1

Quantitative Error Bounds of Polynomial ChaosExpansion and their Application to

Resonant SystemsEduard Frick, David Dahl, Member, IEEE, Katharina Klioba, Christian Seifert, Marko Lindner,

and Christian Schuster, Senior Member, IEEE

Abstract—The polynomial chaos expansion (PCE) has becomea method of significant research interest for uncertainty quantifi-cation in the context of electrical and electromagnetic systems.This is due to several advantages including the fast convergencerate of PCE that is observed for many applications. In resonantsystems however the PCE convergence slows down significantlyfor frequencies in the vicinity of the resonances. This problemis well known but a quantitative analysis that would allow for aprediction of the PCE convergence speed has not been presentedyet. This paper introduces quantitative upper bounds for thePCE approximation error resulting from a general applicationwith a single random uniformly or Gaussian distributed inputparameter. Many resonant systems can be locally approximatedby a RLC parallel circuit. In some of these systems the upperbounds can be used for the estimation of the polynomial degreethat is required for the reduction of the PCE error below aprescribed precision. This is demonstrated on the contour integralmethod (CIM) where a non-intrusive PCE is used for modelingthe absolute value of a stochastic crosstalk impedance in a viaarray.

Index Terms—polynomial chaos expansion, resonant systems,parallel circuit, contour integral method, convergence, errorbounds.

I. INTRODUCTION

VARIOUS mathematical methods, with varying degreesof generality, numerical efficiency, and ease of practical

application, have been developed in the past for the purposeof uncertainty quantification in electrical and electromag-netic systems. One of the most popular methods is MonteCarlo (MC) sampling due to its relatively high generalityand simplicity of implementation and application. One of itsmajor deficiencies is the (in general) low numerical efficiency.When model evaluations need significant times, more efficientmethods are desired.

One of these methods which has attracted significant interestin recent years is the Polynomial Chaos Expansion (PCE)method that has been discussed in detail e.g. in [1], [2]. PCEhas been used recently for modeling various sub-types ofelectrical and electromagnetic components and sub-systems.

Eduard Frick, Katharina Klioba, Christian Seifert, and Marko Lindnerare with the Institute of Mathematics, Hamburg University of Technology,Hamburg, Germany, e-mail: eduard.frick, katharina.klioba, christian.seifert,lindner@tuhh.de

David Dahl and Christian Schuster are with the Institute of ElectromagneticTheory, Hamburg University of Technology, Hamburg, Germany, e-mail:david.dahl, schuster@tuhh.de

Manuscript received XXXX XX, 20XX; revised XXXX XX, 20XX. Thiswork was supported in part by the German Research Foundation (DFG).

It has found applications in modeling via transitions in termsof scattering parameters [3] and the equivalent characteris-tic impedances [4]. Other modeled passive components in-clude cascaded transmission line segments [5], on-chip multi-conductor transmission lines [6], coupled transmission linesand RC-circuits [7], microstrip lines and filters [8], transmis-sion lines with edge-roughness which are modeled as non-uniform ones [9], and planar power distributions networks[10]. PCE has further been applied to power amplifiers [11],[12], and other non-linear circuits [13], [14], [15], and trans-mission lines with connected non-linear circuits [16], [17].In the signal integrity context, it has also been shown to beapplicable for studying the variability of eye openings of links[18], [19].

Mathematically the uncertainty quantification techniquescan be described as the evaluation of a deterministic function fin a stochastic parameter x ∈ R. The PCE method often givesa fast and accurate way to model the uncertainty propagationto the output f(x), provided certain integrability conditionsare met. The speed of convergence however depends stronglyon the function f and on the assumed distribution of theinput parameter, see e.g. [2, Theorem 3.6]. In some cases,the convergence is very slow and then a complete failure ofthe method is possible due to numerical errors. An importantcase in which the application of the PCE becomes challengingis related to the phenomenon of resonances. Here we speakof a resonance when a quantity of the system varies stronglywithin a narrow range of frequencies, showing a local ex-tremum or even a singularity. It is well known that uncertaintyquantification near resonances becomes problematic for thePCE. However, in most relevant applications damping appears,replacing a singularity by a more or less steep but finitepeak, so that the PCE remains applicable although requiringsufficiently high polynomial degrees. The problem of higherrequired polynomial degrees is mentioned in the literature butmostly limited to the statement of the highest used polyno-mial order and the obtained deviations. The relation betweenreduced accuracy and frequency ranges in which resonancesoccur has been discussed in [20] and [21]. In [21] polynomialorders up to 10 are used for the modeling of a realistic link andsignificant deviations can be observed in the resonant regimewhile good agreement is found outside.

The main contribution of this paper is the introduction ofanalytically derived infinite sequences of upper bounds for thePCE approximation error resulting from a general application

2

with a single random uniformly or Gaussian distributed inputparameter. The derivation of these bounds is based on theapproach proposed in [2, Theorem 3.6] and has been appliedin [22]. In contrast to [2, Theorem 3.6], the derived boundsare quantitative, i.e they can be explicitly determined for everyapplication of PCE with a single random uniformly or Gaus-sian distributed input parameter. This is the main improve-ment compared to the asymptotic result in [2, Theorem 3.6].Typically the upper bounds overestimate the PCE error. An-other contribution of this paper is a technique that can beused in applications showing resonance for the estimation ofthe polynomial degree that is required for the reduction ofthe PCE approximation error below a prescribed precision.Some resonant applications can be locally approximated bya RLC parallel circuit. The error estimation technique forsuch systems is based on the quantitative error bounds and anadditional empirical assumption.

The remainder of this paper is organized as follows: Sec-tion II provides the mathematical theory and framework forPCE and demonstrates the convergence behavior of the methodin a short mathematical example. In Section III quantita-tive error bounds are derived and applied to the stochasticRLC parallel circuit. A technique for the estimation of thepolynomial degree that is required for the convergence ofthe PCE approximation error in general applications showingresonance that is based on these error bounds is presented inSection IV-A. This estimation technique is then demonstratedin Section IV-B, where PCE is applied to a contour integralmethod for modeling of the stochastic crosstalk impedance ina via array with a stochastic pitch. A conclusion of the paperand an outlook to future work is presented in Section V. Ap-pendix A provides mathematical background for the precedingsections.

