IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. XX, NO. XX, XX 2015 1

Enabling High-Dimensional HierarchicalUncertainty Quantification by ANOVA and

Tensor-Train DecompositionZheng Zhang, Xiu Yang, Ivan V. Oseledets, George Em Karniadakis, and Luca DanielAccepted by IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems

Abstract—Hierarchical uncertainty quantification can reducethe computational cost of stochastic circuit simulation by em-ploying spectral methods at different levels. This paper presentsan efficient framework to simulate hierarchically some chal-lenging stochastic circuits/systems that include high-dimensionalsubsystems. Due to the high parameter dimensionality, it ischallenging to both extract surrogate models at the low levelof the design hierarchy and to handle them in the high-levelsimulation. In this paper, we develop an efficient ANOVA-based stochastic circuit/MEMS simulator to extract efficientlythe surrogate models at the low level. In order to avoid the curseof dimensionality, we employ tensor-train decomposition at thehigh level to construct the basis functions and Gauss quadraturepoints. As a demonstration, we verify our algorithm on astochastic oscillator with four MEMS capacitors and184 randomparameters. This challenging example is simulated efficiently byour simulator at the cost of only 10 minutes in MATLAB on aregular personal computer.

Index Terms—Uncertainty quantification, hierarchical uncer-tainty quantification, generalized polynomial chaos, stochasticmodeling and simulation, circuit simulation, MEMS simulation,high dimensionality, analysis of variance (ANOVA), tensor train.

I. I NTRODUCTION

PROCESS variations have become a major concern insubmicron and nano-scale chip design [2]–[6]. In or-

der to improve chip performances, it is highly desirable todevelop efficient stochastic simulators to quantify the un-certainties of integrated circuits and microelectromechanicalsystems (MEMS). Recently, stochastic spectral methods [7]–[12] have emerged as a promising alternative to Monte Carlotechniques [13]. The key idea is to represent the stochasticsolution as a linear combination of some basis functions(e.g., generalized polynomial chaos [8]), and then computethesolution by stochastic Galerkin [7], stochastic collocation [9]–[12] or stochastic testing [14]–[16] methods. Due to the

Some preliminary results of this work have been reported in [1]. This workwas funded by the MIT-SkolTech program. I. Oseledets was alsosupportedby the Russian Science Foundation under Grant 14-11-00659.

Z. Zhang and L. Daniel are with the Research Laboratory of Electronics,Massachusetts Institute of Technology (MIT), Cambridge, MA02139, USA(e-mail: z zhang@mit.edu, luca@mit.edu).

X. Yang was with the Division of Applied Mathematics, Brown University,Providence, RI 02912. Now he is with the Pacific Northwest NationalLaboratory, Richland, WA 99352, USA (e-mail: xiu.yang@pnnl.gov).

G. Karniadakis is with the Division of Applied Mathematics, Brown Univer-sity, Providence, RI 02912, USA (e-mail: georgekarniadakis@brown.edu).

Ivan V. Oseledets is with the Skolkovo Institute of Science and Technology,Skolkovo 143025, Russia (e-mail: ivan.oseledets@gmail.com).

fast convergence rate, such techniques have been successfullyapplied in the stochastic analysis of integrated circuits [14]–[20], VLSI interconnects [21]–[25], electromagnetic [26]andMEMS devices [1], [27], achieving significant speedup overMonte Carlo when the parameter dimensionality is small ormedium.

Since many electronic systems are designed in a hierarchicalway, it is possible to exploit such structure and simulate a com-plex circuit by hierarchical uncertainty quantification [28]1.Specifically, one can first utilize stochastic spectral methodsto extract surrogate models for each block. Then, circuitequations describing the interconnection of blocks may besolved with stochastic spectral methods by treating each blockas a single random parameter. Typical application examplesinclude (but are not limited to) analog/mixed-signal systems(e.g., phase-lock loops) and MEMS/IC co-design. In our pre-liminary conference paper [1], this method was employed tosimulate a low-dimensional stochastic oscillator with9 randomparameters, achieving250× speedup over the hierarchicalMonte-Carlo method proposed in [32].

Paper Contributions. This paper extends the recentlydeveloped hierarchical uncertainty quantification method[28]to the challenging cases that include subsystems with highdimensionality (i.e., with a large number of parameters). Dueto such high dimensionality, it is too expensive to extract asurrogate model for each subsystem by any standard stochasticspectral method. It is also non-trivial to perform high-levelsimulation with a stochastic spectral method, due to the high-dimensional integration involved when computing the basisfunctions and Gauss quadrature rules for each subsystem. Inorder to reduce the computational cost, this work developssome fast numerical algorithms to accelerate the simulationsat both levels:

• At the low level, we develop a sparse stochastic testingsimulator based on adaptive anchored ANOVA [33]–[37]to efficiently simulate each subsystem. This approachexploits the sparsity on-the-fly, and it turns out to besuitable for many circuit and MEMS problems. Thisalgorithm was reported in our preliminary conferencepaper [1] and was used for the global sensitivity analysisof analog integrated circuits.

1Design hierarchy can be found in many engineering fields. In the recentwork [29] a hierarchical stochastic analysis and optimization framework basedon multi-fidelity models [30], [31] was proposed for aircraft design.

• In the high-level stochastic simulation, we accelerate thethree-term recurrence relation [38] by tensor-train decom-position [39]–[41]. Our algorithm has a linear complexitywith respect to the parameter dimensionality, generatinga set of basis functions and Gauss quadrature pointswith high accuracy (close to the machine precision). Thisalgorithm was not reported in [1].

II. BACKGROUND REVIEW

This section first reviews the recently developed stochastictesting circuit/MEMS simulator [14]–[16] and hierarchicaluncertainty quantification [28]. Then we introduce some back-ground about tensor and tensor decomposition.

A. Stochastic Testing Circuit/MEMS Simulator

Given a circuit netlist (or a MEMS 3D schematic file),device models and process variation descriptions, one can setup a stochastic differential algebraic equation:

d~q(

~x(t, ~ξ), ~ξ)

dt+ ~f

(

~x(t, ~ξ), ~ξ, u(t))

= 0(1)

where~u(t) is the input signal,~ξ=[ξ1, · · · , ξd] ∈ Ω ⊆ Rd are

d mutually independent random variables describing processvariations. The joint probability density function of~ξ is

ρ(~ξ) =d∏

k=1

ρk (ξk), (2)

whereρk (ξk) is the marginal probability density function ofξk ∈ Ωk. In circuit analysis,~x∈Rn denotes nodal voltagesand branch currents;~q∈Rn and ~f∈Rn represent charge/fluxand current/voltage, respectively. In MEMS analysis, Eq. (1) isthe equivalent form of a commonly used2nd-order differentialequation [1], [42];~x includes displacements, rotations andtheir first-order derivatives with respect to timet.

When~x(~ξ, t) has a bounded variance and smoothly dependson ~ξ, we can approximate it by a truncated generalizedpolynomial chaos expansion [8]

~x(t, ~ξ) ≈ x(t, ~ξ) =∑

~α∈Px~α(t)H~α(~ξ) (3)

where x~α(t) ∈ Rn denotes a coefficient indexed by vector

~α = [α1, · · · , αd] ∈ Nd, and the basis functionH~α(~ξ) is a

multivariate polynomial with the highest order ofξi beingαi.In practical implementations, it is popular to set the highesttotal degree of the polynomials asp, then P = ~α| αk ∈N, 0 ≤ α1 + · · · + αd ≤ p and the total number of basisfunctions is

K =

(

p+ dp

)

=(p+ d)!

p!d!. (4)

For any integerj in [1,K], there is a one-to-one correspon-dence betweenj and ~α. As a result, we can denote a basisfunction asHj(~ξ) and rewrite (3) as

~x(t, ~ξ) ≈ x(t, ~ξ) =

K∑

j=1

xj(t)Hj(~ξ). (5)

surrogate model

Higher-Level Eq.

surrogate model

surrogate model

lower-level parameters

1ξr

1y

iξr

iy qy

qξr

... ...

hr

lower-level parameters

lower-level parameters

Fig. 1. Demonstration of hierarchical uncertainty quantification.

