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The Wiener path integral (WPI) technique for determining the stochastic response of diverse nonlinear systems is enhanced herein based on a Bayesian compressive sampling (CS) treatment. Specifically, first, sparse expansions for the system response joint probability density function (PDF) are employed. Next, exploiting the localization capabilities of the WPI technique for direct evaluation of specific PDF points leads to an underdetermined linear system of equations for the expansion coefficients. Further, relying on a Bayesian CS solution formulation yields a posterior distribution for the expansion coefficient vector. In this regard, a significant advantage of the herein developed methodology relates to the fact that the uncertainty of the response PDF estimates obtained by the WPI technique is quantified. Furthermore, an adaptive scheme is proposed based on the quantified uncertainty of the estimates for optimal selection of PDF sample points. This yields considerably fewer boundary value problems to be solved as part of the WPI technique, and thus, the associated computational cost is significantly reduced. Two indicative numerical examples pertaining to a Duffing nonlinear oscillator and to an oscillator with asymmetric nonlinearities are considered for demonstrating the capabilities of the developed technique. Comparisons with pertinent Monte Carlo simulation data are included as well.
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Uncertainty quantification of nonlinear system
stochastic response estimates based on the Wiener path
integral technique: A Bayesian compressive sampling
treatment
Maria I. Katsidoniotakia, Apostolos F. Psarosb, Ioannis A. Kougioumtzogloua,
aDepartment of Civil Engineering and Engineering Mechanics, Columbia University, USA
bDivision of Applied Mathematics, Brown University, USA
Abstract
The Wiener path integral (WPI) technique for determining the stochastic re-
sponse of diverse nonlinear systems is enhanced herein based on a Bayesian
compressive sampling (CS) treatment. Specifically, first, sparse expansions for
the system response joint probability density function (PDF) are employed.
Next, exploiting the localization capabilities of the WPI technique for direct
evaluation of specific PDF points leads to an underdetermined linear system
of equations for the expansion coefficients. Further, relying on a Bayesian CS
solution formulation yields a posterior distribution for the expansion coefficient
vector. In this regard, a significant advantage of the herein developed method-
ology relates to the fact that the uncertainty of the response PDF estimates
obtained by the WPI technique is quantified. Furthermore, an adaptive scheme
is proposed based on the quantified uncertainty of the estimates for optimal
selection of PDF sample points. This yields considerably fewer boundary value
problems to be solved as part of the WPI technique, and thus, the associated
computational cost is significantly reduced. Two indicative numerical examples
pertaining to a Duffing nonlinear oscillator and to an oscillator with asym-
metric nonlinearities are considered for demonstrating the capabilities of the
Corresponding author
Email address: ikougioum@columbia.edu (Ioannis A. Kougioumtzoglou)
Preprint submitted to Journal of L
A
T
E
X Templates December 7, 2021
developed technique. Comparisons with pertinent Monte Carlo simulation data
are included as well.
Keywords: Path integral, Nonlinear system, Stochastic dynamics, Sparse
representations, Compressive sampling
1. Introduction
Developing mathematical techniques for treating the problem of uncertainty
propagation in the field of stochastic engineering dynamics has been a persis-
tent challenge for more than six decades. In fact, complex nonlinear behaviors,
high-dimensionality, and sophisticated excitation modeling require novel potent5
tools for solving the governing equations of motion accurately and in a compu-
tationally efficient manner. In this regard, diverse solution methodologies have
been developed for determining system response statistics with varying degrees
of success; see, indicatively, [1–3] for a broad perspective.
One of the promising stochastic engineering dynamics techniques, recently10
developed by Kougioumtzoglou and co-workers, pertains to the notion of path
integral. From a mathematics point of view, the path integral concept refers to
the generalization of integral calculus to functionals. It was first introduced by
Wiener [4] (see also preliminary work by Daniell [5]), and was reinvented in a
different form by Feynman [6] leading eventually to a reformulation of quantum15
mechanics [7]. In the field of stochastic engineering dynamics, the Wiener path
integral (WPI) technique has demonstrated a high degree of accuracy [8], is
capable of handling diverse system and excitation modeling [9–14], and can
readily treat high-dimensional systems at a relatively low computational cost
[15].20
Further, the computational efficiency of the WPI technique was enhanced
recently. This was done by resorting to compressive sampling (CS) concepts and
tools in conjunction with appropriate expansions for the joint response prob-
ability density function (PDF) [16, 17]; see also [18] for a broad perspective.
