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Big-Data Tensor Recovery for High-DimensionalUncertainty Quantification of Process VariationsZheng Zhang, Member, IEEE, Tsui-Wei Weng, Student Member, IEEE, and Luca Daniel, Member, IEEE

(Invited Paper)

Abstract— Fabrication process variations are a major source ofyield degradation in the nanoscale design of integrated circuits(ICs), microelectromechanical systems (MEMSs), and photoniccircuits. Stochastic spectral methods are a promising techniqueto quantify the uncertainties caused by process variations. Despitetheir superior efficiency over Monte Carlo for many design cases,stochastic spectral methods suffer from the curse of dimensional-ity, i.e., their computational cost grows very fast as the number ofrandom parameters increases. In order to solve this challengingproblem, this paper presents a high-dimensional uncertaintyquantification algorithm from a big data perspective. Specifically,we show that the huge number of (e.g., 1.5 × 1027) simulationsamples in standard stochastic collocation can be reduced to avery small one (e.g., 500) by exploiting some hidden structures ofa high-dimensional data array. This idea is formulated as a tensorrecovery problem with sparse and low-rank constraints, and it issolved with an alternating minimization approach. The numericalresults show that our approach can efficiently simulate some IC,MEMS, and photonic problems with over 50 independent randomparameters, whereas the traditional algorithm can only deal witha small number of random parameters.

Index Terms— High dimensionality, integrated circuits (ICs),integrated photonics, microelectromechanical system (MEMS),polynomial chaos, process variation, stochastic simulation, tensor,uncertainty quantification.

I. INTRODUCTION

FABRICATION process variations (surface roughness ofinterconnects and nano-photonic devices and random

doping effects of transistors) have become a critical issuein nanoscale design, because they can significantly influencechip performance and decrease product yield [2]. Efficientstochastic modeling and simulation algorithms should bedeveloped and implemented in electronic design automation(EDA) software in order to estimate and control the uncer-tainties in a nanoscale chip design. For several decades,Monte Carlo techniques [3], [4] have been the mainstreamstochastic simulators in commercial tools due to their easeof implementation. Nevertheless, they have a slow conver-gence rate, and thus generally require a large number of

Manuscript received August 29, 2016; revised November 4, 2016; acceptedNovember 7, 2016. This work was supported in part by the NSF NEEDSProgram and in part by the AIM Photonics Program. Recommended for pub-lication by Associate Editor S. Grivet-Talocia upon evaluation of reviewers’comments.

The authors are with the Research Laboratory of Electronics,Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:z [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCPMT.2016.2628703

repeated simulations. In recent years, the emerging stochasticspectral methods [5], [6] have been investigated, and theyprove efficient for many design cases with a small number ofparameters, including integrated circuits (ICs) [7]–[17], micro-electromechanical systems (MEMSs) [18], [19], and photoniccircuits [20], [21].

The key idea of stochastic spectral methods is to approx-imate a stochastic solution (e.g., the uncertain voltage orpower dissipation of a circuit) as the linear combination ofsome specialized basis functions such as generalized polyno-mial chaos [22]. Two main classes of simulators have beenimplemented to obtain the coefficients of each basis functions.In an intrusive (i.e., nonsampling) simulator such as stochasticGalerkin [5] and stochastic testing [9], a deterministic equationis constructed such that the unknown coefficients can bedirectly computed by a single simulation. Generally, stochastictesting [9] is more efficient than stochastic Galerkin for manyapplications, since the resulting Jacobian matrix can be decou-pled and the step sizes in transient analysis can be selectedadaptively. In a sampling-based simulator such as stochasticcollocation [6], a few solution samples are first computed byrepeated simulations. Then, some postprocessing techniquesare used to reconstruct the unknown coefficients. The methodsin [15] and [16] reduce the complexity by selecting criticalsamples or critical basis functions. When the number ofrandom parameters is small, these solvers can provide highlyaccurate solutions with significantly (e.g., 100× to 1000×)higher efficiency than Monte Carlo. Unfortunately, stochasticspectral methods suffer from the curse of dimensionality,i.e., their computational cost grows very fast as the numberof random parameters increases.

A. Related Work

Several advanced uncertainty quantification algorithmsthat solve high-dimensional problems have been reportedin the literature. The following are some representativehigh-dimensional solvers for IC and MEMS applications.

1) Sparse techniques: In a high-dimensional polynomialchaos expansion, very often the coefficients of mostbasis functions are close to zero. In [18], this propertywas exploited for analog IC applications using adap-tive analysis of variance (ANOVA) [23]–[25]. In [26],compressed sensing [27] was employed to minimize the�1-norm of the coefficient vector.

2) Matrix low-lank approach: The intrusive solverreported in [28] recognizes that the polynomial chaos

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expansion can be expressed by an order of magni-tude smaller specialized polynomial basis and computesits coefficients and polynomials iteratively by a non-linear optimization starting from the most dominantterms.

3) Model order reduction: In [29], an efficient reducedmodel was used to obtain most solution samples withina sampling-based solver. The reduced model is con-structed by refinements. When a parameter value isdetected for which the reduced model is inaccurate,the original large-scale equation is solved to update themodel on the fly.