In order to compute~x(t, ~ξ), stochastic testing [14]–[16]substitutesx(t, ~ξ) into (1) and forces the residual to zero atK testing samples of~ξ. This gives a deterministic differentialalgebraic equation of sizenK

dq(x(t))dt

+ f (x(t), u(t)) = 0, (6)

where the state vectorx(t) contains all coefficients in (3).Stochastic testing then solves Eq. (6) with a linear complexityof K and with adaptive time stepping, and it has shown higherefficiency than standard stochastic Galerkin and stochasticcollocation methods in circuit simulation [14], [15].

1) Constructing Basis Functions:The basis functionH~α(~ξ)is constructed as follows (see Section II of [16] for details):

• First, forξi one constructs a set of degree-αi orthonormalunivariate polynomials

ϕiαi(ξi)

p

αi=0according to its

marginal probability densityρi(ξi).• Next, based on the obtained univariate polynomials of

each random parameter one constructs the multivariatebasis function:H~α(~ξ)=

∏di=1 ϕ

iαi(ξi).

The obtained basis functions are orthonormal polynomialsin the multi-dimensional parameter spaceΩ with the densitymeasureρ(~ξ). As a result, some statistical information can beeasily obtained. For example, the mean value and variance of~x(t, ~ξ) are x0(t) and

∑

~α6=0

(x~α(t))2, respectively.

2) Testing Point Selection:The selection of testing pointsinfluence the numerical accuracy of the simulator. In stochastictesting, the testing points~ξjKj=1 are selected by the followingtwo steps (see Section III-C of [14] for details):

• First, compute a set of multi-dimensional quadraturepoints. Such quadrature points should give accurate re-sults for evaluating the numerical integration of any mul-tivariate polynomial of~ξ over Ω [with density measureρ(~ξ)] when the polynomial degree is≤ 2p.

• Next, among the obtained quadrature points, we selecttheK samples with the largest quadrature weights underthe constraint thatV ∈ R

K×K is well-conditioned. The(j, k) element ofV is Hk(~ξ

j).

B. Hierarchical Uncertainty Quantification

Consider Fig. 1, where an electronic system hasq sub-systems. The outputyi of a subsystem is influenced bysome process variations~ξi ∈ R

di , and the output~h of the

whole system depends on all random parameters~ξi’s. Forsimplicity, in this paper we assume thatyi only dependson ~ξi and does not change with time or frequency. Directlysimulating the whole system can be expensive due to the largeproblem size and high parameter dimensionality. Ifyi’s aremutually independent and smoothly dependent on~ξi’s, we canaccelerate the simulation in a hierarchical way [28]:

• First, perform low-level uncertainty quantification. Weuse stochastic testing to simulate each block, obtaininga generalized polynomial expansion for eachyi. In thisstep we can also employ other stochastic spectral methodssuch as stochastic Galerkin or stochastic collocation.

• Next, perform high-Level uncertainty quantification. Bytreating yi’s as the inputs of the high-level equation,we use stochastic testing again to efficiently compute~h. Sinceyi has been assumed independent of time andfrequency, we can treat it as a random parameter.

In order to apply stochastic spectral methods at the highlevel, we need to compute a set ofspecializedorthonormalpolynomials and Gauss quadrature points/weights for eachinput random parameter. For the sake of numerical stability, wedefine a zero-mean unit-variance random variableζi for eachsubsystem, by shifting and scalingyi. The intermediate vari-ables~ζ = [ζ1, · · · , ζq] are used as the random parameters inthe high-level equation. Dropping the subscript for simplicity,we denote a general intermediate-level random parameter byζand its probability density function byρ(ζ) (which is actuallyunknown), then we can constructp+1 orthogonal polynomialsπj(ζ)pj=0 via a three-term recurrence relation [38]

πj+1(ζ) = (ζ − γj)πj(ζ)− κjπj−1(ζ),π−1(ζ) = 0, π0(ζ) = 1, j = 0, · · · , p− 1

(7)

with

γj =

∫

R

ζπ2j (ζ)ρ(ζ)dζ

∫

R

π2j (ζ)ρ(ζ)dζ

, κj+1 =

∫

R

π2j+1(ζ)ρ(ζ)dζ

∫

R

π2j (ζ)ρ(ζ)dζ

(8)

and κ0 = 1, whereπj(ζ) is a degree-j polynomial with aleading coefficient 1. The firstp+1 univariate basis functionscan be obtained by normalization:

φj(ζ) =πj(ζ)√

κ0κ1 · · ·κj, for j = 0, 1, · · · , p. (9)

The parametersκj ’s and γj ’s can be further used to form asymmetric tridiagonal matrixJ ∈ R

(p+1)×(p+1):

J (j, k) =

γj−1, if j = k√κj , if k = j + 1√κk, if k = j − 1

0, otherwise

for 1 ≤ j, k ≤ p+1. (10)

Let J = UΣUT be an eigenvalue decomposition, whereU isa unitary matrix. Thej-th quadrature point and weight ofζareΣ(j, j) and (U(1, j))

2, respectively [43].Challenges in High Dimension.When di is large, it is

difficult to implement hierarchical uncertainty quantification.First, it is non-trivial to obtain a generalized polynomialchaosexpansion foryi, since a huge number of basis functions and

Fig. 2. Demonstration of a vector (left), a matrix (center) anda 3-modetensor (right).

samples are required to obtainyi. Second, when high accuracyis required, it is expensive to implement (7) due to the non-trivial integrals when computingκj andγj . Since the densityfunction of ζi is unknown, the integrals must be evaluated inthe domain of~ξi, with a cost growing exponentially withdiwhen a deterministic quadrature rule is used.

C. Tensor and Tensor Decomposition

Definition 1 (Tensor). A tensorA ∈ RN1×N2×···×Nd is a

multi-mode (or multi-way) data array. The mode (or way) isd, the number of dimensions. The size of thek-th dimensionis Nk. An element of the tensor isA(i1, · · · , id), where thepositive integerik is the index for thek-th dimension and1 ≤ik ≤ Nk. The total number of elements ofA isN1×· · ·×Nd.

As a demonstration, we have shown a vector (1-modetensor) inR

3×1, a matrix (2-mode tensor) inR3×3 and a3-mode tensor inR3×3×4 in Fig. 2, where each small cuberepresents a scalar.

Definition 2 (Inner Product of Two Tensors). For A,B ∈R

N1×N2×···×Nd , their inner product is defined as the sum oftheir element-wise product

〈A,B〉 =∑

i1,··· ,idA (i1, · · · id)B (i1, · · · id). (11)

Definition 3 (Frobenius Norm of A Tensor). For A ∈R

N1×N2×···×Nd , its Frobenius norm is defined as

‖A‖F =√

〈A,A〉. (12)

Definition 4 (Rank-One Tensors). A d-mode tensorA ∈R

N1×···×Nd is rank one if it can be written as the outer productof d vectors

A = v(1) v(2) · · · v(d), with v

(k) ∈ RNk (13)

where denotes the outer product operation. This means that

A(i1, · · · , id) =d∏

k=1

v(k)(ik) for all 1 ≤ ik ≤ Nk. (14)

Herev(k)(ik) denotes theik-th element of vectorv(k).