2
Specifically, first, an appropriately selected basis was considered for expanding25
the joint response PDF by utilizing only a few nonzero terms. Next, a relatively
small number of PDF points were determined directly by relying on the local-
ization capabilities of the WPI technique. In this regard, an underdetermined
linear system of equations was formulated for the sparse expansion coefficient
vector. Furthermore, CS procedures in conjunction with group sparsity concepts30
and appropriate optimization algorithms were employed for solving efficiently
the underdetermined system of equations and computing the coefficients of the
joint response PDF expansion.
In this paper, the WPI technique is extended based on a Bayesian CS treat-
ment (e.g., [19–22]). In particular, compared to the deterministic coefficient35
vector estimate obtained in the standard CS framework, Bayesian CS yields a
posterior distribution for the expansion coefficient vector. Clearly, this provides
a tool for uncertainty quantification associated with the estimated system re-
sponse PDF. Moreover, this additional information regarding the estimates is
exploited herein for reducing further the computational cost associated with the40
WPI technique. Indeed, based on the quantified uncertainty of the estimates,
an adaptive scheme is developed for optimal selection of PDF sample points;
thus, yielding fewer boundary value problems (BVPs) to be solved as part of
the WPI technique. Two indicative numerical examples pertaining to a Duffing
nonlinear oscillator and to a nonlinear oscillator with an asymmetric response45
PDF are considered for demonstrating the capabilities of the developed tech-
nique. Comparisons with pertinent Monte Carlo simulation (MCS) data are
included as well.
2. Wiener path integral technique: selected aspects
2.1. Wiener path integral solution formulation50
The dynamics of a q-degree-of-freedom (q-DOF) nonlinear system subject to
external stochastic excitation is governed by the equation
M¨
x+C˙
x+Kx +g(˙
x,x, t) = w(t) (1)
3
where x(t)=[x1(t), ..., xq(t)]Tis the response displacement vector and M,
C,Kdenote the mass, damping and stiffness matrices, respectively. Also,
g(˙
x,x, t) represents an arbitrary nonlinear function. The excitation w(t) =55
[w1(t), ..., wq(t)]Tis modeled next, for notation simplicity, as a white noise
vector process with E[w(tl)] = E[w(tl+1)] = 0 and Ew(tl)wT(tl+1)=
Swδ(tl+1 tl), where tl, tl+1 are two arbitrary time instants and SwRq×q
+
denotes a constant power spectrum matrix. Note that alternative, more com-
plex, excitation modeling as a non-white and non-Gaussian stochastic vector60
process can be also readily accounted for by the WPI technique; see [11] for
more details.
As shown in [11, 23] (see also [7] for a broader perspective), the joint response
transition PDF p(xf,˙
xf, tf|xi,˙
xi, ti) corresponding to the system of Eq. (1) can
be expressed as a functional integral in the form65
p(xf,˙
xf, tf|xi,˙
xi, ti) = Z
C
exp(Ztf
ti
L(x,˙
x,¨
x) dt)[dx(t)] (2)
where C={xi,˙
xi, ti;xf,˙
xf, tf}is the set of all possible paths with initial con-
dition {xi,˙
xi, ti}and final condition {xf,˙
xf, tf}, dx(t) denotes a functional
measure, and Lrepresents the Lagrangian of the system given by
L(x,˙
x,¨
x) = 1
2{M¨
x+C˙
x+Kx+g(˙
x,x, t)}TS1
w{M¨
x+C˙
x+Kx+g(˙
x,x, t)}(3)
Due to considerable difficulties in evaluating analytically the functional in-
tegral in Eq. (2), researchers resort routinely to approximate schemes involving70
the concept of the “most probable path” [7]. This is the trajectory xc(t) for
which the integral of the Lagrangian Rtf
tiL(x,˙
x,¨
x) dt, known also as stochastic
action, is minimized. This leads to the Euler-Lagrange equations
Lxjd
dtL˙xj+d2
dt2L¨xj= 0, j ∈ {1, ..., q}(4)
subject to 4 ×qboundary conditions
xj(ti) = xj,i,˙
xj(ti) = ˙
xj,i,xj(tf) = xj,f and ˙
xj(tf) = ˙
xj,f (5)
4
Eqs. (4) and (5) constitute a BVP to be solved for determining the most75
probable path xc(t). Following solution of Eq. (4) and substituting xc(t) into
Eq. (2), a specific point of the joint response transition PDF is determined
approximately as
p(xf,˙
xf, tf|xi,˙
xi, ti)C exp(Ztf
ti
L(xc,˙
xc,¨
xc) dt) (6)
where C is a normalization constant. Although it is clear by comparing Eq. (2)
and (6) that only one trajectory (i.e., the most probable path xc(t)) is ac-80
counted for in the evaluation of the WPI, it has been shown in various diverse
applications (e.g., [9–14]) that the accuracy degree exhibited by this kind of
approximation is relatively high. In fact, as proved in [24], for the case of lin-
ear multi-DOF systems the most probable path approximation yields the exact
joint response PDF. Further, note that the accuracy degree of the WPI tech-85
nique has been enhanced recently by considering a quadratic approximation to
account also for fluctuations around the most probable path; see [8] for details.