4) Hierarchical Approach: Using generalized polynomialchaos expansions to describe devices and subsystems,the tensor-train hierarchical uncertainty quantificationframework in [19] was able to handle complex sys-tems with many uncertainties. The basic idea is totreat the stochastic output of each device/subsystemas a new random input. As a result, the system-leveluncertainty quantification has only a small numberof random parameters when new basis functions areused.

B. Contributions

This paper presents a sampling-based high-dimensionalstochastic solver from a big data perspective. The standardstochastic collocation approach is well known to be affected bythe curse of dimensionality, and it was applicable only to prob-lems with a few random parameters. In this paper, we representthe huge number of required solution samples as a tensor,which is a high-dimensional generalization of a matrix or avector [30].1 In order to overcome the curse of dimension-ality in stochastic collocation, we suggest a tensor recoveryapproach: We use a small number of simulation samples toestimate the whole tensor. This idea is implemented by exploit-ing the hidden low-rank property of a tensor and the sparsity ofa generalized polynomial chaos expansion. Numerical methodsare developed to solve the proposed tensor recovery problem.We also apply this framework to simulate some IC, MEMS,and photonic design cases with lots of process variationsand compare it to standard-sampling-based stochastic spectralmethods.

C. Paper Organization

This paper is organized as follows. Section II briefly reviewsstochastic collocation and tensor computation. Section IIIdescribes our tensor recovery model to reduce the com-putational cost of high-dimensional stochastic collocation.Numerical techniques are described in Section IV to solvethe resulting optimization problem. Section V explains howto obtain a generalized polynomial chaos expansion fromthe obtained tensor factors. The simulation results of somehigh-dimensional IC, MEMS, and photonic circuit examplesare reported in Section VI. Finally, Section VII concludes thispaper and points out future work.

1Tensor is an efficient tool to reduce the computational and memory costof many problems (e.g., deep learning and data mining) in big data analysis.

II. PRELIMINARIES

A. Uncertainty Quantification Using Stochastic Collocation

Let the vector ξ = [ξ1, . . . , ξd ] ∈ Rd denote a set

of mutually independent random parameters that describeprocess variations (e.g., deviation of transistor threshold volt-age and thickness of a dielectric layer in MEMS fabrica-tion). We intend to estimate the uncertainty of an output ofinterest y(ξ). This parameter-dependent output of interest candescribe, for instance, the power consumption of an analogcircuit or the frequency of a MEMS resonator.

1) Generalized Polynomial Chaos: Assuming that ysmoothly depends on ξ and that y has a bounded variance,2

we apply a truncated generalized polynomial chaos expan-sion [22] to approximate the stochastic solution

y(ξ) ≈p∑

|α|=0

cα�α(ξ), with E[�α (ξ ) �β(ξ)] = δα,β . (1)

Here, the operator E denotes the expectation, δ denotes a Deltafunction, and the basis functions {�α(ξ )} are orthonormalpolynomials, where α = [α1, . . . , αd ] ∈ N

d is a vectorindicating the highest polynomial order of each parame-ter in the corresponding basis. The total polynomial order|α| = |α1| + . . . + |αd | is bounded by p, and thus the totalnumber of basis functions is K = (p + d)!/(p!d!). Since ξ

are mutually independent, for each parameter ξk , one can firstconstruct a set of univariate orthonormal polynomials φk,αk (ξk)with αk = 0, . . . , p. Then, the multivariate polynomial basisfunction with index α is

�α(ξ) =d∏

k=1

φk,αk (ξk). (2)

The univariate polynomial functions can be obtained by thethree-term recurrence relation in [31], and the main steps aresummarized in Appendix A.

2) Stochastic Collocation: Since all basis functions in (1)are orthonormal to each other, the coefficient cα can beobtained by a projection framework

cα =∫

Rdy(ξ)�α(ξ )ρ(ξ )dξ , with ρ(ξ ) =

d∏

k=1

ρk(ξk). (3)

Note that ρ(ξ ) is the joint probability density function of vec-tor ξ and ρk(ξk) is the marginal density of ξk . The above inte-gral needs to be evaluated with a proper numerical technique.Popular integration techniques include randomized approachessuch as Monte Carlo [3] and deterministic approaches liketensor product and sparse grid [32]. Monte Carlo is feasiblefor extremely high-dimensional problems, but its numericalaccuracy is low. Deterministic approaches can generate veryaccurate results using a low-order quadrature rule, but they arefeasible only for problems with a small or medium number ofrandom parameters due to the curse of dimensionality. Thispaper considers the tensor product implementation, which wasregarded as much less efficient than sparse grid techniques inalmost all previous publications.

2In this paper, we assume that y is a scalar.


ZHANG et al.: BIG-DATA TENSOR RECOVERY FOR HIGH-DIMENSIONAL UNCERTAINTY QUANTIFICATION OF PROCESS VARIATIONS 3

Fig. 1. (a) 2-D data array (e.g., a medical image) is a matrix. (b) 3-D dataarray (e.g., multiple slices of images) is a tensor.