Definition 5 (Tensor Rank). The rank ofA ∈ RN1×···×Nd

is the smallest positive integerr, such that

A =r

∑

j=1

v(1)j v(2)

j · · · v(d)j , with v

(k)j ∈ R

Nk . (15)

It is attractive to perform tensor decomposition: given asmall integerr < r, approximateA ∈ R

N1×···×Nd by arank-r tensor. Popular tensor decomposition algorithms in-clude canonical decomposition [44]–[46] and Tuker decom-position [47], [48]. Canonical tensor decomposition aims toapproximateA by the sum ofr rank-1 tensors [in the formof (15)] while minimizing the approximation error, which isnormally implemented with alternating least square [45]. Thisdecomposition scales well with the dimensionalityd, but itis ill-posed for d ≥ 3 [49]. Tucker decomposition aims torepresent a tensor by a small core tensor and some matrixfactors [47], [48]. This decomposition is based on singularvalue decomposition. It is robust, but the number of elementsin the core tensor still grows exponentially withd.

Alternatively, tensor-train decomposition [39]–[41] approx-imatesA ∈ R

N1×···×Nd by a low-rank tensorA with

A (i1, · · · id) = G1 (:, i1, :)G2 (:, i1, :) · · ·Gd (:, id, :) . (16)

Here Gk ∈ Rrk−1×Nk×rk , and r0 = rd = 1. By fixing the

second indexik, Gk(:, ik, :)∈Rrk−1×rk becomes a matrix (orvector whenk equals1 or d). To some extent, tensor-traindecomposition have the advantages of both canonical tensordecomposition and Tuker decomposition: it is robust sinceeach core tensor is obtained by a well-posed low-rank matrixdecomposition [39]–[41]; it scales linearly withd since storingall core tensors requires onlyO(Nr2d) memory if we assumeNk = N and rk = r for k = 1, · · · , d − 1. Given an errorboundε, the tensor train decomposition in (16) ensures

∥

∥

∥A− A

∥

∥

∥

F≤ ε ‖A‖F (17)

while keepingrk ’s as small as possible [39].

Definition 6 (TT-Rank ). In tensor-train decomposition (16)Gk ∈ R

rk−1×Nk×rk for k = 1, · · · d. The vector~r =[r0, r1, · · · , rd] is called TT-rank.

Recently, tensor decomposition has shown promising appli-cations in high-dimensional data and image compression [50]–[53], and in machine learning [54], [55]. In the uncertaintyquantification community, some efficient high-dimensionalstochastic PDE solvers have been developed based on canon-ical tensor decomposition [56]–[58] (which is called “ProperGeneralized Decomposition” in some papers) and tensor-train decomposition [59]–[62]. In [63], a spectral tensor-train decomposition is proposed for high-dimensional functionapproximation.

III. ANOVA-B ASED SURROGATEMODEL EXTRACTION

In order to accelerate the low-level simulation, this sectiondevelops a sparse stochastic circuit/MEMS simulator basedonanchored ANOVA (analysis of variance). Without of loss ofgenerality, lety = g(~ξ) denote the output of a subsystem. Weassume thaty is a smooth function of the random parameters~ξ ∈ Ω ⊆ R

d that describe the process variations.

A. ANOVA and Anchored ANOVA Decomposition

1) ANOVA: With ANOVA decomposition [34], [64],y canbe written as

y = g(~ξ) =∑

s⊆Igs(~ξs), (18)

where s is a subset of the full index setI = 1, 2, · · · , d.Let s be the complementary set ofs such thats ∪ s = Iand s ∩ s = ∅, and let |s | be the number of elements ins .When s =

i1, · · · , i|s|

6= ∅, we setΩs = Ωi1 ⊗ · · · ⊗ Ωi|s| ,~ξs = [ξi1 , · · · , ξi|s| ] ∈ Ωs and have the Lebesgue measure

dµ(~ξs) =∏

k∈s

(ρk (ξk) dξk). (19)

Then,gs(~ξs) in ANOVA decomposition (18) is defined recur-sively by the following formula

gs(~ξs) =

E

(

g(~ξ))

=∫

Ω

g(~ξ)dµ(~ξ) = g0, if s = ∅

gs(~ξs)−∑

t⊂s

gt(~ξt) , if s 6= ∅.(20)

Here E is the expectation operator,gs(~ξs) =∫

Ωs

g(~ξ)dµ(~ξs),

and the integration is computed for all elements except thosein ~ξs . From (20), we have the following intuitive results:

• g0 is a constant term;• if s=j, then gs(~ξs) = gj(ξj), gs(~ξs) = gj(ξj) =gj(ξj)− g0;

• if s=j, k and j < k, then gs(~ξs) = gj,k(ξj , ξk) andgs(~ξs) = gj,k(ξj , ξk)− gj(ξj)− gk(ξk)− g0;

• both gs(~ξs) andgs(~ξs) are|s |-variable functions, and thedecomposition (18) has2d terms in total.

Example 1. Considery = g(~ξ) = g(ξ1, ξ2). SinceI = 1, 2,its subset includes∅, 1, 2 and 1, 2. As a result, thereexist four terms in the ANOVA decomposition (18):

• for s = ∅, g∅(~ξ∅) = E

(

g(~ξ))

= g0 is a constant;

• for s = 1, g1(ξ1)=g1(ξ1) − g0, and g1(ξ1) =∫

Ω2

g(~ξ)ρ2(ξ2)dξ2 is a univariate function ofξ1;

• for s = 2, g2(ξ2)=g2(ξ2) − g0, and g2(ξ2) =∫

Ω1

g(~ξ)ρ1(ξ1)dξ1 is a univariate function ofξ2;

• for s=1, 2, g1,2(ξ1, ξ2)=g1,2(ξ1, ξ2) − g1(ξ1) −g2(ξ2)− g0. Sinces=∅, we haveg1,2(ξ1, ξ2) = g(~ξ),which is a bi-variate function.

Since all terms in the ANOVA decomposition are mutuallyorthogonal [34], [64], we have

Var

(

g(~ξ))

=∑

s⊆IVar

(

gs(~ξs))

(21)

whereVar(•) denotes the variance over the whole parameterspaceΩ. What makes ANOVA practically useful is that formany engineering problems,g(~ξ) is mainly influenced by theterms that depend only on a small number of variables, andthus it can be well approximated by a truncated ANOVAdecomposition

g(~ξ) ≈∑

|s|≤deff

gs(~ξs), s ⊆ I (22)

wheredeff d is called theeffective dimension.

Example 2. Consider y = g(~ξ) with d = 20. In the fullANOVA decomposition (18), we need to compute over106

terms, which is prohibitively expensive. However, if we setdeff = 2, we have the following approximation

g(~ξ) ≈ g0 +

20∑

j=1

gj(ξj) +∑

1≤j<k≤20

gj,k(ξj , ξk) (23)

which contains only221 terms.

Unfortunately, it is still expensive to obtain the truncatedANOVA decomposition (22) due to two reasons. First, thehigh-dimensional integrals in (20) are expensive to compute.Second, the truncated ANOVA decomposition (22) still con-tains lots of terms whend is large. In the following, weintroduce anchored ANOVA that solves the first problem. Thesecond issue will be addressed in Section III-B.