II. MATHEMATICAL DESCRIPTION OF THE PROBLEM

In this section we introduce the Polynomial Chaos Expan-sion and demonstrate its convergence behavior. The math-ematical definition of the PCE and of the correspondingconvergence errors is given in Section II-A and is based ondefinitions given in Appendix A. It is important to know thatthe approximation error consists of two parts, the analyticaland the numerical. PCE can be applied to every functionf ∈ L2(R, ρ) if the numerical part of the error is sufficientlysmall. The same is true for the MC method. However incontrast to MC, the convergence rate of PCE errors dependsmore strongly on some additional properties of f . For somechoices of f ∈ L2(R, ρ) the convergence of the analytical PCEerror can be very slow (we postpone the quantification of thisobservation to Section III) and the presence of the numericalerror can then result in the divergence of the method. Theapplication of PCE and MC and the presence of both parts ofthe PCE approximation error is demonstrated in an illustrativemathematical example in Section II-B.

A. Polynomial Chaos Expansion and its Analytical and Nu-merical Errors

Let X be the random input variable and ρ be the corre-sponding probability density function. Then, following the way

described in Appendix A, every function f ∈ L2(R, ρ) can beexpanded as an infinite series (27) w.r.t. an orthogonal basis(ψn) of polynomials. For p ∈ N0, its finite truncation of orderp,

Ipf :=

p∑n=0

fnψn =

p∑n=0

1

γn〈f, ψn〉ρ ψn

, is then called the polynomial chaos expansion (PCE) of fof order p.

Its analytical approximation error is given by

Ep(f) := ‖Ipf − f‖ρ =

( ∞∑n=p+1

|fn|2γn

) 12

(1)

with the norm defined in (25) and converges towards zero asp → ∞. The rate of convergence will be discussed in moredetail in Section III-A. We also define the relative analyticalerror

ep :=Ep(f)

‖f‖ρ. (2)

In practice, one typically cannot compute the inner products〈f, ψn〉ρ in fn analytically, but approximates the correspond-ing integrals on R with some numerical scheme, e.g. Gaussianquadrature. Thus one obtains approximate generalized Fouriercoefficients fn ≈ fn resulting, for p ∈ N0, in an approxima-tion Ipf =

∑pn=0 fnψn of Ipf , which introduces an additional

(now numerical) error. So the total relative error

ep :=

∥∥∥f − Ipf∥∥∥ρ

‖f‖ρ≤ ep +

∥∥∥Ipf − Ipf∥∥∥ρ

‖f‖ρ, (3)

is bounded by the sum of the relative analytical error (2)which stems from the truncation of the generalized Fourierseries and the relative numerical error from the computationalapproximation of the integrals. If ep is small, then all requiredstatistical information about f(X) can be extracted fromits analytical PCE expansion Ipf . The mean value and thevariance of Ipf are then given by

E(Ipf(X)) = f0, (4a)

V(Ipf(X)) =

p∑n=1

|fn|2 γn. (4b)

If in addition the relative numerical error, and thus alsothe total error ep is small, then it is sufficient to use theapproximations

E(Ipf(X)) = f0 (5a)

V(Ipf(X)) =

p∑n=1

|fn|2γn. (5b)

Note that the analytical variance error equals the squaredanalytical error, i.e.

|V(f(X))− V(Ipf(X))| =∞∑n=1

|fn|2 γn −p∑

n=1

|fn|2 γn

=

∞∑n=p+1

|fn|2 γn = E2p(f).

(6)

FRICK et al.: QUNTITATIVE ERROR BOUNDS OF POLYNOMIAL CHAOS EXPANSION AND THEIR APPLICATION TO RESONANT SYSTEMS 3

Consequently the relative analytical variance error is given by

|V(f(X))− V(Ipf(X))|V(f(X))

=E2p(f)

V(f(X))= e2

p ·‖f‖2ρ

V(f(X)). (7)

This means that the errors (2) and (7) can be easily convertedinto each other. On the other hand we always have

E(f(X))− E(Ipf(X)) = 0 (8)

for the analytical mean error. This means that an analyticalPCE method of degree p = 0 is already sufficient for thecalculation of the exact mean value. In other words, if oneobserves problems in obtaining the mean value from a PCEsimulation, then these problems are always due to numericalerrors.

B. Illustration of Errors and Convergence Behavior

We define the function fa,b by

fa,b : R→ R ∪ ∞, fa,b(x) =1√

(x− a)2 + b2(9)

for parameters a, b ∈ R and consider two different probabilitydistributions, the uniform distribution with X ∼ U(−1, 1) andthe Gaussian distribution with X ∼ N (0, 1). The uniformdensity ρ is defined by (29) and has a bounded supportsupp (ρ) = [−1, 1], whereas the Gaussian density ρ is definedby (30) and one has supp (ρ) = R. Note that for b = 0,the function fa,0 has a singularity in x = a and thus onehas fa,0 /∈ L2(R, ρ) if a ∈ supp (ρ), which is always truefor the Gaussian distribution and only true for the uniformdistribution if |a| ≤ 1. On the other hand b 6= 0 alwaysimplies fa,b ∈ L2(R, ρ), which then in theory makes theapplication of both PCE and MC possible. The parameterb 6= 0 has the effect of damping the singularity. A smallvalue of |b| means that

∣∣∣f ′a,b(x0)∣∣∣ becomes large for all x0

in the vicinity of a. As we will see, this will slow downthe PCE convergence and to a lesser extent also the MCconvergence. Because of the numerical part of the PCE errorone can sometimes even observe divergence of the PCE. At thesame time the convergence problems resulting from the smalldamping |b| ≈ 0 can be reduced if the parameter a is selectedin such a way that ρ(x0) ≈ 0 for all x0 with large values∣∣∣f ′a,b(x0)

∣∣∣. To demonstrate this we consider the parametersa ∈ 0, 6 and b ∈ 0.01, 0.9. Fig. 1 shows the central partsof the graphs of the corresponding functions fa,b together withthe graphs of the uniform and Gaussian densities.

We will now investigate the convergence of the relativevariance errors for PCE and MC (see (7) and Appendix A). Forthe determination of the analytical variance V(fa,b(X)) theright hand side of (31) is computed with symbolic integrationin Matlab [23]. The symbolic Matlab integration is alsoused for the determination of the analytical PCE coefficients,that are required for the computation of the variance of theanalytical PCE expansion according to (4b). This is fairlyexact (so that the second error term in (3) is close to zero)as long as the polynomial degree p is not too large; we willuse p ≤ 20 in the following. It is then justified to see thesymbolic integration as an analytical method.