Clearly, in the general case, the Euler-Lagrange Eqs. (4) and (5) are not
amenable to an analytic solution treatment, and therefore, numerical schemes
are required. In this regard, adopting a brute-force solution approach, for a90
specific time instant tfthe values of the joint response PDF are computed
based on Eq. (6) over a discretized PDF domain of Npoints in each dimension.
This yields N2qBVPs to be solved for a q-DOF system.
2.2. Joint response PDF sparse representations
It can be readily seen that, by relying on the aforementioned brute-force so-95
lution approach, the WPI technique becomes computationally prohibitive with
increasing number of dimensions. This is due to the fact that the number N2q
of BVPs to be solved increases exponentially with respect to q. To address this
challenge, a polynomial expansion of the log-PDF was employed in [25] of the
form100
log(p(x,˙
x))
n
X
i=1
cibi(x,˙
x) (7)
5
where ciare the polynomial expansion coefficients to be determined and bi
are the basis functions, with i∈ {1, ..., n}. Further, following the selection of
nlocations to perform the approximation, Eq. (7) takes the form of a linear
system of nequations, i.e.,
y0=Bc (8)
where y0Rn×1is a vector of nsamples of log(p(x,˙
x)), BRn×nis the105
basis matrix and c= [c1, ..., cn]TRn×1is the expansion coefficient vector.
Therefore, in comparison to the brute-force approach, which requires N2qBVPs
to be solved numerically, the approach based on the PDF representation of
Eq. (7) requires the solution of only n=(p+2q)!
(2q)!p!BVPs, where pis the order of
the polynomial approximation. For the vast majority of applications, n << N2q,110
and thus, the gain in terms of computational efficiency is significant.
In [16], it was shown that the computational efficiency of the WPI technique
can be further enhanced by resorting to sparse representations and compressive
sampling concepts and tools; see also [18] for a broad perspective. Specifically, it
was shown that for a large class of nonlinear systems the joint PDF expansion of115
Eq. (7) exhibits group sparsity, and thus, only m<nPDF samples are required
via the WPI technique. In this regard, Eq. (8) takes the form
y=Dy0=DBc =Φc (9)
where yRm×1is a vector of m<nsamples of log(p(x,˙
x)) and the matrix D
Rm×ndeletes rows randomly from the polynomial basis B. The product DB
yields the matrix ΦRm×nand cRn×1is the sparse polynomial coefficient120
vector to be determined. Eq. (9) constitutes an underdetermined linear system
of equations, which can be solved based on an l1-norm minimization formulation.
This leads to an unconstrained minimization problem in the form
ˆ
c=arg minc{kyΦck2
2+λkck1}(10)
where λis a hyperparameter. Solving the problem of Eq. (10) yields the sparse
coefficient vector to be substituted into the expansion of Eq. (9) for approxi-125
6
mating the system joint response PDF. The interested reader is also directed to
[16–18] for a more detailed presentation.