2) Anchored ANOVA: In order to avoid the expensivemultidimensional integral computation, [34] has proposedanefficient algorithm which is called anchored ANOVA in [33],[35], [36]. Assuming thatξk ’s have standard uniform distri-butions, anchored ANOVA first chooses a deterministic pointcalled anchored point~q = [q1, · · · , qd] ∈ [0, 1]d, and thenreplaces the Lebesgue measure with the Dirac measure

dµ(~ξs) =∏

k∈s

(δ (ξk − qk) dξk). (24)

As a result,g0 = g(~q), and

gs(~ξs) = g(

ξ)

, with ξk =

qk, if k ∈ s

ξk, otherwise.(25)

Here ξk denotes thek-th element ofξ ∈ Rd, qk is a fixed

deterministic value, andξk is a random variable. AnchoredANOVA was further extended to Gaussian random parametersin [35]. In [33], [36], [37], this algorithm was combined withstochastic collocation to efficiently solve high-dimensionalstochastic partial differential equations.

Example 3. Considery=g(ξ1, ξ2). With an anchored point~q = [q1, q2], we haveg0 = g(q1, q2), g1(ξ1) = g(ξ1, q2),g2(ξ2) = g(q1, ξ2) and g1,2(ξ1, ξ2) = g(ξ1, ξ2). Com-puting these quantities does not involve any high-dimensionalintegrations.

B. Adaptive Anchored ANOVA for Circuit/MEMS Problems

1) Extension to General Cases:In many circuit and MEMSproblems, the process variations can be non-uniform and non-Gaussian. We show that anchored ANOVA can be applied tosuch general cases.

Observation: The anchored ANOVA in [34] can be appliedif ρk(ξk) > 0 for any ξk ∈ Ωk.

Proof: Let uk denote the cumulative density function forξk, then uk can be treated as a random variable uniformlydistributed on[0, 1]. Since ρk(ξk) > 0 for any ξk ∈ Ωk,there existsξk = λk(uk) which mapsuk to ξk. Therefore,

g(ξ1, · · · , ξd) = g (λ1(u1), · · · , λd(ud)) = ψ(~u) with ~u =[u1, · · · , ud]. Following (25), we have

ψs(~us) = ψ (u) , with uk =

pk, if k ∈ s

uk, otherwise,(26)

where~p = [p1, · · · , pd] is the anchor point for~u. The aboveresult can be rewritten as

gs(~ξs) = g(

ξ)

, with ξk =

λk(qk), if k ∈ s

λk(ξk), otherwise,(27)

from which we can obtaings(~ξs) defined in (20). Conse-quently, the decomposition forg(~ξ) can be obtained by using~q = [λ1(p1), · · · , λd(pd)] as an anchor point of~ξ.

Anchor point selection. It is is important to select a properanchor point [36]. In circuit and MEMS applications, we findthat ~q = E(~ξ) is a good choice.

2) Adaptive Implementation:In order to further reducethe computational cost, the truncated ANOVA decomposition(22) can be implemented in an adaptive way. Specifically, inpractical computation we can ignore those terms that havesmall variance values. Such a treatment can produce a highlysparse generalized polynomial-chaos expansion.

For a given effective dimensiondeff d, let

Sk = s |s ⊂ I, |s | = k , k = 1, · · · deff (28)

contain the initialized index sets for allk-variate terms inthe ANOVA decomposition. Given an anchor point~q and athresholdσ, starting fromk=1, the main procedures of ourANOVA-based stochastic simulator are summarized below:

1) Computeg0, which is a deterministic evaluation;2) For everys ∈ Sk, compute the low-dimensional function

gs(~ξs) by stochastic testing. The importance ofgs(~ξs)is measured as

θs =Var

(

gs

(

~ξs

))

k∑

j=1

∑

s∈Sj

Var

(

gs

(

~ξs

))

. (29)

3) Update the index sets ifθs < σ for s ∈ Sk. Specifically,for k < j ≤ deff , we check its index sets ′ ∈ Sj . If s′

contains all elements ofs , then we removes′ from Sj .Onces′ is removed, we do not need to evaluategs′(~ξs′)in the subsequent computation.

4) Setk= k+1, and repeat steps 2) and 3) untilk = ddef .

Example 4. Let y=g(~ξ), ~ξ ∈ R20 and deff = 2. Anchored

ANOVA starts with

S1 = jj=1,··· ,20 and S2 = j, k1≤j<k≤20 .

For k=1, we first utilize stochastic testing to calculategs(~ξs)and θs for everys ∈ S1. Assume

θ1 > σ, θ2 > σ, and θj < σ for all j > 2,

implying that only the first two parameters are important tothe output. Then, we only consider the coupling ofξ1 and ξ2in S2, leading to

S2 = 1, 2 .

Algorithm 1 Stochastic Testing Circuit/MEMS SimulatorBased on Adaptive Anchored ANOVA.

1: Initialize Sk ’s and setβ = 0;2: At the anchor point, run a deterministic circuit/MEMS

simulation to obtaing0, and sety = g0;3: for k = 1, · · · , deff do4: for eachs ∈ Sk do5: run stochastic testing simulator to get the generalized

polynomial-chaos expansion ofgs(~ξs) ;6: get the generalized polynomial-chaos expansion of

gs(~ξs) according to (20);

7: updateβ = β +Var

(

gs(~ξs))

;

8: updatey = y + gs(~ξs);9: end for

10: for eachs ∈ Sk do11: θs = Var

(

gs(~ξs))

/β;

12: if θs < σ13: for any index sets ′ ∈ Sj with j > k, remove

s′ from Sj if s ⊂ s

′.14: end if15: end for16: end for

Consequently, fork = 2 we only need to calculate one bi-variate functiong1,2(ξ1, ξ2), yielding

g(

~ξ)

≈ g0 +∑

s∈S1

gs

(

~ξs

)

+∑

s∈S2

gs

(

~ξs

)

= g0 +20∑

j=1

gj (ξj) + g1,2 (ξ1, ξ2) .

The pseudo codes of our implementation are summarized inAlg. 1. Lines10 to 15 shows how to adaptively select the indexsets. Let the final size ofSk be |Sk| and the total polynomialorder in the stochastic testing simulator bep, then the totalnumber of samples used in Alg. 1 is

N = 1 +

deff∑

k=1

|Sk|(k + p)!

k!p!. (30)

Note that all univariate terms in ANOVA (i.e.,|s | = 1) are keptin our implementation. For most circuit and MEMS problems,setting the effective dimension as2 or 3 can achieve a highaccuracy due to the weak couplings among different randomparameters. For many cases, the univariate terms dominate theoutput of interest, leading to a near-linear complexity withrespect to the parameter dimensionalityd.

Remarks. Anchored ANOVA works very well for a largeclass of MEMS and circuit problems. However, in practicewe also find a small number of examples (e.g., CMOS ringoscillators) that cannot be solved efficiently by the proposedalgorithm, since many random variables affect significantlythe output of interest. For such problems, it is possibleto reduce the number of dominant random variables by alinear transform [65] before applying anchored ANOVA. Othertechniques such as compressed sensing can also be utilized toextract highly sparse surrogate models [66]–[69] in the low-level simulation of our proposed hierarchical framework.

3) Global Sensitivity Analysis:Since each termgs(ss) iscomputed by stochastic testing, Algorithm 1 provides a sparsegeneralized polynomial-chaos expansion for the output ofinterest:y=

∑

|~α|≤p

y~αH~α(~ξ), where most coefficients are zero.