Fig. 1. Graphs of fa,b(x) = 1√(x−a)2+b2

for all combinations of the values

a ∈ 6, 0 and b ∈ 0.9, 0.01 and both uniform and Gaussian densities ρ.The distributions themselves are also shown.

To show the significance of the second error term in (3) thePCE coefficients will also be determined numerically by theGaussian quadrature. In the considered example we use the 22roots of the orthogonal basis polynomial ψ22 as integrationnodes needed to approximate the n-th projection coefficient(fa,b)n and compute the variance of the truncated numericalPCE expansion according to (5b).

The relative variance errors from the analytical and numeri-cal PCE simulation are shown in Figs. 2a and 2b. As expected,the analytical PCE converges for all considered choices ofparameters a and b and both distributions. In all cases theerrors from the uniform PCE simulation decrease faster thanthe errors from the Gaussian simulation. For both distributionswe observe that the damping parameter b has a large impacton the convergence rate if a = 0. The convergence rate of theanalytical PCE is very slow and the numerical PCE does notconverge at all if a = 0 and b = 0.01. While for a = 6 andX ∼ U(−1, 1) there is no significant difference between theresults corresponding to b = 0.9 and b = 0.01, the decreaseof damping from b = 0.9 to b = 0.01 results in a slowerconvergence of the Gaussian PCE errors. To explain this wego back to the definition (1) of the PCE error and split thecoefficients (fa,b)n on the right hand side of (1) into two parts

(fa,b)n =1

γn

3∫−∞

fa,b(x)ψn(x)ρ(x) dx

+1

γn

∞∫3

fa,b(x)ψn(x)ρ(x) dx.

(10)

Keeping Fig. 1 in mind we can now think of the PCEerrors as consisting of two separate parts corresponding tothe two different parts of the integral in (10). In the case ofX ∼ U(−1, 1) we have ρ(x) = 0 for x > 3 and thus thesecond integral in (10) does not contribute to the error. Forthe Gaussian distribution the second part does not vanish butit is damped by small values of ρ(x). In Fig. 1 we see that thefunctions f6,b, b ∈ 0.9, 0.01, are very smooth and almostidentical on [−3, 3]. This results in a fast convergence of thefirst part of the error. For X ∼ N (0, 1) this means that aftera few polynomial orders the error is dominated by the secondpart of the integral in (10). Note that the damped singularities

4

of the functions f6,b, b ∈ 0.9, 0.01, have their centers atx = 6 and thus the error corresponding to the second part of(10) has a much lower convergence rate for b = 0.01 than forb = 0.9.

For the MC experiment we take N random uniformlyand N random Gaussian distributed samples x1, . . . , xN andcompute the sample variances Vmc(N) of the function valuesfa,b(x1), . . . , fa,b(xN ), for N ∈ 101,102, . . . ,108. Thisexperiment is repeated M = 10000 times, resulting in Mdifferent relative variance errors for every number of samplesN . To demonstrate the expected MC convergence rate wethen take the mean values over M relative variance errors forevery number of samples N . The log-log plots Figs. 2c and2d show the convergence of these mean values together withthe function 1√

N. Except for the slight deviation of the error

corresponding to a = 6, b = 0.01 and X ∼ N (0, 1) all errorsare of the order O( 1√

N). This is much slower than the observed

convergence of PCE for a = 6 and b = 0.9. On the other hand,even though the MC method requires more samples to achievethe same precision in the case of low damping, its convergencerate remains unchanged which makes MC more attractive thanPCE for some choices of parameters a and b.

III. QUANTITATIVE ERROR BOUNDS AND THEIRAPPLICATION TO THE PARALLEL RESONANT CIRCUIT

In this section upper bounds for the relative analyticalerror ep (2) will be introduced. The bounds are derived inSection III-A. They are quantitative, i.e they can be explicitlydetermined for every application of PCE with a single randomuniformly or Gaussian distributed input variable X . In SectionIII-B these bounds will be applied to a resonant stochasticRLC parallel circuit which can be seen as a model for amore general resonant system. The random output in such asystem is modeled by a certain function Z : R→ R. In SectionII-B a strong dependence of the PCE convergence rate on theparameters a and b in the definition (9) of the function fa,bwas demonstrated. It will be shown in Section III-B that Z,modeling the absolute value of the impedance of the RLCparallel circuit at a fixed angular frequency ω (see (20)), hasexactly the form of (9) apart from a scaling factor. This meansthat the parameters a and b will have the same influence onthe PCE convergence rate as was already observed for the fa,bfunction in Section II-B.

Our goal is now to replace the parameters a and b by morereasonable and reproducible measures, the relative resonancedistance ω and the quality factor Q, that will be defined in thefollowing. Determination of the deterministic values of Z forall frequencies ω results in a deterministic curve which has alocal maximum or a local minimum at the resonance angularfrequency ω0. We will then denote by ∆ω the bandwidth at1/√

2 (or√

2) times the local maximum (or local minimum)of the deterministic curve. The relative resonance distance isdefined by

ω :=ω − ω0

ω0· 100 % (11)

which is the percentage share of ω − ω0 with respect to ω0.The amount of damping in our system is measured by thequality factor

Q(ω0) :=ω0

∆ω. (12)

As will be demonstrated in application examples in SectionsIII-B and IV-B, the rate of the PCE convergence depends in thesame way on ω and 1

Q as it depends on a and b in Sec. II-B.

A. Quantitative Upper Bounds of the Analytical Error

The derivation of quantitative upper bounds for the relativeanalytical error ep as defined in (2) will require an additionalregularity assumption on the function Z. The bounds dependon the order of the polynomial chaos expansion p and convergetowards zero as p converges towards ∞.

The existence of qualitative upper bounds is well known.For all n ∈ N0 there exist non-negative constants C(n) andΛ

(n)p , depending only on n and ρ, such that for all Z ∈ Hn

ρ

the error is bounded by

Ep(Z) ≤ C(n) · Λ(n)p · ‖Z‖Hnρ . (13)

(For notations of Sobolev spaces, we remind the reader ofthe Mathematical Appendix.) For the uniform distribution thisis shown in [2, Theorem 3.6], where one has Λ

(n)p = p−n.