We briefly introduce the idea of tensor product numericalintegration. Let {(ξ ik

k , wikk )}n

ik=1 be n pairs of 1-D quadra-ture points (or samples) and weights for parameter ξk . Suchquadrature points and weights can be obtained by variousnumerical techniques, which can be found in [33]. In thispaper, we use the Gauss quadrature rule [34] to generate such1-D samples and weights, as summarized in Appendix B.A Gauss quadrature rule with n samples can generate exactresults when the univariate integrand is a polynomial functionof ξk and when the highest polynomial degree is not higherthan 2n −1. By tensorizing all 1-D quadrature points/weights,the d-dimensional integral in (3) can be evaluated as

cα =∑

1≤i1,...,id ≤n

y(ξ i1 ...id )�α(ξ i1 ...id )wi1 ...id . (4)

Here, ξ i1 ...id = [ξ i11 , . . . , ξ

idd ] and wi1 ...id = wi1

1 . . . widd are the

resulting multidimensional quadrature samples and weights,respectively. Obtaining each solution sample y(ξ i1 ...id ) mayrequire a time-consuming numerical simulation. For instance,a periodic steady-state solver may be called to compute thefrequency of an oscillator. In device modeling, a large-scalesolver must be called to solve a complex partial differentialequation or integral equation for each quadrature sample.The numerical implementation (4) requires nd times of suchexpensive device or circuit simulations.

B. Tensor and Tensor Decomposition

1) Tensor: Tensor is a high-dimensional generalization ofmatrix. A matrix X ∈ R

n1×n2 is a second-order tensor, and itselement indexed by i = (i1, i2) can be denoted by xi1i2 or X(i).For a general dth-order tensor X ∈ R

n1×...nd , its elementindexed by i = (i1, . . . , id) can be denoted by xi1...id or X (i).Here, the integer k ∈ [1, d] is the index for a mode of X .Fig. 1 shows a matrix and a third-order tensor.

Given any two tensors X and Y of the same size, theirinner product is defined as

〈X ,Y〉 :=∑

i1...id

xi1...id yi1...id .

The Frobenius norm of tensor X is further defined as||X ||F := (〈X ,X 〉)1/2.

2) Tensor Decomposition: A tensor X is rank-1 if it can bewritten as the outer product of some vectors

X = u1 ◦ · · · ◦ ud ⇔ xi1...id = u1(i1) · · · ud(id) (5)

where uk(ik) denotes the ik th element of vector uk ∈ Rnk .

Similar to matrices, a low-rank tensor can be written as a

canonical decomposition [35], which expresses X as the sumof some rank-1 tensors

X = Tcp(U(1), . . . , U(d)) :=r∑

j=1

u j1 ◦ · · · ◦ u j

d . (6)

Here, U(k) = [u1k, . . . , ur

k ] ∈ Rnk×r is a matrix including all

factors corresponding to mode k, operator Tcp converts allmatrix factors to a tensor represented by canonical decom-position, and the minimum integer r that ensures (6) to holdis called tensor rank. As a demonstration, Fig. 2 shows thelow-rank factorizations of a matrix and third-order tensor,respectively. Tensor decomposition (6) can significantly reducethe cost of storing high-dimensional data arrays. For simplic-ity, let us assume nk = n. Directly representing tensor Xrequires storing nd scalars, whereas only ndr scalars need tobe stored if the above low-rank factorization exists.

Note that there are other kinds of tensor factorizationssuch as Tucker decomposition [36] and tensor-train decom-position [37]. We introduce only canonical decomposition inthis paper because we will use it to solve high-dimensionaluncertainty quantification problems in the subsequent sections.Interested readers are referred to [30] for a detailed surveyof tensor decompositions, as well as [38] for a tutorial withapplications in EDA.

III. TENSOR RECOVERY APPROACH

Formulation (4) was applicable to only problems with fiveor six random parameters due to the nd simulation samples.This section describes our tensor recovery approach that cansignificantly reduce the computational cost of tensor productstochastic collocation. With this framework, (4) can be moreefficient than sparse grid approaches and Monte Carlo simu-lation for many high-dimensional design cases.

A. Reformulating Stochastic Collocation With Tensors

We first define the following two tensors:

1) tensor Y ∈ Rn1×...×nd , with nk = n and each element

being yi1...id = y(ξ i1...id );2) tensor Wα ∈ R

n1×...×nd , with nk = n and its elementindexed by (i1, . . . id) being �α(ξ i1 ...id )wi1 ...id .

Tensor Wα depends only on the basis function in (1) andthe multidimensional quadrature weights in (4). Furthermore,according to (2), it is straightforward to see that Wα is arank-1 tensor with the following canonical decomposition:

Wα = vα11 ◦ · · · ◦ vαd

d

with

vαkk = [

φk,αk

(ξ1

k

)w1

k , . . . , φk,αk

(ξn

k

)wn

k

]T ∈ Rn×1. (7)

Note that ξikk and w

ikk are the ik th 1-D quadrature point

and weight for parameter ξk , respectively, as described inSection II-A.

With the above two tensors, (4) can be written in thefollowing compact form:

cα = 〈Y,Wα〉. (8)



Fig. 2. Top: low-rank factorization of a matrix. Bottom: canonical decomposition of a third-order tensor.