From this result, we can identify how much each parametercontributes to the output by global sensitivity analysis. Twokinds of sensitivity information can be used to measure theimportance of parameterξk: the main sensitivitySk and totalsensitivityTk, as computed below:

Sk =

∑

αk 6=0,αj 6=k=0

|y~α|2

Var(y), Tk =

∑

αk 6=0

|y~α|2

Var(y). (31)

IV. ENABLING HIGH-LEVEL SIMULATION BY

TENSOR-TRAIN DECOMPOSITION

In this section, we show how to accelerate the high-levelnon-Monte-Carlo simulation by handling the obtained high-dimensional surrogate models with tensor-train decomposi-tion [39]–[41].

A. Tensor-Based Three-Term Recurrence Relation

In order to obtain the orthonormal polynomials and Gaussquadrature points/weights ofζ, we must implement the three-term recurrence relation in (7). The main bottleneck is tocompute the integrals in (8), since the probability densityfunction of ζ is unknown.

For simplicity, we rewrite the integrals in (8) asE(q(ζ)),with q(ζ) = φ2j (ζ) or q(ζ) = ζφ2j (ζ). Since the probabilitydensity function ofζ is not given, we compute the integral inthe parameter spaceΩ:

E (q (ζ)) =

∫

Ω

q(

f(

~ξ))

ρ(~ξ)dξ1 · · · dξd, (32)

wheref(~ξ) is a sparse generalized polynomial-chaos expan-sion for ζ obtained by

ζ = f(~ξ) =(y − E(y))√

Var(y)=

∑

|~α|≤p

y~αH~α(~ξ). (33)

We compute the integral in (32) with the following steps:1) We utilize a multi-dimensional Gauss quadrature rule:

E (q (ζ)) ≈m1∑

i1=1

· · ·md∑

id=1

q(

f(

ξi11 , · · · , ξidd))

d∏

k=1

wikk

(34)wheremk is the number of quadrature points forξk,(ξikk , w

ikk ) denotes theik-th Gauss quadrature point and

weight.2) We define twod-mode tensorsQ, W ∈ R

m1×m2···×md ,with each element defined as

Q (i1, · · · id) = q(

f(

ξi11 , · · · , ξidd))

,

W (i1, · · · id) =d∏

k=1

wikk ,

(35)

for 1 ≤ ik ≤ mk. Now we can rewrite (34) as the innerproduct ofQ andW :

E (q (ζ)) ≈ 〈Q,W〉 . (36)

For simplicity, we setmk=m in this manuscript.

The cost of computing the tensors and the tensor innerproduct isO(md), which becomes intractable whend is large.Fortunately, bothQ and W have low tensor ranks in ourapplications, and thus the high-dimensional integration (32)can be computed very efficiently in the following way:

1) Low-rank representation of W . W can be written asa rank-1 tensor

W = w(1) w(2) · · · w(d), (37)

where w(k) = [w1

k; · · · ;wmk ] ∈ R

m×1 contains allGauss quadrature weights for parameterξk. Clearly, nowwe only needO(md) memory to storeW .

2) Low-rank approximation for Q. Q can be well ap-proximated byQ with high accuracy in a tensor-trainformat [39]–[41]:

Q (i1, · · · id) = G1 (:, i1, :)G2 (:, i1, :) · · ·Gd (:, id, :)(38)

with a pre-selected error boundε such that∥

∥

∥Q− Q

∥

∥

∥

F≤ ε ‖Q‖F . (39)

For many circuit and MEMS problems, a tensor trainwith very small TT-ranks can be obtained even whenε =10−12 (which is very close to the machine precision).

3) Fast computation of (36). With the above low-ranktensor representations, the inner product in (36) can beaccurately estimated as

⟨

Q,W⟩

= T1 · · ·Td, with Tk =m∑

ik=1

wikk Gk (:, ik, :)

(40)Now the cost of computing the involved high-dimensional integration dramatically reduces toO(dmr2), which only linearly depends the parameterdimensionalityd.

B. Efficient Tensor-Train Computation

Now we discuss how to obtain a low-rank tensor train.An efficient implementation calledTT cross is describedin [41] and included in the public-domain MATALB packageTT Toolbox [70]. In TT cross, Skeleton decomposition isutilized to compress the TT-rankrk by iteratively searching arank-rk maximum-volume submatrix when computingGk. Amajor advantage ofTT cross is that we do not need to knowQ a-priori. Instead, we only need to specify how to evaluatethe elementQ(i1, · · · , id) for a given index(i1, · · · , id). Asshown in [41], with Skeleton decompositions a tensor-traindecomposition needsO(ldmr2) element evaluations, wherelis the number of iterations in a Skeleton decomposition. Forexample, whenl = 10, d = 50, m = 10 and r = 4 we mayneed up to105 element evaluations, which can take aboutone hour since each element ofQ is a high-order polynomialfunction of many bottom-level random variables~ξ.

In order to make the tensor-train decomposition ofQ fast,we employ some tricks to evaluate more efficiently eachelement ofQ. The details are given below.

• Fast evaluation ofQ(i1, · · · , id). In order to reduce thecost of evaluatingQ(i1, · · · , id), we first construct a low-rank tensor trainA for the intermediate-level randomparameterζ, such that∥

∥

∥A− A

∥

∥

∥

F≤ ε ‖A‖F , A (i1, · · · , id) = f

(

ξi11 , · · · , ξidd)

.

OnceA is obtained,Q(i1, · · · , id) can be evaluated by

Q (i1, · · · , id) ≈ q(

A (i1, · · · , id))

, (41)

which reduces to a cheap low-order univariate polynomialevaluation. However, computingA(i1, · · · , id) by di-rectly evaluatingA(i1, · · · , id) in TT crosscan be time-consuming, sinceζ = f(~ξ) involves many multivariatebasis functions.

• Fast evaluation of A(i1, · · · , id). The evaluation ofA (i1, · · · , id) can also be accelerated by exploiting thespecial structure off(~ξ). It is known that the generalizedpolynomial-chaos basis of~ξ is

H~α

(

~ξ)

=

d∏

k=1

ϕ(k)αk

(ξk), ~α = [α1, · · · , αd] (42)

whereϕ(k)αk (ξk) is the degree-αk orthonormal polynomial

of ξk, with 0 ≤ αk ≤ p. We first construct a3-modetensorX ∈ R

d×(p+1)×m indexed by(k, αk +1, ik) with

X (k, αk + 1, ik) = ϕ(k)αk

(

ξikk)

(43)

where ξikk is the ik-th Gauss quadrature point for pa-rameterξk [as also used in (34)]. Then, each element ofA (i1, · · · , id) can be calculated efficiently as

A (i1, · · · , id) =∑

|~α|<p

~y~α

d∏

k=1

X (k, αk + 1, ik) (44)

without evaluating the multivariate polynomials. Con-structingX does not necessarily needd(p + 1)m poly-nomial evaluations, since the matrixX (k, :, :) can bereused for any other parameterξj that has the same typeof distribution withξk.

In summary, we compute a tensor-train decomposition forQ as follows: 1) we construct the3-mode tensorX definedin (43); 2) we call TT cross to computeA as a tensor-train decomposition ofA, where (44) is used for fast elementevaluation; 3) we callTT cross again to computeQ, where(41) is used for the fast element evaluation ofQ. With theabove fast tensor element evaluations, the computation time ofTT cross can be reduced from dozens of minutes to severalseconds to generate some accurate low-rank tensor trains forour high-dimensional surrogate models.