3. A Bayesian compressive sampling solution treatment
In this section, the WPI technique is enhanced by developing a solution
framework based on Bayesian CS (e.g., [18, 19]). Specifically, a significant ad-130
vantage of the herein developed technique relates to the fact that not only an
estimate is obtained for the coefficient vector cin Eq. (9), but also the uncer-
tainty of this estimate is quantified. Further, this capability for uncertainty
quantification of the system response PDF estimates motivates in this section
also the development of an adaptive scheme for optimal selection of PDF sample135
points. In this regard, the associated computational cost can be significantly
reduced. In fact, it is shown that fewer BVPs need to be solved as part of the
WPI technique without compromising, in general, the exhibited accuracy.
3.1. Bayesian modeling
Considering noise in the samples vector y, Eq. (9) is expressed as140
y=Φc +e(11)
where the components of the noise vector eRm×1are modeled as indepen-
dent and identically distributed Gaussian random variables with zero mean and
unknown variance σ2
e. Equivalently, yRm×1follows a Gaussian likelihood
model given by
p(y|c, σ2
e) = (2πσ2
e)m
2exp(kyΦck2
2
2σ2
e
) (12)
Next, the objective of the Bayesian solution treatment relates to obtaining145
the coefficient vector cbased on the available samples vector y. To this aim,
first, a sparsity-promoting prior distribution is selected for c. In this regard,
employing a hierarchical modeling approach (e.g., [26]), a Gaussian multivariate
prior is considered for the coefficients cin the form
p(c|σ2
c) =
n
Y
i=1
N(ci|0, σ2
ci) (13)
7
where σ2
c={σ2
ci}n
i=1 is a hyperparameter vector with nindependent weights.150
Note that modeling approaches based on such Gaussian hierarchical priors are
typically referred to as Gaussian automatic relevance determinators (ARD) in
the Bayesian literature (e.g., [27]). ARDs adopt the following rationale for
identifying the “relevant” parameters ci. During the tuning process of σ2
c(see
sections 3.2-3.3), if a non-zero value is obtained for the unknown hyperparameter155
σ2
ci, then the corresponding parameter ciis considered active in the approxima-
tion of y, whereas if σ2
ci 0, cibecomes inactive.
Further, hyperpriors are introduced for the hyperparmeters σ2
cand σ2
e.
Specifically, the inverse Gamma hyperprior constitutes a convenient choice for
p(σ2
c) from an analytical treatment perspective, since it is the conjugate prior of160
the Gaussian distributions in Eq. (13). Consequently, the convolution integral
p(c) = Zp(c|σ2
c)p(σ2
c)dσ2
c(14)
leads to a sparsity-promoting Student-tprior distribution for c. Alternatively,
the hyperpriors can become non-informative by setting inverse uniform distribu-
tions as the priors of σ2
cand σ2
e, i.e., p(σ2
ci ), p(σ2
e)1. In this case, Eq. (14)
yields165
p(c)
n
Y
i=1
1
|ci|(15)
In fact, selecting uniform hyperpriors for σ2
cand σ2
enot only simplifies the
computations in the ensuing analysis, but also renders the predictions scale-
invariant; i.e., independent of the measurement units of the data. Furthermore,
the distribution of the coefficient vector p(ci)1
|ci|exhibits a sharp peak at the
origin, similarly to the Laplace prior p(ci)exp(−|ci|) that is widely used as a170
sparse prior in the literature [28]. Therefore, although p(c|σ2
c) is Gaussian, and
thus, relatively less sparsity-promoting, the resulting p(c) following integration
over the hyperparamaters via Eq. (14) exhibits significant sparsity-promoting
behavior as shown in Fig. 1.
8
(a) Eq. (15): p(c1, c2)Q2
i=1
1
|ci|(b) Laplace: p(c1, c2)Q2
i=1 exp(−|ci|)
(c) Gaussian: p(c1, c2) = Q2
i=1 N(ci|0,1) (d) Comparison for p= 0.01
Figure 1: Comparison between (a) prior of Eq. (15) with uniform hyperprior (b) Laplace prior
(c) Gaussian distribution and (d) their contours for p= 0.01.
3.2. Posterior inference175
In this section, the posterior distribution p(c|y) defined as
p(c|y) = Zp(c|y,σ2
c, σ2
e)p(σ2
c, σ2
e|y)dσ2
cdσ2
e(16)
is obtained analytically. Applying Bayes’ theorem, p(c|y,σ2
c, σ2
e) in Eq. (16) is
expressed as
p(c|y,σ2
c, σ2
e) = p(y|c, σ2
e)p(c|σ2
c)
p(y|σ2
c, σ2
e)(17)
where p(y|σ2
c, σ2
e) is the marginal likelihood of y. Obviously, once the distribu-
tion p(c|y) is known, the evaluation of p(y0|y) is straightforward via the linear180
relationship of Eq. (8). Thus, the uncertainty of the WPI-based joint response
PDF estimate can be quantified.
