC), withthe increase in the number of random variables and the numberof Hermite coefficients used to represent the circuit responsein each random variable. The proposed technique is based onderiving a closed-form formula for the structure of the augmentedmatrices generated by the PC approach, and then shows that thisstructure can be approximated with a different structure that canbe decoupled easily.

I. INTRODUCTION

One of the main challenges in nano-scale design is pre-

dicting the effect of the inherent process variability of geo-

metrical and physical parameters on the general performance

of integrated circuits. The lack of predictability arises mainly

from the difficulty of controlling the physical and geometrical

parameters during the fabrication process. This effectively

makes the numerical values for those parameters subject to

significant uncertainty, which, in turn, produces uncertainty in

the electric performance of the circuit.

Traditionally, Monte Carlo (MC) simulations have been

used in commercial circuit and Electromagnetic (EM) simula-

tions for predicting the statistical distribution of the circuit and

system performance. However, the slow convergence for MC

has become a computational burden especially in simulating

large circuits. This fact has prompted wide interest in exploring

alternative approaches to the problem of statistical analysis of

the performance of electronic circuits.

A recent approach based on the notion of polynomial

chaos (PC) was developed and showed great computational

advantage over the standard MC-based analysis [1]. This

approach has been used in estimating the statistical properties

of different types of circuits. Earlier PC approaches focused

on characterizing the variability analysis of interconnects [2]

and multi-conductor transmission lines in the presence of

process variations [3]. Further work addressed the problem of

variability in generic multiport linear circuits [4]. It was also

used in to handle the problem of variability due to uncertainty

in physical parameters of Carbon Nanotube interconnects [5].

The basic idea of the PC approach is to expand the circuit

response in a series of the Askey-Wiener type of orthogonal

polynomials, e.g, Hermite polynomials. Subsequent to that,

a Galerkin projection process is carried out to construct a

deterministic system of equations in the coefficients of the

series of Hermite polynomials.

However, one of the main issues that still hinders widening

the domain of applications of the PC-based variability analysis

to more problems is the issue of the computational cost for

general problems. This issue arises from the fact that the

computational complexity does not scale favorably with the

involved number of random variables and the number of

Hermite coefficients used with each random variable. The

difficulty therein stems from the fact that the Galerkin pro-

jection process always results in a system of equations that

couples the Hermite coefficients of each random variable for

each component of the circuit response. This fact makes the

augmented matrix that must be factorized significantly larger

and without the desirable sparsity patterns that characterize

general circuits.

The goal of this paper is to address this issue through

presenting a scheme whose computational complexity scales

in linear manner with the growth of the random variables

or the number of Hermite coefficients used to represent

the components of circuit response. The proposed approach

is derived through introducing a new general formula that

characterizes the structure of augmented matrix that arises

from a Hermite-based PC. It then shows that this structure

can be approximated with a different structure that can be

decoupled into smaller components and solved independently,

or in parallel. The focus of this paper is rather placed on

handling circuits whose response is a function of only one

normalized random variable, where the proposed approach

shows how the Hermite coefficients of each component in the

circuit response is decoupled. Addressing the case of multi-

random variables will be presented in future works.

The rest of the paper is organized as follows. Section

II reviews briefly the preliminaries of the PC approach to

variability analysis, while Section III provides the circuit

context for its application. Section IV presents a new lemma

that characterizes the general structure of the augmented ma-

trices obtained from a Hermite-based PC application to circuit

equations. Section V provides the approximate structures,

whereas Section VI provides a numerical example to validate

the accuracy of the proposed technique.

978-1-4673-5679-4/13/$31.00 ©2013 IEEE

II. PC-BASED VARIABILITY ANALYSIS

PC-based variability analysis is based on expanding a

stochastic process X as

X(ξ) =

K∑

k=0

αkφk(ξ) (1)

where ξ is a vector of normalized random variables, αi scalar

coefficients, and φ(ξ) are multi-dimensional polynomials that

are orthogonal with respect to a probability measure w(ξ) with

support on Ω, where

< φi(ξ), φj(ξ) >=

∫

Ω

φi(ξ)φj(ξ)w(ξ)dξ = κiδij (2)

and δij is the Kronecker delta function. If the probability

density function (PDF) of ξ is one of the standard distributions

(e.g. Gaussian, Uniform, or Beta), the optimal basis functions

are of the Askey-Wiener type (i.e. Hermite, Lengendre, or

Laguerre) which are orthogonal with respect to the weighting

function given by the PDF. The key advantage of the PC-based

variability analysis is that it enables representing the statistical

properties of the stochastic process, e.g. mean µ and variance

σ2, analytically. For example, the mean µ is given by

µ = α0 (3)

while the standard deviation, σ, is given by

σ =

K∑

i=1

α2i (4)