Similar results, describing the asymptotic PCE convergence,are also known for other commonly used distributions. Therelative error ep defined in (2) is then bounded by

ep ≤ C(n) · Λ(n)p ·

‖Z‖Hnρ‖Z‖ρ

. (14)

For the quantification of the PCE convergence the knowledgeof all constants on the right hand side of (14) is mandatory.In most cases the constant Λ

(n)p is known and the norms

‖Z‖Hnρ and ‖Z‖ρ (defined in (26) and (25), respectively) canbe calculated analytically or approximated by a numericalor statistical method. Note that one has ep ≤ 1 for alldistributions and because of ‖Z‖H0

ρ= ‖Z‖ρ the inequality

(14) holds for n = 0 with C(0) = Λ(0)p = 1. However, to our

knowledge, the constant C(n) has never been quantified in thenon trivial case of n > 0.

By following the same procedure as in the proof of [2,Theorem 3.6] and applying some additional analysis discussedin [22], the following result for the uniform and Gaussiandistributions have been obtained. Consider first the case of theuniform distribution. For all k ∈ N0 we set

β :=2

3 ·√

3

and

Sk =

4 + 2β, k = 0,

53 + (76 + 20β) ·√

21+2β60+16β , k = 1,

456 + (424 + 36β) ·√

110+4β220+24β , k = 2,

(16k4 + 16k3 + 20k2 + 1) +(2 ·√

8k3 + 10k2 + 3k + 2β ·√32k3 − 8k2 + (8β − 12)k + 2− 2β

), k > 2.

FRICK et al.: QUNTITATIVE ERROR BOUNDS OF POLYNOMIAL CHAOS EXPANSION AND THEIR APPLICATION TO RESONANT SYSTEMS 5

(a) (b)

(c) (d)

Fig. 2. Relative variance errors from the analytical and numerical PCE simulation for fa,b(X) as a function of polynomial degree p and mean valuesfrom 10000 MC relative variance errors for fa,b(X) as a function of the number of MC samples N . All combinations of the parameters a ∈ 6, 0 andb ∈ 0.9, 0.01 are considered. For a = 6 the difference between the analytical and numerical PCE results is insignificant. (a) PCE relative variance errorsfor the distribution X ∼ U(−1, 1). (b) PCE relative variance errors for the distribution X ∼ N (0, 1). (c) MC relative variance errors for the distributionX ∼ U(−1, 1). (d) MC relative variance errors for the distribution X ∼ N (0, 1).

Then (14) holds for all even, positive integers

n ∈ 2N = 2, 4, 6, . . .

and

C(n) =

n2−1∏k=0

√Sk, Λ(n)

p = (p2 + 3 · p+ 2)−n2 .

In the case of the Gaussian distribution we set S0 = 1, S1 = 9and

Sk = 9k2 + 1 + 2 ·√

3 ·√

6k − 7 ·√

2k − 1

for k ≥ 2. Then (14) holds for all

n ∈ 3N− 1 = 2, 5, 8, . . .

and

C(n) =

n+13 −1∏k=0

√Sk, Λ(n)

p = (p+ 1)−n+13 .

The procedure above gives rise to two different (with respectto the distribution) infinite sequences, one for the uniform, onefor the Gaussian distribution, of bounds given by the right handside of (14). We denote the corresponding right hand side of(14) by

b(m)p , m ∈ N0, (15)

where the new enumeration m ∈ N0 corresponds to n ∈ N0

from (14) via Table I. As the error ep of the expansion of

TABLE IORDERING OF THE UPPER BOUNDS (14). THE ORDERING DEPENDS ON THE

DISTRIBUTION OF THE RANDOM INPUT VARIABLE X . EACH INDEXm ∈ N0 IN THE REORDERED SEQUENCE (15) CORRESPONDS TO A

SOBOLEV NORM n OF OF THE UPPER BOUND IN (14).

m = 0 1 2 3 · · ·

X ∼ U(−1, 1), n = 0 2 4 6 · · ·X ∼ N (0, 1), n = 0 2 5 8 · · ·

order p ∈ N0 is bounded by each b(m)p , it is also bounded by

the minimum of all bounds, i.e.

ep ≤ minm∈N0

b(m)p . (16)

In the following we will restrict ourselves to the computationof only two bounds, b(1) and b(2), for each of the distributions.

Let us further denote the minimal polynomial degree thatis required for the reduction of the error ep at a fixed angularfrequency ω below a prescribed precision δ > 0 by

Pδ(ep) := min p ∈ N0 : ep ≤ δ. (17)

Let b(m)p be one of the upper bounds, that is given by the right

hand side of (14). Then the corresponding minimal polynomialdegree

Pδ(b(m)p ) := min p ∈ N0 : b(m)

p ≤ δ (18)

6

overestimates Pδ(ep) by a certain factor. We will denote thisfactor by

λ(m)δ :=

Pδ(ep)

Pδ(b(m)p )

(19)

and refer to it as a convergence quotient.

B. Application to Error Bounds for RLC Parallel CircuitsConsider an RLC parallel circuit, composed of an in-

ductance L, a capacitance C, and a resistance R. For thefurther discussion it is more convenient to use the conductanceG = 1/R. Our uncertain input parameter is deliberatelychosen to be the capacitance C, distributed with mean C0 =5.12× 10−12 F and standard deviation σ = 0.0625C0, being6.25 % of the mean value. We standardize C by a linear trans-form. Then the random input is modeled by C = C0 + σX ,where we have σ = σ and X ∼ N (0, 1) in case of theGaussian distribution, and σ =

√3 · σ and X ∼ U(−1, 1)

in case of the uniform distribution. The complex impedanceas a function of the stochastic parameter of the circuit dependson the angular frequency ω and is given by

Z(C) =1

G+ j(ω · C − 1

ωL

) ,where j is the imaginary unit. The stochastic output variableof our choice is the absolute value |Z(C)|, that is modelledby the function Z : R→ R given by

Z(x) = |Z(C0 + σx)| = 1√G2 + (ω(C0 + σx)− 1

ωL )2.

(20)Our goal is to apply the PCE method and to compare

the convergence rate of the error ep to the convergence rateof its upper bounds. We will restrict ourselves to the upperbounds b(1)

p and b(2)p (see Section III-A) and compute the error

and the bounds analytically, i.e. using symbolic integrationin MATLAB [23]. As in the case of the fa,b function (9)in Section II-B we limit ourselves to the polynomial degreesp ≤ 20 for the determination of ep. This is done to avoidnumerical errors and computational cost and is not requiredin the case of b(1)

p and b(2)p , where, as one can see from (14),

only the p-dependent part of the constant has to be recomputedif the polynomial degree p is increased. The latter is notcomputationally expensive and does not cause considerablenumerical errors.