Since Wα is straightforward to obtain, the main computationalcost is to compute Y . Once Y is computed, the cα valuesand thus the generalized polynomial chaos expansion (1)can be obtained easily. Unfortunately, directly computing Yis impossible for high-dimensional cases since it requiressimulating a specific design case nd times.

B. Tensor Recovery Problem (Ill-Posed)

We define two index sets.

1) Let I include all indices (i1, . . . , id) for the elementsof Y . The number of elements in I, denoted by |I|,is nd .

2) Let be a small subset of I, with || |I|. For eachindex (i1, . . . , id) ∈ , the corresponding solution sam-ple yi1...id is already obtained by a numerical simulator.

In order to reduce the computational cost, we aim at estimatingthe whole tensor Y using the small number of availablesimulation data specified by . With the sampling set ,a projection operator P is defined for Y

B = P(Y) ⇔ bi1...id ={

yi1...id , if (i1, . . . , id) ∈

0, otherwise.

(10)

We want to find a tensor X such that it matches Y for theelements specified by

‖P(X − Y)‖2F = 0. (11)

However, this problem is ill-posed because any value can beassigned to xi1...id if (i1, . . . , id ) /∈ .

C. Regularized Tensor Recovery Model

In order to make the tensor recovery problem well posed,we add the following constraints based on heuristic observa-tions and practical implementations.

1) Low-Rank Constraint: Very often we observe that thehigh-dimensional solution data array Y has a low tensorrank. Therefore, we expect that its approximation X hasa low-rank decomposition described in (6).

2) Sparse Constraint: As shown in the previous workof ANOVA decomposition [18] and compressed sens-ing [26], most of the coefficients in a high-dimensionalgeneralized polynomial chaos expansion have a verysmall magnitude. This implies that the �1-norm of avector collecting all coefficients cα, which is computedas

p∑

|α|=0

|cα| ≈p∑

|α|=0

|〈X ,Wα〉| (12)

should be very small.

Finalized Tensor Recovery Model: Combining the abovelow-rank and sparse constraints together, we suggest thefinalized tensor recovery model (9), as shown at the bottomof this page, to compute X as an estimation of Y . In thisformulation, X is assumed to have a rank-r decompositionand we compute its matrix factors U(k) instead of the wholetensor X . This treatment has a significant advantage: Thenumber of unknown variables is reduced from nd to dnr ,which is now a linear function of parameter dimensionality d .

D. Cross Validation

An interesting question is how accurate is X compared tothe exact tensor Y . Our tensor recovery formulation enforcesconsistency between X and Y at the indices specified by .It is desired that X also has a good predictive behavior—xi1...idis also close to xi1...id for (i1, . . . , id) /∈ . In order to measurethe predictive property of our results, we define a heuristicprediction error

εpr =√∑

(i1,...,id )∈′ (xi1...id − yi1...id )2 wi1...id∑(i1,...,id )∈′ (yi1...id )2 wi1...id

.

Here, ′ ⊂ I is a small-size index set such that ′ ∩ = ∅.The solution X is regarded as a good approximate to Y if εpris small; then, (1) can be obtained accurately using (8), and thestatistical behavior (e.g., probability density function) of y(ξ)can be well predicted. Estimating εpr requires simulating thedesign problem at some extra quadrature samples. However,a small-size ′ can provide a good heuristic estimation.

minU(1),...,U(d)∈Rn×r

f (U(1), . . . , U(d)) = 1

2‖P(Tcp(U(1), . . . , U(d)) − Y)‖2

F + λ

p∑

|α|=0

|〈Tcp(U(1), . . . , U(d)),Wα〉| (9)



Algorithm 1 Alternating Minimization for Solving (9)

1: Initialize: U(k),0 ∈ Rn×r for k = 1, . . . d;

2: for l = 0, 1, . . .3: for k = 1, . . . , d do4: solve (13) by Alg. 2 to obtain U(k),l+1 ;5: end for6: break if converged;7: end for8: return U(k) = U(k),l+1 for k = 1, . . . , d .

At present, we do not have a rigorous approach to findthe optimal values of λ and r . In practice, their valuesare chosen heuristically. Specifically, we increment λ and runtil εpr becomes small enough. Occasionally, optimizationproblem (9) may be solved several times for different valuesof λ and r . However, like other sampling-based stochasticsolvers, the computational cost of postprocessing [i.e., solv-ing (9)] is generally negligible compared to the cost ofsimulating solution samples indexed by .

IV. SOLVE PROBLEM (9)

The optimization problem (9) is solved iteratively in ourimplementation. Specifically, starting from a provided initialguess of the low-rank factors {U(k)}d

k=1, alternating minimiza-tion is performed recursively using the result of the previousiteration as a new initial guess. Each iteration of alternatingminimization consists of d steps. At the kth step, the kth-modefactor matrix U(k) corresponding to parameter ξk is updated bykeeping all other factors fixed and by solving (9) as a convexoptimization problem.