III. APPLICATIONS OF PC TO CIRCUIT VARIABILITY

ANALYSIS

In the context of circuit or EM simulation, random variables

are those physical or electrical components whose values

are subject to significant uncertainty, while the stochastic

processes are the node voltages, branch currents, and capacitor

and inductors charges and fluxes, respectively. This notion

is captured by using the modified nodal analysis (MNA) [6]

to represent the circuit as a system of differential-algebraic

equations (DAE),

(G+ sC)X(s) = U(s) (5)

where G,C ∈ RN×N are matrices describing the memoryless

and memory elements in the circuit, respectively, X(s) ∈ CN

is a vector of the voltages of circuit nodes, currents in the

inductors or independent voltage sources, and U(s) ∈ CN is a

vector representing the independent stimulus of the circuit, all

given in the Laplace-domain, with N representing the number

of the variables in the circuit response and s representing the

frequency variable. In order to capture the idea of uncertainty

in the circuit response due to uncertainty in the parameters of

the fabrication process, we modify the MNA formulation in

(5) as follows

(G(ξ) + sC(ξ))X(s, ξ) = U(s) (6)

where ξ is a parameter that represents the randomness or the

uncertainty of the manufacturing process.

The starting point of the Hermite-based PC variability

analysis is to expand the all the ξ-dependent terms in (6) as a

series of Hermite polynomials in ξ. Thus we have,

G =

H∑

i=0

GiHi(ξ) (7)

C =

H∑

i=0

CiHi(ξ) (8)

X(s, ξ) =

M∑

i=0

Xi(s)Hi(ξ) (9)

for H and M sufficiently large integers, and Hi(ξ) is the ith-

order Hermite polynomial.

The Galerkin projection process is carried out through the

following steps,

• substitution from (7)-(9) into (6),

• multiplication by Hk(ξ)e−ξ2/2,

• integration from −∞ to ∞ with respect to ξ, and

• substitution using the closed-form of the integral of the

triple product given by

θi,j,k :=

∫ ∞

−∞Hi(ξ)Hj(ξ)Hk(ξ)e

−ξ2/2dξ

=

√2πi!j!k!

(s−i)!(s−j)!(s−k)!

i+ j + k = even

i+ j ≥ k

k + i ≥ j

j + k ≥ i

0 otherwise

(10)

with s = i+j+k2 .

The above process typically results in the following system

of equations

H∑

i=0

M∑

j=0

GiXj + s

H∑

i=0

M∑

j=0

CiXj

θi,j,k =

√2πU0(s) k = 0

0 k > 0(11)

Noting that the above system of equation has only N equations

while involving (M + 1)N unknowns, we repeat the above

process by sweeping the index k from 0 to M generating the

following system of equations(G+ sC

)X(s) = U(s) (12)

where G, C ∈ RN(M+1)×N(M+1) are the augmented matri-

ces, and X(s) ∈ RN(M+1) is given by

X(s) =[X1(s)

⊤ X2(s)⊤ · · · XN (s)⊤

]⊤(13)

with Xi ∈ CM+1 is a vector grouping the M + 1 Hermite

coefficients of the ith circuit variable, and ⊤ denotes the

transpose operator.

IV. STRUCTURAL CHARACTERIZATION OF THE

AUGMENTED MATRICES

The first step in describing the proposed decoupling ap-

proach will present a formal characterization for the structure

of the augmented matrices G and C. For this purpose, we

introduce the following lemma.

Lemma 1: Let K0 be an (M+1)×(M+1) identity matrix,

and let Ki (i > 0) be given by

Ki = ALKi−1 +Ki−1AU (14)

where

AL =

0 0 0 · · · 0 01 0 0 · · · 00 1 0 · · · 00 0 1 0 · · · 0... 0

. . .

0 0 0 · · · 1 0

(15)

AU =

0 1 0 0 · · · · · · 00 0 2 0 0 · · · 00 0 0 3 0 · · · 0...

. . .... 0

0 0 · · · 0 0 M

0 0 · · · 0 0 0

(16)

then

G =

H∑

i=0

Gi ⊗Ki (17)

C =

H∑

i=0

Ci ⊗Ki (18)

and ⊗ denotes the matrix Kronecker product operator.

It is fairly easy to see, using successive substitution, that

Ki can be written as

Ki =

i∑

j=0

i!

(i− j)!j!A

jUA

i−jL (19)

which is equivalent to the binomial expansion of the term

(AU + AL)i, had AL and AU been commuting matrices.

However, given that this is not the case, we then recourse to

approximate analysis by first defining

Ki = (AU +AL)i

(20)

and then using Ki to construct the augmented matrices in

(17)-(18) instead of Ki.