Consider a variation of the frequency ν from 17.08 GHz to27.4 GHz in steps of 40 MHz. To observe a resonance exactlyat frequency ν0 = 22.24 GHz the inductance L is set such thatwe get

ω0C0 −1

ω0L= 0,

where ω0 = 2π · ν0. Our choice of the frequency range thenincludes the variation of the relative resonance distance ω ofmore than 20 %.

We want to construct a configuration in dependence of thequality factor Q defined in (12). For a fixed Q factor theconductance G is then obtained as

G =ω0C0

Q.

Fig. 3. Deterministic curves of the absolute value of the impedance Zcomputed for the RLC parallel circuit for different quality factors Q.

Figure 3 shows the deterministic curves of Z for differentchoices of Q.

By defining the parameters a, b, and K as

a :=C0

σ

(ω2

0

ω2− 1

), b :=

ω0

ω· C0

Qσ, K :=

1

ωσ, (21)

one can rewrite Z in (20) as

Z(x) = K · 1√(x− a)2 + b2

(22)

and see that, apart from the scaling factor K, Z has the formof the function fa,b from Section II-B. It turns out that Q isinversely proportional to b. This means that a large qualityfactor is expected to cause the same convergence problemsas a small parameter b did in Section II-B. There it has alsobeen shown that the impact of the damping parameter b onthe convergence rate is dependent on |a|. In this section wewill look at the relative resonance distance ω instead of theparameter a. As one can see from (21) and (11), ω = 0 isequivalent to a = 0 whereas an increase or decrease of |ω|implies an increase or decrease of |a| and vice versa.

Consider now the minimal polynomial degrees (17) and(18), computed for δ = 0.01 and δ = 0.05 for differentfrequencies and quality factors. The results are shown inFigures 4 and 5. Figure 4 shows the plots of Pδ(b

(1)p ), Pδ(b

(2)p ),

and Pδ(ep) for all quality factors

Q ∈ 1, 2, 4, 6, 8, 10 ∪ 20, 30, 40 . . . , 100

and three fixed frequencies with corresponding relative res-onance distances ω = 0 %, ω = 10.6 %, and ω = 20 %.In Figure 5 the minimal polynomial degrees are plotted overthe relative resonance distance ω for a fixed quality factorQ = 40. As expected one always has Pδ(ep) ≤ Pδ(b

(1)p ) and

Pδ(ep) ≤ Pδ(b(2)p ). While the convergence behavior of the

upper bounds is very similar to that of the error ep, the error isoverestimated by both upper bounds, i.e. Pδ(b

(1)p ) and Pδ(b

(2)p )

are larger than Pδ(ep) for most parameter configurations ofδ, Q, and ω. We also see that the convergence of the errorand of the bounds is slower in the case of the Gaussiandistribution than in the case of the uniform distribution. Anincreasing quality factor Q is significantly slowing down theconvergence rates for both distributions for frequencies in thevicinity of resonances, i.e. for small absolute values of therelative resonance distance |ω|, while the influence of theconvergence rate on Q is decreasing with increasing |ω|. This

FRICK et al.: QUNTITATIVE ERROR BOUNDS OF POLYNOMIAL CHAOS EXPANSION AND THEIR APPLICATION TO RESONANT SYSTEMS 7

decrease is expected to happen faster for X ∼ U(−1, 1) thanfor X ∼ N (0, 1), which was already observed and explainedin Section II-B and is due to the bounded support of theuniform distribution.

IV. APPLICATIONS TO A RESONANT SYSTEM

The main goal of this section is to present a techniquefor the estimation of the minimal polynomial degree Pδ(ep)(see (17)) that is required for the reduction of the error epbelow a desired precision δ > 0 when applying PCE toa resonant system for angular frequencies in a vicinity ofa resonance angular frequency w0. A single uniformly orGaussian distributed random input parameter is assumed andthe damping of the resonance is described by the quality factorQ as defined in (12). The success of this technique is basedon the empirical assumption that the convergence quotientsobtained from the application of the definition (19) to theconsidered resonant system, are in good agreement with theconvergence quotients obtained from the application of thedefinition (19) to a RLC parallel circuit with the same qualityfactor and the same distribution of the random input parameter(random capacitance C) as in the considered system. Thetechnique for the estimation of the required polynomial degreePδ(ep), cf. (17), is described in Section IV-A and demonstratedin Section IV-B, where the PCE is applied to the contourintegral method (CIM).

A. Adaptation to General Systems

Consider a resonant system with a single uniformly orGaussian distributed random input parameter. The goal isto apply the PCE at a fixed angular frequency ω with anerror ep (2) that is bounded by δ for some δ > 0. Therequired polynomial degree Pδ(ep) can then be estimated froma successive application of the following steps.

1) Determine the relative resonance distance ω and thequality factor Q according to the definitions (11) and(12).

2) Apply the definition (18) to determine the minimalpolynomial degree Pδ(b

(1)p ) for the first upper bound

b(1)p (see (14), (15) and Table I).

3) Use the convergence quotients λ(1)δ (19), computed for

the RLC parallel circuit for the same values of Q, ω, δand the same distribution of the random input parameter(random capacitance C) as in the considered system, toobtain the approximation of Pδ(ep):

Pδ(ep) ≈ minq ∈ N0 : Pδ(b(1)p ) · λ(1)

δ ≤ q. (23)

The convergence quotients λ(1)δ can be calculated and

tabulated in a preparation step. They are defined for eachfrequency, but instead we will use the maximal convergencequotients, where the maximum is taken over all frequen-cies ω with their relative frequency distance bounded byω1 ≤ |ω| ≤ ω2 for different variations of the values ω1, ω2.Table II shows the maximal convergence quotients.

An alternative approach for the estimation of Pδ(ep) hasbeen introduced in [24]. This alternative method does not

require the computation of the analytically derived upperbounds and can be applied to multi-resonant systems. It isalso motivated by the resonant RLC parallel circuit. But incontrary to the method in this paper, one does not simplyreplace the system by a RLC parallel circuit. Instead one looksdirectly at the stochastic output of the considered system. Fora fixed frequency this output is represented by a certain realvalued function. The function that represents the output is thenreplaced by a fit function which is constructed as a finite linearcombination of simple analytically defined functions of thefrom (9). Each one of the functions in this linear combinationhas a form of a stochastic output of a certain parallel resonantcircuit. Thus indirectly the stochastic system is approximatedby a superposition of stochastic parallel circuits. The minimalpolynomial degree that is required for the reduction of the PCEerror in the considered system below a prescribed precisionis then estimated by the minimal polynomial degree that isrequired for the reduction of the error resulting from theapplication of the PCE to the constructed fit functions. Apartfrom the estimated minimal polynomial degree, that is oftenin good agreement with the required polynomial degree, thealternative method also provides proven and thus alwaysreliable upper and lower bounds for the required minimalpolynomial degree. The advantage of the alternative method isits generality and reliability. Its main disadvantage comparedto the method presented in this paper is its complexity. Theconstruction of the fit functions can be time-intensive and cannot be done in a preprocessing step. Both methods have beenderived only for the case of a single random input parameter.