A. Outer Loop: Alternating Minimization

1) Algorithm Flow: We use an iterative algorithm tosolve (9). Let U(k),l be the mode−k factors of X afterl iterations. Starting from an initial guess {U(k),0}d

k=1, weperform the following iterations.

i) At iteration l + 1, we use {U(k),l}dk=1 as an initial

guess and obtain updated tensor factors {U(k),l+1}dk=1

by alternating minimization.ii) Each iteration consists of d steps; at the kth step,

U(k),l+1 is obtained by solving

U(k),l+1 = arg minX

f (. . . , U(k−1),l+1, X, U(k+1),l, . . .).

(13)

Since all factors except that of mode k are fixed, (13) becomesa convex optimization problem and its global minimum canbe computed by the solver in Section IV-B. The alternatingminimization method ensures that the cost function decreasesmonotonically to a local minimal. The pseudocodes are sum-marized in Algorithm 1.

2) Convergence Criteria: With matrices {U(k),l}dk=1

obtained after l iterations of the outer loops of Algorithm 1,

we define

fl := f (U(1),l, . . . , U(d),l)

Xl := Tcp(U(1),l, . . . , U(d),l)

clα := 〈Xl ,Wα〉.

The first term is the updated cost function of (9), the secondterm is the updated tensor solution, and the last term is theupdated coefficient corresponding to basis function �α(ξ )in (1). Let cl = [. . . , cl

α, . . .] ∈ RK collect all coefficients

in (1), then we define the following quantities for errorcontrol:

i) Relative update of the tensor factors:

εl,tensor =√√√√

∑dk=1 ‖U(k),l − U(k),l−1‖2

F∑dk=1 ‖U(k),l−1‖2

F

.

ii) Relative update of c = [. . . , cα, . . .]εl,gPC = ‖cl − cl−1‖/‖cl−1‖.

iii) Relative update of the cost function:

εl,cost = | fl − fl−1|/| fl−1|.The computed factors U(1),l, . . . , U(d),l are regarded as a localminimal, and thus Algorithm 1 terminates if εl,tensor, εl,gPC,and εl,cost are all small enough.

B. Inner Loop: Subroutine for Solving (13)

Following the procedures in Appendix C, we rewriteProblem (13) as a generalized LASSO problem:

vec(U(k),l+1) = arg minx

1

2‖Ax − b‖2

2 + λ|Fx| (14)

where A ∈ R||×nr , F ∈ R

K×nr , and b ∈ R||×1 and

x = vec(X) ∈ Rnr×1 is the vectorization of X [i.e., the

( jn−n+i)th element of x is X(i, j) for any integer 1 ≤ i ≤ nand 1 ≤ j ≤ r ]. Note that || is the number of available sim-ulations samples in tensor recovery and K = (p + d)!/(p!d!)is the total number of basis functions in (1).

We solve (14) by the alternating direction method of mul-tipliers (ADMM) [39]. Problem (14) can be rewritten as

minx,z

1

2‖Ax − b‖2

2 + λ|z| s.t. Fx − z = 0.

By introducing an auxiliary variable u and starting with initialguesses x0, u0 = z0 = Fx0, the following iterations areperformed to update x and z:

x j+1 = (AT A + sFT F)−1(AT b + sFT (z j − u j ))

z j+1 = shrinkλ/s(Fx j+1 + z j + u j )

u j+1 = u j + Fx j+1 − z j+1. (15)

Here, s > 0 is an augmented Lagrangian parameter, and thesoft thresholding operator is defined as

shrinkλ/s(a) =

⎧⎪⎨

⎪⎩

a − λ/s, if a > λ/s

0, if |a| < λ/s

a + λ/s, if a < −λ/s.

The pseudocodes for solving (13) are givenin Algorithm 2.



Fig. 3. Numerical results of the MEMS capacitor, with r = 3 and λ = 0.01. Top-left: relative error of the generalized polynomial chaos coefficients initerations. Top-right: decrease of the cost function in (9). Bottom-left: sparsity of the obtained generalized polynomial chaos expansion. Bottom-right: obtainedprobability density function compared to that from Monte Carlo.

Algorithm 2 ADMM for Solving (13)1: Initialize: form A, F and b according to Appendix C,

specify initial guess x0, u0 and z0;2: for j = 0, 1, . . . do3: compute x j+1, z j+1 and u j+1 according to (15);4: break if ‖Fx j+1 − z j+1‖ < ε1 & ‖FT (z j+1 − z j )‖ < ε2;5: end for6: return U(k),l+1 = reshape(x j+1, [n, r ]).

C. Remarks

The cost function of (9) is nonconvex, and therefore, it isnontrivial to obtain a globally optimal solution with theoreticalguarantees. Theoretically speaking, the numerical solution of anonconvex optimization problem depends on the given initialguess. Although researchers and engineers are very oftensatisfied with a local minimal, the obtained result may notbe good enough for certain cases. In our examples, we findthat using random initial guess works well for most cases.However, novel numerical solvers are still highly desired tocompute the globally optimal solution of (9) with theoreticalguarantees.