V. PROPOSED DECOUPLING TECHNIQUE

The proposed decoupling technique is based on using Ki to

construct the augmented matrices and obtain an approximation

to the Hermite coefficients of the state variables X(s, ξ). To

this end, we rewrite (12), using Lemma 1, while employing

Ki instead of Ki,(H∑

i=0

Gi ⊗ Ki + s

H∑

i=0

Gi ⊗ Ki

)X(s) = U(s) (21)

where X(s) is used instead of X(s) to indicate that the

outcome of solving (21) is an approximation to the original

solution of (12). Next we note that K1 = K1 = AL +AU ,

and representing this matrix by its eigen-decomposition, K1 =V λV −1 allows us to write the eigen-decomposition for a

general Ki as follows,

Ki = V λiV −1 (22)

Substitution from (22) into (21), yields

Γ

(H∑

i=0

Gi ⊗ λi + s

H∑

i=0

Gi ⊗ λi

)Γ−1X(s) = U(s) (23)

where Γ = I⊗V , Γ−1 = I⊗V −1, and I is an identity matrix

of size N . It can be shown that the coefficients matrix in the

above system is block diagonal, by performing the change of

variables Y (s) = Γ−1X(s). This in fact achieves the required

decoupling, since it allows us to perform a sequence of M+1factorizations on matrices of size N ×N , each.

It should be stressed, that the matrix of eigenvectors V is

problem-independent, and therefore can be computed offline,

stored and used for all problems. This fact implies that the

transformation matrix Γ is also readily available, problem-

independent, and does not need to be computed.

It is also to be noted that the proximity between Ki and

the original Ki can be improved by adding specific correction

matrices to Ki without altering its fundamental property in

(22 ), which is the key to the decoupling technique. However,

those details have been omitted due to lack of space.

VI. NUMERICAL EXAMPLE

In this section, the proposed algorithm is implemented to

the variability analysis of the four coupled transmission lines

structure shown in Fig. 1. In this example, each TL has a length

of 5 cm and its macromodel is modeled using 100 sections of

lumped RLGC segmentation. The first line is excited by an

AC voltage source with 1V amplitude at its near-end while

the other lines are quiet.

Macromodel

of a 4-conductor TL circuit

50 Ω

50 Ω

Vfar,1

Vfar,2

50 Ω

50 Ω

Vfar,3

Vfar,4

AC

Fig. 1. Schematic of a four-coupled TL circuit.

The source of uncertainty here is assumed to be the resis-

tivity ρ of the line conductors and the separation d between

the plates of load capacitors at the far-end of TLs, which are

expressed as functions of the same normalized random variable

ξ in the following form:

ρ = ρ(1 + σ1ξ), d = d(1 + σ2ξ) (24)

where ξ has a Gaussian distribution with zero mean value and

unit variance, σ1 = 0.1 and σ2 = 0.1 are the normalized

standard deviations, while ρ and d are the mean values of the

parameters, ρ = 15.9nΩ.m and d = 10.5A.

0 1 2 3 4 5 6 7 8 9 10

x 109

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Freq(Hz)

Magnitude(V

)

coupled PC method

decoupled PC method

Fig. 2. Statistical analysis for the far-end voltage of line 1. Solid blacklines: nominal value and ±3σ tolerances obtained by coupled method; crosses:nominal value and ±3σ tolerances obtained by decoupled method

0 1 2 3 4 5 6 7 8 9 10

x 109

0

0.005

0.01

0.015

0.02

0.025

0.03

Freq(Hz)

Magnitude(V

)

coupled PC method

decoupled PC method

Fig. 3. Statistical analysis for the far-end voltage of line 3. Solid blacklines: nominal value and ±3σ tolerances obtained by coupled method; crosses:nominal value and ±3σ tolerances obtained by decoupled method

0 1 2 3 4 5 6 7 8 9 10

x 109

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Freq(Hz)

Magnitude(V

)

MC method from HSPICE

decoupled PC method

Fig. 4. Statistical analysis for the far-end voltage of line 1. Solid black lines:nominal value and ±3σ tolerances obtained by MC from HSPICE; crosses:nominal value and ±3σ tolerances obtained by decoupled method

Figs. 2 and 3 show the nominal value and the 3σ tolerances

of the far-end voltages of line 1 and line 3, respectively,

0 1 2 3 4 5 6 7 8 9 10

x 109

0

0.005

0.01

0.015

0.02

0.025

0.03

Freq(Hz)

Magnitude(V

)

MC method from HSPICE

decoupled PC method

Fig. 5. Statistical analysis for the far-end voltage of line 3. Solid black lines:nominal value and ±3σ tolerances obtained by MC from HSPICE; crosses:nominal value and ±3σ tolerances obtained by decoupled method

obtained from both the proposed (decoupled) and standard

(coupled) PC approach, while Figs. 4 and 5 provide a com-

parison, for the same results, between a Monte Carlo-based

simulation using HSPICE and the proposed algrotihm.

As can be seen, the nominal value and the 3σ boundaries

obtained by the proposed PC expansion accurately matched

with the nominal value and the 3σ boundaries obtained by

the standard PC expansion as well as the Monte-Carlo from

HSPICE.

VII. CONCLUSION

This paper presented a new approach to decouple the system

of equations arising from the Hermite-based Polynomial Chaos

approach to variability analysis into smaller systems. The

paper also presented a new theoretical result to characterize

the structure of the augmented matrices and show its proximity

to a different structure that lends itself more naturally to de-

coupling. A numerical example has been presented to validate

the proposed technique by comparing the results obtained from

the original coupled, and the proposed decoupled PC approach

along with Monte Carlo simulations. Future works will address

the general case of multi-random variables.

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