B. Application to Crosstalk Analysis of Via Arrays

To test the technique for the estimation of Pδ(ep) fromSection IV-A we will apply the non-intrusive PCE to thecontour integral method (CIM) [25], [26] for modeling theabsolute value of a stochastic crosstalk impedance in a viaarray. The test configuration is depicted in Figure 6. Theuncertain input parameter is selected to be the pitch d. A meanvalue of d0 = 80 mil and a standard deviation of σ = 5 mil,being 6.25 % of the mean value, is assumed. The pitch dis then modeled by a random variable d0 + σX , where wehave σ = σ and X ∼ N (0, 1) in case of the Gaussiandistribution, and σ =

√3 · σ and X ∼ U(−1, 1) in case

of the uniform distribution, and the absolute value of thecrosstalk impedance |Z12| is modeled by the random elementZ(d), where Z := |Z12|. Consider a frequency variation from18 GHz to 32 GHz in steps of 40 MHz. The deterministiccurves that are obtained for the constant pitch value d = 80 milshow damped resonances at two different frequencies locatedat about 22.24 GHz and 31.1 GHz, i.e. frequencies for which|Z12| has a local maximum.

The damping of the resonances depends on the loss tan-gents. We will consider the loss tangents values 0.09, 0.03and 0.007. The three steps of the estimation technique fromSection IV-A will now be applied for all considered valuesof the loss tangents to all frequencies between 17.08 GHzand 27.4 GHz. The estimation technique from Section IV-Ais developed for systems with a single resonance frequency.

8

1 20 40 60 80 1000100101102103104

1 20 40 60 80 1000100101102103104

1 20 40 60 80 1000

100101102103104

(a)

1 20 40 60 800

5

10

15

1001 20 40 60 8005

10152025

1 20 40 60 80 1000

100

101

102

(b)

Fig. 4. The minimal polynomial degrees Pδ(ep), Pδ(b(1)p ) and Pδ(b

(2)p ) (see (17) and (18)) obtained for the RLC parallel circuit with a random capacitance

C, distributed with mean C0 = 5.12× 10−12 F and standard deviation σ = 0.0625C0, are plotted over the quality factor Q for three different frequencies ν.The random input is modeled by C0+ σX , where we have σ = σ and X ∼ N (0, 1) in case of the Gaussian distribution, and σ =

√3 ·σ and X ∼ U(−1, 1)

in case of the uniform distribution. (a) Results for X ∼ N (0, 1) and δ = 0.01. (b) Results for X ∼ U(−1, 1) and δ = 0.01.

-20 -15 -10 -5 0 5 10 15 20

100200300400500600700

-20 -15 -10 -5 000

0

5 10 15 20

50100150200250300

(a)

010

20

30

40

50

-20 -15 -10 -5 0 5 10 15 2005

1015202530

(b)

Fig. 5. The minimal polynomial degrees Pδ(ep), Pδ(b(1)p ), and Pδ(b

(2)p ) (see (17) and (18) ) for the RLC parallel circuit with a random capacitance C,

distributed with mean C0 = 5.12× 10−12 F and standard deviation σ = 0.0625C0, are plotted over the relative resonance distance ω (see (11)) for thequality factors Q ∈ 10, 40 and δ ∈ 0.01, 0.05. The random input is modeled by C0 + σX , where we have σ = σ and X ∼ N (0, 1) in case of theGaussian distribution, and σ =

√3 · σ and X ∼ U(−1, 1) in case of the uniform distribution. (a) Results for X ∼ N (0, 1), Q = 40 and δ ∈ 0.01, 0.05.

(b) Results for X ∼ U(−1, 1), Q = 40 and δ ∈ 0.01, 0.05.

y

h

d

= sources and sinks of radial waves

= Cylindrical scattererGround via

Cylindrical signal via ports

1 2

x

zd

ddd

Fig. 6. Configuration considered in the CIM simulation: a pair of signalvias surrounded by 10 ground vias which act as scatterers in the two-dimensional structure bounded by two infinite and perfectly conducting planesof separation h = 10mil. Each via is represented by a radial port with a radiusof 8mil. The dielectric permittivity equals 3.5 and the pitch d has a nominalvalue of 80mil.

However in the considered example the PCE error is affectedmostly by the first large resonance. We will neglect thesecond resonance and use the first resonance angular frequency

Fig. 7. Deterministic curves of the absolute value of the crosstalk impedanceZ = |Z12| computed with CIM for different quality factors Q. The qualityfactors have been determined by applying the definition (12), where theresonance angular frequency at about ω0 = 2π· 22.24GHz has been used.

when applying the estimation technique. The first resonancefrequency will be denoted by ω0, i.e. ω0 ≈ 2π· 22.24 GHz.

In the first step the definitions (11) and (12) are applied toω0. The considered frequency range includes the variation of

FRICK et al.: QUNTITATIVE ERROR BOUNDS OF POLYNOMIAL CHAOS EXPANSION AND THEIR APPLICATION TO RESONANT SYSTEMS 9

(a) (b)

(c) (d)

Fig. 8. The required and estimated minimal polynomial degrees Pδ(ep) for the quality factors Q ∈ 8.7, 20.6, 39.7 and δ ∈ 0.01, 0.05 correspondingto the application of the non intrusive PCE to CIM that is used for modeling of a stochastic crosstalk impedance with a random pitch d, distributed with amean of d0 = 80mil and a standard deviation of σ = 0.0625 d0. The random input is modeled by d0 + σX , where we have σ = σ and X ∼ N (0, 1)in case of the Gaussian distribution, and σ =