V. GENERATING STOCHASTIC MODEL (1)

Assuming that the low-rank factors U(1), . . . , U(d) of Xhave been computed, we are ready to compute the coefficientcα for each basis function in (1). Specifically, replacing Y withX in (8) and exploiting the rank-1 property of Wα in (7),we can easily compute cα by

cα ≈ 〈X ,Wα〉 =r∑

j=1

(d∏

k=1

⟨u j

k , vαkk

⟩)

Fig. 4. Schematic of an RF MEMS capacitor [41].

where vαkk is a low-rank factor of Wα in (7). The above

expression can be computed by efficient vector inner products.Once the generalized polynomial chaos expansion (1) is

obtained, various statistical information of the performancemetric y(ξ) can be obtained. For instance, the expectationand standard deviation of y(ξ ) can be obtained analytically;the density function of y(ξ) can be obtained by sampling (1)or using the maximum entropy algorithm [40].

VI. NUMERICAL RESULTS

In order to verify the effectiveness of our tensor recoveryuncertainty quantification framework, we show the simulationresults of three examples ranging from ICs, MEMSs, and pho-tonic circuits. Since our focus is to solve high-dimensionalproblems, we simply assume that all process variations aremutually independent to each other, although they are likelyto be correlated in practice. All codes are implemented inMATLAB and run on a Macbook with a 2.5-GHz CPU and a16-GB memory.

A. MEMS Example (With 46 Random Parameters)

We first consider the MEMS device in Fig. 4, which wasdescribed in detail in [41]. This example has 46 randomparameters describing the material and geometric uncertainties



Fig. 5. Numerical results of the ring oscillator, with r = 3 and λ = 0.1. Top-left: relative error of the tensor factors for each iteration. Top-right: decreasein the cost function in (9). Bottom-left: sparsity of the obtained generalized polynomial chaos expansion. Bottom-right: obtained density function comparedto that from Monte Carlo using 5000 samples.

TABLE I

COMPARISON OF SIMULATION COST FOR THE MEMS CAPACITOR

in CMOS fabrication. The capacitance of this device dependson both bias voltage and process parameters. We assumethat a fixed dc voltage is applied to this device, such thatwe can approximate the capacitance as a second-order gen-eralized polynomial chaos expansion of 46 random parame-ters. Assume that we use three Gauss quadrature points foreach parameter. Consequently, as shown in Table I, a tensorproduct integration requires 346 ≈ 8.9 × 1021 simulationsamples and the Smolyak sparse grid technique still requires4512 simulation samples.

We simulate this device using only 300 quadrature sam-ples randomly selected from the tensor product integrationrules, and then our tensor recovery method estimates thewhole tensor Y [which contains all 346 samples for theoutput y(ξ )]. The relative approximation error for the wholetensor is about 0.1% (measured by cross validation). As shownin Fig. 3, our optimization algorithm converges with less than70 iterations and the generalized polynomial chaos coefficientsare obtained with a small relative error (below 10−4), and theobtained model is very sparse and the obtained density func-tion of the MEMS capacitor is almost identical with that fromMonte Carlo. Note that the number of repeated simulationsin our algorithm is only about quarter of the total number ofbasis functions.

B. Multistage CMOS Ring Oscillator (With 57 Parameters)

We continue to consider the CMOS ring oscillator inFig. 6. This circuit has seven stages of CMOS inverters;

Fig. 6. Schematic of a CMOS ring oscillator.

TABLE II

COMPARISON OF SIMULATION COST FOR THE RING OSCILLATOR

57 random parameters are used to describe the variations ofthreshold voltages, gate-oxide thickness, and effective gatelength/width. We intend to obtain a second-order polynomialchaos expansion for its oscillation frequency by calling a peri-odic steady-state simulator repeatedly. The required number ofsimulations for different algorithms is listed in Table II, whichclearly shows the superior efficiency of our approach for thisexample.

We simulate this circuit using only 500 samples randomlyselected from the 357 ≈ 1.6 × 1027 tensor product integra-tion samples, and then our algorithm estimates the wholetensor Y with a 1% relative error. As shown in Fig. 5,our optimization algorithm converges after 46 iterations andthe tensor factors are obtained with less than 1% relativeerrors, and the obtained model is very sparse and the obtaineddensity function of the oscillator frequency is almost identicalwith that from Monte Carlo. Note that the number of our



Fig. 7. Numerical results of the photonic bandpass filter, with r = 3 and λ = 0.1. Top-left: relative error of the tensor factors for each iteration.Top-right: decrease in the cost function in (9). Bottom-left: sparsity of the obtained generalized polynomial chaos expansion. Bottom-right: obtained densityfunction of the filter bandwidth compared to that from Monte Carlo using 5000 samples.

simulations (i.e., 500) is much smaller than the total numberof basis functions (i.e., 1711) in the generalized polynomialchaos expansion.

C. Photonic Bandpass Filter (With 41 Parameters)

Finally, we consider the photonic bandpass filter in Fig 8.This Chebyshev-type filter has 20 ring resonators and wasoriginally designed to have a 3-dB bandwidth of 20 GHz,a 26-dB minimum return loss, a 400-GHz free spectral range,and a 1.55-μm operation wavelength. A total of 41 randomparameters are used to describe the variations of the effectivephase index (neff ) of each ring as well as the gap (g)between adjoint rings and between the first/last ring andthe bus waveguides. These parameters are assumed to beindependent Gaussian variables, with neff,i = 2.2315585 +N (0, 5 × 10−6) and gi = 0.3 + N (0, 10−3)μm. We intendto obtain a second-order polynomial chaos expansion for the3-dB bandwidth at the DROP port of this filter. The requirednumber of simulations for different algorithms is listed inTable III. Similar to the results of the previous two examples,our tensor recovery approach is significantly more efficientthan the standard tensor product stochastic collocation and thesparse grid implementation.