C. Algorithm Summary

Given the Gauss quadrature rule for each bottom-levelrandom parameterξk, our tensor-based three-term recurrencerelation for an intermediate-level random parameterζ is sum-marized in Alg. 2. This procedure can be repeated for allζi’sto obtain their univariate generalized polynomial-chaos basis

Algorithm 2 Tensor-based generalized polynomial-chaos basisand Gauss quadrature rule construction forζ.

1: Initialize: φ0(ζ) = π0(ζ) = 1, φ1(ζ) = π1(ζ) = ζ, κ0 =κ1 = 1, γ0 = 0, a = 1;

2: Compute a low-rank tensor trainA for ζ;3: Compute a low-rank tensor trainQ for q(ζ) = ζ3, and

obtainγ1 =⟨

Q,W⟩

via (40);4: for j = 2, · · · , p do5: get πj(ζ) = (ζ − γj−1)πj−1(ζ)− κj−1πj−2(ζ) ;6: construct a low-rank tensor trainQ for q(ζ) = π2

j (ζ),

and computea =⟨

Q,W⟩

via (40) ;

7: κj = a/a, and updatea = a ;8: construct a low-rank tensor trainQ for q(ζ) = ζπ2

j (ζ),

and computeγj =⟨

Q,W⟩

/a ;

9: normalization:φj(ζ) =πj(ζ)√κ0···κj

;10: end for11: Form matrixJ in (10);12: Eigenvalue decomposition:J = UΣUT ;13: Compute the Gauss-quadrature abscissaζj = Σ(j, j) and

weightwj = (U(1, j))2 for j = 1, · · · , p+ 1 ;

functions and Gauss quadrature rules, and then the stochastictesting simulator [14]–[16] (and any other standard stochasticspectral method [7]–[9]) can be employed to perform high-level stochastic simulation.

Remarks. 1) If the outputs of a group of subsystems areidentically independent, we only need to run Alg. 2 onceand reuse the results for the other subsystems in the group.2) When there exist many subsystems, our ANOVA-basedstochastic solver may also be utilized to accelerate the high-level simulation.

V. NUMERICAL RESULTS

In this section, we verify the proposed algorithm on aMEMS/IC co-design example with high-dimensional randomparameters. All simulations are run in MATLAB and executedon a 2.4GHz laptop with 4GB memory.

A. MEMS/IC Example

In order to demonstrate the application of our hierarchicaluncertainty quantification in high-dimensional problems,weconsider the oscillator circuit shown in Fig. 3. This oscillatorhas four identical RF MEMS switches acting as tunable ca-pacitors. The MEMS device used in this paper is a prototypingmodel of the RF MEMS capacitor reported in [71], [72].

Since the MEMS switch has a symmetric structure, weconstruct a model for only half of the design, as shown inFig. 4. The simulation and measurement results in [42] showthat the pull-in voltage of this MEMS switch is about37 V.When the control voltage is far below the pull-in voltage,the MEMS capacitance is small and almost constant. In thispaper, we set the control voltage to2.5 V, and thus the MEMSswitch can be regarded as a small linear capacitor. As already

C

R

Vctrl

Vdd

L

(W/L)n

10.83fF

110Ω

0-2.5V

2.5V

Parameter Value

8.8nH

4/0.25

LL R

VctrlCm

Rss

MnMn

Iss

Cm

C

Vdd

V1 V2

0.5mAIss

Rss 106Ω

CmCm

Fig. 3. Schematic of the oscillator circuit with4 MEMS capacitors (denotedasCm), with 184 random parameters in total.

RF Conducting

Path

Secondary

ActuatorRF Capacitor

Contact Bump Primary

Actuator

Fig. 4. 3-D schematic of the RF MEMS capacitor.

shown in [73], the performance of this MEMS switch can beinfluenced significantly by process variations.

In our numerical experiments, we use46 independent ran-dom parameters with Gaussian and Gamma distributions todescribe the material (e.g, conductivity and dielectric con-stants), geometric (e.g., thickness of each layer, width andlength of each mechanical component) and environmental(e.g., temperature) uncertainties of each switch. For eachrandom parameter, we assume that its standard deviation is3% of its mean value. In the whole circuit, we have184random parameters in total. Due to such high dimensionality,simulating this circuit by stochastic spectral methods is achallenging task.

In the following experiments, we simulate this challengingdesign case using our proposed hierarchical stochastic spectralmethods. We also compare our algorithm with other two kindsof hierarchical approaches listed in Table I. In Method 1,both low-level and high-level simulations use Monte Carlo,assuggested by [32]. In Method 2, the low-level simulation usesour ANOVA-based sparse simulator (Alg. 1), and the high-level simulation uses Monte Carlo.

B. Surrogate Model Extraction

In order to extract an accurate surrogate model for theMEMS capacitor, Alg. 1 is implemented in the commercial

TABLE IISURROGATE MODEL EXTRACTION WITH DIFFERENTσ VALUES.

σ # |s|= 1 # |s|= 2 # |s|= 3 # ANOVA terms # nonzero gPC terms # samples0.5 46 0 0 47 81 185

0.1 to 10−3 46 3 0 50 90 21510−4 46 10 1 58 112 30510−5 46 21 1 69 144 415

TABLE IDIFFERENT HIERARCHICAL SIMULATION METHODS.

Method Low-level simulation High-level simulationProposed Alg. 1 stochastic testing [15]

Method 1 [32] Monte Carlo Monte CarloMethod 2 Alg. 1 Monte Carlo

5.5 6 6.5 7 7.5 8 8.5 90

0.2

0.4

0.6

0.8

1

1.2

1.4

MEMS capacitor (fF)

surrogate (σ=10−2)Monte Carlo (5000)

Fig. 5. Comparison of the density functions obtained by our surrogate modeland by5000-sample Monte Carlo analysis of the original MEMS equation.

network-based MEMS simulation tool MEMS+ [74] of Coven-tor Inc. Each MEMS switch is described by a stochasticdifferential equation [c.f. (1)] with consideration of processvariations. In order to compute the MEMS capacitor, we canignore the derivative terms and solve for the static solutions.

By settingσ = 10−2, our ANOVA-based stochastic MEMSsimulator generates a sparse 3rd-order generalized polynomialchaos expansion with only90 non-zero coefficients, requiringonly 215 simulation samples and8.5 minutes of CPU timein total. This result has only3 bivariate terms and no three-variable terms in ANOVA decomposition, due to the very weakcouplings among different random parameters. Settingσ =10−2 can provide a highly accurate generalized polynomialchaos expansion for the MEMS capacitor, which has a relativeerror around10−6 (in theL2 sense) compared to that obtainedby settingσ = 10−5.

By evaluating the surrogate model and the original model(by simulating the original MEMS equation) with5000 sam-ples, we have obtained the same probability density curvesshown in Fig. 5. Note that using the standard stochastictesting simulator [14]–[16] requires18424 basis functions andsimulation samples for this high-dimensional example, whichis prohibitively expensive on a regular computer. When theeffective dimensiondeff is set as3, there should be16262terms in the truncated ANOVA decomposition (22). However,due to the weak couplings among different random parameters,only 90 of them are non-zero.

We can get surrogate models with different accuracies by

0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

Parameter index

main sensitivitytotal sensitivity

Fig. 6. Main and total sensitivities of different random parameters for theRF MEMS capacitor.