√3 · σ and X ∼ U(−1, 1) in case of the uniform distribution. The quality factors have been determined by

applying the definition (12), where the resonance angular frequency at about ω0 = 2π· 22.24GHz has been used. The resulting approximation error ep (see(2)) is used for the determination of the required Pδ(ep), which is computed according to the definition (17). The estimated Pδ(ep) is obtained by followingthe steps described in Section IV-A. (a) Results for d ∼ 1mil · U(80, 52) and Q = 39.7. (b) Results for d ∼ 1mil · U(80, 52) and Q = 20.6. (c) Resultsfor d ∼ 1mil · U(80, 52) and Q = 8.7. (d) Results for d ∼ 1mil · N (80, 52) and Q = 8.7. The solid red line shows the estimated Pδ(ep) computed withquotients λ(1)δ from Table II. The dashed red line shows the estimated Pδ(ep) computed with λ(1)δ · 2.5 instead of λ(1)δ .

the relative resonance distance ω (11) from -20% to 20%. Theobtained quality factors Q = 8.7, Q = 20.6 and Q = 39.7correspond to the loss tangents values of 0.09, 0.03 and 0.007.Figure 7 shows the deterministic curves. In the second stepof the estimation technique we compute the first upper boundb(1)p = C(2) · Λ(2)

p · ‖Z‖H2ρ/‖Z‖ρ. The second part of this bound,

i.e. ‖Z‖H2ρ/‖Z‖ρ does not depend on the polynomial degree

and is calculated only once for each frequency. Recalling thedefinitions (25) and (26), we use the global adaptive Gaussianquadrature method in Matlab [23] combined with a centralfinite difference method with a fixed meshsize to compute theintegrals ∫

R

∣∣∣Z(k)(x)∣∣∣2 ρ(x) dx, k ∈ 0, 1, 2.

The first polynomial degree dependent part of the bound

i.e. C(2) · Λ(2)p is computed analytically, without mentionable

numerical errors or computational cost. To obtain Pδ(b(1)p )

according to the definition (18) we increase the polynomialdegree p beginning with p = 0 and calculate b

(1)p until we

reach the minimal p with b(1)p < δ. In the third and final step

we use the maximal convergence quotients from Table II tocalculate the estimated polynomial degrees according to (23).

Figure 8 shows the obtained estimated polynomial degreestogether with the required polynomial degrees Pδ(ep). Notethat Pδ(ep) is defined by (17) for the analytical PCE approx-imation error ep. Here we use the global adaptive Gaussianquadrature method in Matlab [23] to obtain the error epaccording to the definition (2), which results in an additionalnumerical error. This error has an influence on the requiredpolynomial degrees Pδ(ep) which are computed according to

10

TABLE IIMAXIMAL CONVERGENCE QUOTIENTS λ

(1)δ (SEE (19)) FOR THE

UNIFORMLY AND GAUSSIAN DISTRIBUTED RANDOM INPUT (MODELED BYA RANDOM VARIABLE X ) WITH A STANDARD DEVIATION σ = 0.0625.

THE MAXIMUM IS TAKEN OVER ALL RELATIVE FREQUENCIES ω THAT AREBOUNDED BY ω1 ≤ |ω| ≤ ω2 FOR DIFFERENT VARIATIONS OF THEVALUES ω1, ω2 ∈ [−20, 20]. THE CONVERGENCE QUOTIENTS ARE

COMPUTED FOR δ = 0.01 AND δ = 0.05 AND DIFFERENT VARIATIONS OFTHE QUALITY FACTOR Q.

X ∼ U(−1, 1) X ∼ N (0, 1)

Q |ω| ∈[ω1,ω2] δ = 0.05 δ = 0.01 δ = 0.05 δ = 0.01

6 [0, 5] 0.33 0.21 0.10 0.0410 [0, 5] 0.33 0.25 0.15 0.0720 [0, 5] 0.40 0.33 0.32 0.1540 [0, 5] 0.47 0.3450 [0, 5] 0.46 0.3360 [0, 5] 0.45 0.3170 [0, 5] 0.44 0.2780 [0, 5] 0.45 0.23

6 [5, 10] 0.17 0.21 0.05 0.0410 [5, 10] 0.33 0.25 0.15 0.0620 [5, 10] 0.33 0.24 0.30 0.1540 [5, 10] 0.29 0.20 0.39 0.1650 [5, 10] 0.29 0.19 0.3760 [5, 10] 0.30 0.19 0.3770 [5, 10] 0.29 0.19 0.3580 [5, 10] 0.29 0.20 0.34

6 [10, 15] 0.17 0.14 0.05 0.0410 [10, 15] 0.33 0.14 0.10 0.0520 [10, 15] 0.33 0.20 0.22 0.1340 [10, 15] 0.33 0.20 0.37 0.1650 [10, 15] 0.33 0.20 0.37 0.1660 [10, 15] 0.33 0.20 0.35 0.1670 [10, 15] 0.33 0.20 0.32 0.1780 [10, 15] 0.33 0.20 0.29 0.15

6 [15, 20] 0.17 0.14 0.05 0.0210 [15, 20] 0.17 0.14 0.10 0.0320 [15, 20] 0.33 0.20 0.14 0.0740 [15, 20] 0.33 0.20 0.18 0.1650 [15, 20] 0.33 0.20 0.26 0.1760 [15, 20] 0.33 0.20 0.24 0.1770 [15, 20] 0.33 0.20 0.28 0.1780 [15, 20] 0.33 0.20 0.29 0.15

the definition (17). In the case of the uniformly distributedpitch the required polynomial degree is well approximated forangular frequencies that have a relative resonance distance|ω| ≤ 5 %. For each of these frequencies the estimatedPδ(ep) provides a good upper bound for the required Pδ(ep)or underestimates the required Pδ(ep) by just one degree,whereas a relatively large discrepancy between the two degreesis observed for many others frequencies with |ω| > 5 %.The unbounded support of the Gaussian distribution results inadditional errors. These errors have not been fully taken intoaccount in the determination of the corresponding convergencequotients λ(1)

δ in Section IV-A. The required and estimatedpolynomial degrees for the Gaussian distribution for δ = 0.01,δ = 0.05 are depicted in Figure 8d for Q = 8.7. The dashedred line showing the estimated Pδ(ep) computed with λ(1)

δ ·2.5instead of λ(1)

δ . It appears to be a good upper bound for therequired polynomial degree for all frequencies with |ω| ≤ 5.