We simulate this photonic circuit using only 500 samplesrandomly selected from the 341 ≈ 3.6 × 1019 tensor productintegration samples, and then our algorithm estimates thewhole tensor Y with a 0.1% relative error. As shown in Fig. 7,our optimization algorithm converges after 32 iterations andthe tensor factors are obtained with less than 1% relativeerrors, and the obtained model is also sparse and the obtaineddensity function of the bandwidth is almost identical with thatfrom Monte Carlo. Note that the number of our simulations(i.e., 500) is much smaller than the total number of basisfunctions (i.e., 903) in the generalized polynomial chaosexpansion.

Fig. 8. Schematic of a photonic bandpass filter, with N = 20.

TABLE III

COMPARISON OF SIMULATION COST FOR THE PHOTONIC CIRCUIT

VII. CONCLUSION

This paper has presented a big data approach forsolving the challenging high-dimensional uncertaintyquantification problem. Our key idea is to estimate thehigh-dimensional simulation data array from an extremelysmall subset of its samples. This idea has been describedas a tensor recovery model with low-rank and sparseconstraints. Detailed numerical methods have beendescribed to solve the resulting optimization problem.The simulation results on a CMOS ring oscillator, a MEMSRF capacitor, and an integrated photonic circuit showthat our algorithm can be easily applied to problems withabout 40 to 60 random parameters. Instead of using a hugenumber of (e.g., about 1027) quadrature samples, our algorithmrequires only several hundreds, which is even much smallerthan the number of basis functions. The proposed algorithmis much more efficient than sparse grid and Monte Carlo for



our tested cases, whereas Monte Carlo used to be the onlyfeasible approach to handle the underlying high-dimensionalnumerical integration.

There exist some open theoretical questions that are worthfurther investigations.

1) First, it is desirable to develop a rigorous frameworksuch that the tensor rank r and the regularization para-meter λ can be determined in an optimal manner.

2) Second, the resulting tensor recovery model is noncon-vex. A framework that can obtain its global optimal orrelax the model to a convex one will be valuable.

3) Third, it is worth improving our method such that it canefficiently handle a vector output y(ξ ).

4) Finally, our framework generates the subset in arandom way. How to generate optimally is stillunclear.

It is also possible to extend our framework to other engineeringapplications, such as power systems and robotics.

APPENDIX ACONSTRUCTING ORTHONORMAL POLYNOMIALS [31]

Consider a single random parameter ξk ∈ R with a prob-ability density function ρk(ξk), one can construct a set ofpolynomial functions subject to the orthonormal condition

∫

R

φk,α(ξk)φk,β (ξk)ρk(ξk)dξk = δα,β

where δα,β is a Delta function and integer α is the highestdegree of φk,α(ξk). Such polynomials can be constructed asfollows [31]. First, one constructs orthogonal polynomials{πk,α(ξk)}p

α=0 with an leading coefficient 1 recursively

πk,α+1(ξk) = (ξk − γα)πk,α(ξk) − καπk,α−1(ξk)

for α = 0, 1, . . . p − 1, with initial conditionsπk,−1(ξk) = 0, πk,0(ξk) = 1, and κ0 = 1. For α ≥ 0,the recurrence parameters are defined as

γα = E(ξkπ

2k,α(ξk)

)

E(π2

k,α(ξk)) , κα+1 = E

(ξkπ

2k,α+1(ξk)

)

E(ξkπ

2k,α(ξk)

) . (16)

Here, E denotes the operator that calculates expectation.Second, one can obtain {φk,α(ξk)}p

α=0 by normalization

φk,α(ξk) = πk,α(ξk)√κ0κ1 . . . κα

, for α = 0, 1, . . . , p.

APPENDIX BGAUSS QUADRATURE RULE [34]

Given ξk ∈ R with a density function ρk(ξk) and a smoothfunction q(ξk), the Gauss quadrature evaluates the integral

∫

R

q(ξk)ρk(ξk)dξk ≈n∑

ik =1

q(ξ

ikk

)w

ikk

with an error decreasing exponentially as n increases. An exactresult is obtained if q(ξk) is a polynomial function ofdegree ≤ 2n − 1. One can obtain {(ξ ik

k , wikk )}n

ik=1 by reusing

the recurrence parameters in (16) to form a symmetric tridi-agonal matrix J ∈ R

n×n

J(i, j) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

γi−1, if i = j√κi , if i = j + 1√κ j , if i = j − 1

0, otherwise

for 1 ≤ i, j ≤ n.

Let J = Q�QT be an eigenvalue decomposition and Q aunitary matrix, then ξ

ikk = �(ik, ik) and w

ikk = (Q(1, ik))

2.