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

9

Parameter index

TT−rank

Fig. 7. TT-rank for the surrogate model of the RF MEMS capacitor.

changing the thresholdσ. Table II has listed the number ofobtained ANOVA terms, the number of non-zero generalizedpolynomial chaos (gPC) terms and the number of requiredsimulation samples for different values ofσ. From this table,we have the following observations:

1) Whenσ is large, only46 univariate terms (i.e., the termswith |s | = 1) are obtained. This is because the varianceof all univariate terms are regarded as small, and thusall multivariate terms are ignored.

2) Whenσ is reduced (for example, to0.1), three dominantbivariate terms (with|s | = 2) are included by consid-ering the coupling effects of the three most influentialrandom parameters. Since the contributions of otherparameters are insignificant, the result does not changeeven if σ is further decreased to10−3.

3) A three-variable term (with|s | = 3) and some bivariatecoupling terms among other parameters can only becaptured whenσ is reduced to10−4 or below. In thiscase, the effect of some non-dominant parameters canbe captured.

Fig. 6 shows the global sensitivity of this MEMS capacitorwith respect to all46 random parameters. The output is

−4 −2 0 2 40

0.2

0.4

0.6

0.8(a)

Gauss quadrature points

wei

ghts

−4 −2 0 2 4−10

−5

0

5

10

ζ

gPC

bas

is fu

nctio

ns

(b)

k=0k=1k=2k=3

Fig. 8. (a) Gauss quadrature rule and (b) generalized polynomial chaos (gPC) basis functions for the RF MEMS capacitor.

dominated by only3 parameters. The other43 parameterscontribute to only2% of the capacitor’s variance, and thus theirmain and total sensitivities are almost invisible in Fig. 6.Thisexplains why the generalized polynomial-chaos expansion ishighly sparse. Similar results have already been observed inthe statistical analysis of CMOS analog circuits [1].

C. High-Level Simulation

The surrogate model obtained withσ = 10−2 is importedinto the stochastic testing circuit simulator described in[14]–[16] for high-level simulation. At the high-level, we have thefollowing differential equation to describe the oscillator:

d~q(

~x(t, ~ζ), ~ξ)

dt+ ~f

(

~x(t, ~ζ), ~ζ, u)

= 0(45)

where the input signalu is constant,~ζ=[ζ1, · · · , ζ4] ∈ R4

are the intermediate-level random parameters describing thefour MEMS capacitors. Since the oscillation periodT (~ζ) nowdepends on the MEMS capacitors, the periodic steady-statecan be written as~x(t, ζ) = ~x(t + T (~ζ), ζ). We simulate thestochastic oscillator by the following steps [15]:

1) Choose a constantT0 > 0 to define an unknown scalingfactor a(~ζ) = T (~ζ)/T0 and a scaled time axisτ =t/α(~ζ). With this scaling factor, we obtain a reshapedwaveform ~z(τ, ~ζ) = ~x(t/a(~ζ), ~ζ). At the steady state,we have~z(τ, ~ζ) = ~z(τ + T0, ~ζ). In other words, thereshaped waveform has a periodT0 independent of~ζ.

2) Rewrite (45) on the scaled time axis:

d~q(

~z(τ, ~ζ), ~ξ)

dτ+ a(~ζ)~f

(

~z(τ, ~ζ), ~ζ, u)

= 0.(46)

3) Approximate~z(τ, ~ζ) and a(~ζ) by generalized polyno-mial chaos expansions of~ζ. Then, convert (46) to alarger-scale deterministic equation by stochastic testing.Solve the resulting deterministic equation by shootingNewton with a phase constraints, which would providethe coefficients in the generalized polynomial-chaosexpansions of~z(τ, ~ζ) andα(~ζ) [15].

4) Map ~z(τ, ~ζ) to the original time axis, we obtain theperiodic steady state of~x(t, ζ).

In order to apply stochastic testing in Step 3), we need tocompute some specialized orthonormal polynomials and Gaussquadrature points for each intermediate-level parameterζi. Weuse9 quadrature points for each bottom-level parameterξk toevaluate the high-dimensional integrals involved in the three-term recurrence relation. This leads to946 function evaluationsat all quadrature points, which is prohibitively expensive.

To handle the high-dimensional MEMS surrogate models,the following tensor-based procedures are employed:

• With Alg. 2, a low-rank tensor train ofζ1 is first con-structed for an MEMS capacitor. For most dimensionsthe rank is only2, and the highest rank is4, as shown inFig. 7.

• Using the obtained tensor train, the Gauss quadraturepoints and generalized polynomial chaos basis functionsare efficiently computed, as plotted in Fig. 8.

The total CPU time for constructing the tensor trainsand computing the basis functions and Gauss quadraturepoints/weights is about40 seconds in MATALB. If we di-rectly evaluate the high-dimensional multivariate generalizedpolynomial-chaos expansion, the three-term recurrence rela-tion requires almost1 hour. The obtained results can bereused for all MEMS capacitors since they are independentlyidentical.

With the obtained basis functions and Gauss quadraturepoints/weights for each MEMS capacitor, the stochastic pe-riodic steady-state solver [15] is called at the high level tosimulate the oscillator. Since there are4 intermediate-levelparametersζi’s, only 35 basis functions and testing samplesare required for a3rd-order generalized polynomial-chaosexpansion, leading to a simulation cost of only56 secondsin MATLAB.

Fig. 9 shows the waveforms from our algorithm at thescaled time axisτ = t/a(~ζ). The high-level simulation gen-erates a generalized polynomial-chaos expansion for all nodalvoltages, branch currents and the exact parameter-dependentperiod. Evaluating the resulting generalized polynomial-chaosexpansion with5000 samples, we have obtained the densityfunction of the frequency, which is consistent with those from

0 0.2 0.4 0.60

2

4

(a)

τ [ns] 0 0.2 0.4 0.60

0.01

0.02

0.03(b)

τ [ns]

0 0.2 0.4 0.65

5.002

5.004x 10

−4 (c)

τ [ns]

A

0 0.2 0.4 0.60

0.5

1

1.5x 10

−4 (d)

τ [ns]

Fig. 9. Simulated waveforms on the scaled time axisτ = t/a(~ζ). (a) and (b): the mean and standard deviation ofVout1 (unit: V), respectively; (c) and (d):the mean and standard deviation of the current (unit: A) fromVdd, respectively.

8.75 8.8 8.85 8.9 8.95 9 9.05 9.1 9.15 9.20

1

2

3

4

5

6

7

8

9

10

Frequency [GHz]

ProposedMethod 1Method 2

Fig. 10. Probability density functions of the oscillation frequency.

Method 1 (using5000 Monte Carlo samples at both levels)and Method 2 (using Alg. 1 at the low level and using5000Monte-Carlo samples at the high level), as shown in Fig. 10.

In order to show the variations of the waveform, we furtherplot the output voltages for100 bottom-level random samples.As shown in Fig. 11, the results from our proposed methodand from Method 1 are indistinguishable from each other.

D. Complexity Analysis

Table III has summarized the performances of all threemethods. In all Monte Carlo analysis,5000 random samplesare utilized. If Method 1 [32] is used, Monte Carlo hasto be repeatedly used for each MEMS capacitor, leading toextremely long CPU time due to the slow convergence. IfMethod 2 is used, the efficiency of the low-level surrogatemodel extraction can be improved due to the employmentof generalized polynomial-chaos expansion, but the high-level simulation is still time-consuming. Since our proposedtechnique utilizes fast stochastic testing algorithms at bothlevels, this high-dimensional example can be simulated at verylow computational cost, leading to92× speedup over Method1 and14× speedup over Method 2.