V. CONCLUSION AND OUTLOOK

An infinite sequence of quantitative analytical upper boundsfor the PCE approximation error has been derived. Thesebounds can be explicitly determined for every applicationof the PCE with a single random uniformly or Gaussiandistributed input parameter. This result is new and quantitativein contrast to the results describing the asymptotic convergenceof the PCE which can be found in the PCE related literature.The derived upper bounds overestimate the PCE error in mostcases. Nevertheless they can be used as a basis for a futuredevelopment of algorithms for the estimation of the PCEconvergence rate in practical applications.

The derivation of the bounds has been applied in [22]and can be easily adopted to other choices of commonlyused distributions of the random input parameter. To do this,the mathematical analysis from [22] has to be modified (insome parts). In [22] this analysis has been performed for twomost commonly used distributions, the uniform, that has beenselected to represent the class of distributions with a boundedsupport, and the Gaussian that stands for a distribution withan unbounded support. The derivation of upper bounds forother distributions with a bounded support would be similarto the derivation of the bounds for the uniform distribution,whereas the derivation of upper bounds for distributions withan unbounded support would be similar to the derivation inthe Gaussian case.

Most of the practical problems include a large varietyof stochastic model parameters, which can posses differentdistribution densities. The expansion of the quantitative upperbounds for such multivarite PCE applications is feasible. Thegeneralized quantitative upper bounds can be derived throughtenzorization of the one-dimensional analysis, provided thatall input parameters are independently distributed. The caseof dependent stochastic input variables remains a task for thefuture research.

APPENDIXMATHEMATICAL APPENDIX

Square Integrable Functions, Orthogonal Polynomials andSobolev Spaces

Let ρ : R → [0,∞) be measurable such that ρ is strictlypositive on some open interval. Then we will call ρ a weightfunction. Later, this ρ will be the probability density functionof our random input.

Let K denote either R or C, depending on the application.We define

L2(R, ρ) :=f : R→ K : f measurable,∫

R|f(x)|2 ρ(x) dx <∞

.

Then L2(R, ρ) is by definition the factor space of L2(R, ρ)modulo the subspace of functions satisfying∫

R|f(x)|2 ρ(x) dx = 0,

FRICK et al.: QUNTITATIVE ERROR BOUNDS OF POLYNOMIAL CHAOS EXPANSION AND THEIR APPLICATION TO RESONANT SYSTEMS 11

i.e. functions vanishing almost everywhere w.r.t. ρ. The sup-port of ρ is the closure of the subset of R where f is non-zeroi.e.

supp (ρ) = x ∈ R : ρ(x) 6= 0. (24)

The space L2(R, ρ) of square integrable functions turns outto be a Hilbert space for the inner product given by

〈f, g〉ρ :=

∫Rf(x)g(x)ρ(x) dx (f, g ∈ L2(R, ρ))

(we ignore the bar in g(x) if K = R) and the correspondingnorm

‖f‖ρ :=√〈f, f〉ρ (f ∈ L2(R, ρ)). (25)

The Sobolev space Hnρ is then defined by

Hnρ :=

f : R→ K :

∂kf

∂xk∈ L2(R, ρ), 0 ≤ k ≤ n

for all n ∈ N0. It is a normed vector space with the norm

‖f‖Hnρ :=

(n∑k=0

∥∥∥∥∂kf∂xk

∥∥∥∥2

ρ

) 12

. (26)

For n ∈ N0 let mn : R → K be defined by mn(x) :=xn (x ∈ R). Let P := lin

mn; n ∈ N0

denote the

space of K-valued polynomials on R. Let us assume thatP ⊆ L2(R, ρ) and that P is dense in L2(R, ρ). Then theGram-Schmidt-orthogonalisation applied to (mn)n∈N0

yieldsa sequence (ψn)n∈N0

of orthogonal polynomials in L2(R, ρ),i.e. 〈ψl, ψr〉ρ = γlδl,r, where δl,r is the Kronecker delta andγl := ‖ψl‖2ρ. Moreover, (ψn) is actually an orthogonal basisfor L2(R, ρ). This means that every f ∈ L2(R, ρ) has theunique representation

f =∑n∈N0

1

γn〈f, ψn〉ρ ψn =

∑n∈N0

fnψn (27)

as a generalized Fourier series w.r.t. (ψn), where fn :=1γn〈f, ψn〉ρ is the n-th generalized Fourier coefficient w.r.t.

(ψn), and (fn) is the corresponding sequence of generalizedFourier coefficients. By Parseval’s identity, we observe that∑

n∈N0

γn |fn|2 =∑n∈N0

1

γn

∣∣∣〈f, ψn〉ρ∣∣∣2 (28)

= ‖f‖2ρ =

∫R|f(x)|2 ρ(x) dx.

Random Variables, Statistics and Monte Carlo Sampling

Let (Ω,F , P ) be a probability space, i.e. Ω is the sam-ple space, F is a σ-field on Ω containing the events andP : F → [0, 1] is a probability measure. Let X : Ω → R bea random variable (i.e. a measurable mapping) modelling therandom input. Then X gives rise to a push-forward measurePX on R defined by PX(A) := P (X−1(A)) for all Borel setsA ⊆ R, which is called the distribution of X . We are theninterested in the random output f(X) for some measurablefunction f : R→ K.

Let us assume that PX is absolutely continuous with respectto the Lebesgue measure, or, put differently, PX has a density

ρ. As two of may typical examples we will consider theuniform distribution U(−1, 1) on [−1, 1] with the density

ρ(x) :=

0.5, |x| ≤ 1,

0, |x| > 1.(x ∈ R) (29)

and the Gaussian distribution X ∼ N (0, 1) with the density

ρ(x) :=1√2πe−x

2/2 (x ∈ R). (30)

Then the procedure above gives rise to orthogonal polynomials(ψn)n∈N0

yielding an orthogonal basis of L2(R, ρ). For f, g ∈L2(R, ρ) we obtain

〈f, g〉ρ =

∫Rf(x)g(x)ρ(x) dx =

∫Ω

f(X(ω))g(X(ω)) dP (ω)

= E(f(X)g(X)),

where E denotes the mean value w.r.t. P . The variance of fw.r.t. P is defined by

V(f(X)) :=

∫R|f(x)− E(f(X))|2 ρ(x) dx. (31)

It is then easy to see that

V(f(X)) = ‖f‖2ρ − E(f(X))2. (32)

One way to obtain information on f(X) is by Monte Carlo(MC) sampling. Let us denote the variance of f(X) obtainedfrom a MC experiment with N samples by Vmc(N), then therelative variance error is given by

|Vmc(N)− V(f(X))|V(f(X))

.

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