APPENDIX CASSEMBLING THE MATRICES AND VECTOR IN (14)

Consider the tensor factors U(1),l+1, . . ., U(k−1),l+1, X,and U(k+1),l , . . ., U(d),l in (13). We denote the (i, j) elementof U(k′),l (or X) by scalar u(k′),l

i, j (or xi, j ), and its j th column

by vector u(k′),lj (or x j ) ∈ R

n×1. Then, the cost function in(13) is

f (. . . , U(k−1),l+1, X, U(k+1),l, . . .)

= 1

2

∑

i∈

⎛

⎝r∑

j=1

xik , jμi, j −Y(i)

⎞

⎠2

+λ∑

|α|≤p

∣∣∣∣∣∣

r∑

j=1

να, j⟨x j , w(k)

αk

⟩∣∣∣∣∣∣

where the scalars μi, j and να, j are computed as follows:

μi, j =k−1∏

k′=1

u(k′),l+1ik′ , j

d∏

k′=k+1

u(k′),lik′ , j

να, j =k−1∏

k′=1

⟨u(k′),l+1

j , w(k′)αk′

⟩ d∏

k′=k+1

⟨u(k′),l

j , w(k′)αk′

⟩.

Since each row (or element) of A (or b) corresponds to anindex i = (i1, . . . , id) ∈ and each row of F corresponds toa basis function �α(ξ ), in this Appendix, we use i as the rowindex (or element index) of A (or b) and α as the row indexof F.3 Now we specify the elements of A, b, and F of (14).

1) For every i ∈ , b(i) = Y(i).2) Since xik , j is the (( j − 1)n + ik)th element of x =

vec(X) ∈ Rnr×1, for every i ∈ , we have

A(i, ( j − 1)n + ik) ={

μi, j , for j = 1, . . . , r

0, otherwise.

3) Since x j includes the elements of x ∈ Rnr×1 ranging

from index ( j − 1)n + 1 to jn, given an index vector α,the corresponding row of F can be specified as

F(α, jn − n + ik) = να, j vαkk (ik) = να, j φk,αk

(ξ

ikk

)w

ikk

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Zheng Zhang (M’15) received the Ph.D. degree inelectrical engineering and computer science fromthe Massachusetts Institute of Technology (MIT),Cambridge, MA, USA, in 2015.

His industrial experiences include Coventor Inc.Cambridge, and Maxim-IC, Colorado Springs, CO,USA, academic visiting experiences include the Uni-versity of California at San Diego, La Jolla, CA,USA, Brown University, Providence, RI, USA, andPolitechnico di Milano, Milan, Italy; governmentlaboratory experiences include the Argonne National

Laboratory, Lemont, IL, USA. Currently, he is a Post-Doctoral Associate withthe Research Laboratory of Electronics, MIT. His current research interestsinclude uncertainty quantification, tensor and model order reduction, withapplication to nanoelectronics, energy, and biomedical problems.

Dr. Zhang was a recipient of the 2016 ACM Outstanding Ph.D. DissertationAward in Electronic Design Automation, the 2015 Doctoral DissertationSeminar Award (i.e., Best Thesis Award) from the Microsystems TechnologyLaboratory of MIT, the 2014 Best Paper Award from the IEEE TRANS-ACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND

SYSTEMS, the 2014 Chinese Government Award for Outstanding StudentsAbroad, and the 2011 Li Ka-Shing Prize from the University of Hong Kong.



Tsui-Wei Weng (S’12) received the B.S. and M.S.degrees in electrical engineering from NationalTaiwan University, Taipei, Taiwan, in 2011 and2013, respectively. She is currently pursuing thePh.D. degree with the Department of ElectricalEngineering and Computer Science, MassachusettsInstitute of Technology, Cambridge, MA, USA.

Her current research interests include mixedinteger programming and nonconvex optimizationproblems in machine learning, as well as uncer-tainty quantification in emerging technology such as

artificial intelligence and nanophotonics.

Luca Daniel (S’98–M’03) received the Ph.D. degreein electrical engineering from the University ofCalifornia at Berkeley (UC Berkeley), Berkeley, CA,USA, in 2003.

His industry experiences include the HP ResearchLaboratory, Palo Alto, Santa Clara, CA, USA,in 1998, and the Cadence Berkeley Laboratory,Berkeley, in 2001. He is currently a Full Professorwith the Electrical Engineering and ComputerScience Department, Massachusetts Institute ofTechnology, Cambridge, MA, USA. His current

research interests include integral equation solvers, uncertainty quantificationand parameterized model order reduction, applied to RF circuits, siliconphotonics, MEMSs, magnetic resonance imaging scanners, and the humancardiovascular system.

Prof. Daniel was a recipient of the 1999 IEEE TRANSACTIONS ON POWER

ELECTRONICS Best Paper Award, the 2003 Best Ph.D. Thesis Awards fromthe Electrical Engineering and the Applied Math departments at UC Berkeley,the 2003 ACM Outstanding Ph.D. Dissertation Award in Electronic DesignAutomation, the 2009 IBM Corporation Faculty Award, the 2010 IEEE EarlyCareer Award in Electronic Design Automation, the 2014 IEEE TRANSAC-TIONS ON COMPUTER-AIDED DESIGN Best Paper Award, and seven bestpaper awards in conferences.

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