0 0.1 0.2 0.3 0.4 0.5 0.60

2

4

6

t [ns]

(a)

0 0.1 0.2 0.3 0.4 0.5 0.60

2

4

6

t [ns]

(b)

Fig. 11. Realization of the output voltages (unit: volt) at100 bottom-levelsamples, generated by (a) proposed method and (b) Method 1.

VI. CONCLUSIONS ANDFUTURE WORK

This paper has proposed a framework to acceleratethe hierarchical uncertainty quantification of stochasticcir-cuits/systems with high-dimensional subsystems. We havedeveloped an ANOVA-based stochastic testing simulator to ac-celerate the low-level simulation, and a tensor-based techniquefor handling high-dimensional surrogate models at the highlevel. Both algorithms have a linear (or near-linear) complexitywith respect to the parameter dimensionality. Our simulatorhas been tested on an oscillator circuit with four MEMS ca-pacitors and totally184 random parameters, achieving highlyaccurate results at the cost of10-min CPU time in MATLAB.In such example, our method is over92× faster than thehierarchical Monte Carlo method developed in [32], and isabout 14× faster than the method that uses ANOVA-basedsolver at the low level and Monte Carlo at the high level.

There are lots of problems worth investigation in the direc-tion of hierarchical uncertainty quantification. Some unsolvedimportant questions include:

1) How to extract a high-dimensional surrogate model such

TABLE IIICPU TIMES OF DIFFERENT HIERARCHICAL STOCHASTIC SIMULATION ALGORITHMS.

Simulation MethodLow level High level

Total simulation costMethod CPU time Method CPU time

Proposed Alg. 1 8.5 min stochastic testing 1.5 minute Low (10 min)Method 1 Monte Carlo 13.2 h Monte Carlo 2.2 h High (15.4 h)Method 2 Alg. 1 8.5 min Monte Carlo 2.2 h Medium (2.3 h)

that the tensor rank is as small as possible (or the tensorrank is below a provided upper bound)?

2) How to perform non-Monte-Carlo hierarchical uncer-tainty quantification when the outputs of different blocksare correlated?

3) How to perform non-Monte-Carlo hierarchical uncer-tainty quantification whenyi depends on some varyingvariables (e.g., time and frequency)?

ACKNOWLEDGMENTS

The authors would like to thank Coventor Inc. for providingthe MEMS+ license and the MEMS switch design files. Wewould like to thank Shawn Cunningham and Dana Dereus ofWispry for providing access to the MEMS switch data. We aregrateful to Dr. Giovanni Marucci for providing the oscillatordesign parameters, as well as Prof. Paolo Maffezzoni and Prof.Ibrahim Elfadel for their technical suggestions.

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Zheng Zhang (S’09) received his B.Eng. degreefrom Huazhong University of Science and Technol-ogy, China, in 2008, and M.Phil. degree from theUniversity of Hong Kong, Hong Kong, in 2010.Currently, he is a Ph.D student in Electrical Engi-neering and Computer Science at the MassachusettsInstitute of Technology (MIT), Cambridge, MA.His research interests include uncertainty quantifi-cation and tensor analysis, with applications in in-tegrated circuits (ICs), microelectromechanical sys-tems (MEMS), power systems, silicon photonics and

other emerging engineering problems.Mr. Zhang received the 2014 IEEE Transactions on CAD of Integrated

Circuits and Systems best paper award, the 2011 Li Ka Shing Prize (universitybest M.Phil/Ph.D thesis award) from the University of Hong Kong, and the2010 Mathworks Fellowship from MIT. Since 2011, he has been collaboratingwith Coventor Inc., working on numerical methods for MEMS simulation.

Xiu Yang received his B.S. and M.S. degrees inapplied mathematics from Peking University, Chinain 2005 and 2008, respectively, and his Ph.D. degreein applied mathematics in 2014 from Brown Univer-sity, Providence, RI.

He is a postdoc research associate with the PacificNorthwest National Laboratory, Richland, WA. Hisresearch interests include uncertainty quantification,sensitivity analysis, rare events and model calibra-tion with application to multiscale modeling, com-putational fluid dynamics and electronic engineering.

Ivan Oseledets received his PhD and Doctor ofSciences (second Russian degree) degrees in 2007and 2012, respectively, both from the Institute ofNumerical Mathematics of the Russian Academyof Sciences (INM RAS) in Moscow, Russia. Hehas worked in the INM RAS since 2003, where heis now a Leading Researcher. Since 2013 he hasbeen an Associate Professor in Skolkovo Institute ofScience and Technology (Skoltech) in Russia.

His research interests include numerical analy-sis, linear algebra, tensor methods, high-dimensional

problems, quantum chemistry, stochastic PDEs, wavelets, data mining. Appli-cations of interest include solution of integral and differential equations onfine grids, construction of reduced-order models for multi-parametric systemsin engineering, uncertainty quantification,ab initio computations in quantumchemistry and material design, data mining and compression.

Dr. Oseledets received the medal of Russian Academy of Sciences for thebest student work in Mathematics in 2005; the medal of Russian Academyof Sciences for the best work among young mathematicians in 2009. He isthe winner of the Dynasty Foundation contest among young mathematiciansin Russia in 2012.

George Karniadakis received his S.M. (1984) andPh.D. (1987) from Massachusetts Institute of Tech-nology (MIT). Currently, he is a Full Professorof Applied Mathematics in the Center for FluidMechanics at Brown University, Providence, RI.He has been a Visiting Professor and Senior Lec-turer of Ocean/Mechanical Engineering at MIT since2000. His research interests include diverse topics incomputational science and engineering, with currentfocus on stochastic simulation (uncertainty quan-tification and beyond), fractional PDEs, multiscale

modeling of physical and biological systems.Prof. Karniadakis is a Fellow of the Society for Industrial and Applied

Mathematics (SIAM), Fellow of the American Physical Society (APS), Fellowof the American Society of Mechanical Engineers (ASME) and AssociateFellow of the American Institute of Aeronautics and Astronautics (AIAA).He received the CFD award (2007) and the J Tinsley Oden Medal (2013) bythe US Association in Computational Mechanics.

Luca Daniel (S’98-M’03) received the Ph.D. degreein electrical engineering and computer science fromthe University of California, Berkeley, in 2003.

He is currently an Associate Professor in the Elec-trical Engineering and Computer Science Depart-ment of the Massachusetts Institute of Technology(MIT). His research interests include developmentof integral equation solvers for very large complexsystems, uncertainty quantification and stochasticsolvers for large number of uncertainties, and au-tomatic generation of parameterized stable compact

models for linear and nonlinear dynamical systems. Applications of interestinclude simulation, modeling and optimization for mixed-signal/RF/mm-wavecircuits, power electronics, MEMs, nanotechnologies, materials, MagneticResonance Imaging Scanners, and the human cardiovascular system.

Prof. Daniel has received the 1999 IEEE Trans. on Power Electronicsbest paper award; the 2003 best PhD thesis awards from both the ElectricalEngineering and the Applied Math departments at UC Berkeley;the 2003ACM Outstanding Ph.D. Dissertation Award in Electronic Design Automation;5 best paper awards in international conferences and 9 additional nominations;the 2009 IBM Corporation Faculty Award; the 2010 IEEE Early Career Awardin Electronic Design Automation; and the 2014 IEEE Trans. On ComputerAided Design best paper award.