Next, according to the Laplace asymptotic approximation [29], p(c|y)
p(c|y,ˆσ2
c,ˆσ2
e), where ˆσ2
c,ˆσ2
eare obtained by maximizing p(σ2
c, σ2
e|y). Neverthe-
less, due to the proportionality relationship p(σ2
c, σ2
e|y)p(y|σ2
c, σ2
e)p(σ2
c)p(σ2
e)185
9
and the uniform distribution modeling of p(σ2
c) and p(σ2
e), this is equivalent to
maximizing p(y|σ2
c, σ2
e). This scheme is referred to in the literature as type-II
maximum likelihood estimation (MLE).
Further, considering Eq. (17), the posterior distribution p(c|y,σ2
c, σ2
e) is eval-
uated analytically and takes the form of a multivariate Gaussian distribution,190
i.e., p(c|y,σ2
c, σ2
e) = N(µ, Σ), where the mean vector and covariance matrix are
expressed as [28]
µ=σ2
eΣΦTy and Σ= (σ2
eΦTΦ+A)1(18)
respectively. In Eq. (18), σ2
eand A=I·σ2
c=diag{σ2
c1, ..., σ2
cn } ∈ Rn×n
are the hyperparameters to be evaluated.
Furthermore, due to the linear relationship between cand y0in Eq. (8), the195
distribution of y0is also Gaussian given by [19]
p(y0|y,σ2
c, σ2
e) = N(Bµ,BΣBT)(19)
Equivalently, the distribution of the WPI-based estimate p(x,˙
x) at an arbitrary
point (x,˙
x) is log-normal based on the relation y0=log(p(x,˙
x)) with
E{p(x,˙
x)}=exp(Bµ+1
2diag(BΣBT)) (20)
V ar{p(x,˙
x)}=exp(2Bµ+diag(BΣBT))(exp(diag(BΣBT)) 1).(21)
Obviously, the uncertainty of the response PDF estimate obtained by the200
WPI technique is quantified herein based on the log-normal distribution de-
scribed by Eqs. (20) and (21).
3.3. Performance assessment of selected relevance vector machine (RVM) algo-
rithms
Considering Eqs. (12) and (13), a type-II MLE scheme for obtaining the opti-205
mal values ˆσ2
c,ˆσ2
epertains to maximizing the logarithm of p(y|σ2
c, σ2
e) expressed
as
log p(y|σ2
c, σ2
e) = 1
2[m·log2π+log|C|+yTC1y] (22)
10
where C=σ2I+ΦA1ΦT. This hyperparameter learning problem can be
solved iteratively by employing a relevance vector machine (RVM) scheme [28,
30]. Specifically, a relatively standard implementation dictates that the values210
σ2
cand σ2
eare obtained at each step as
σ2
ci(new)=µ2
i
γi
and σ2
e(new)=kyΦcsk2
mΣiγi
(23)
where i∈ {1,2, ..., n}, and γi= 1Σii2
ci [0,1] with Σii being the variance of
the ith element. Notably, the set of coefficients cicorresponding to large values
of σ2
ci are the ones contributing to the representation of ythe most, whereas
small values of σ2
ci correspond to rather inactive coefficients [31].215
Further, the Fast RVM scheme [32] represents an enhancement of the stan-
dard RVM that is computationally more efficient and promotes sparser, in gen-
eral, solutions. According to Fast RVM, the expression of p(y|σ2
c, σ2
e) in Eq. (22)
and the matrix Care related to a single hyperparameter σ2
ci as
log p(y|σ2
c, σ2
e) = log p(y|σ2
ci, σ2
e)+ 1
2[log σ2
ci log(σ2
ci +si)+ q2
i
σ2
ci +si
] (24)
where qi=φT
iC1
iyand si=φT
iC1
iφi. The matrix Cis given by C=220
Ci+σ2
ciφiφT
i, implying that the ith row is neglected in Ci. Furthermore, the
maximization of Eq. (24) according to [33] yields the expressions
σ2
ci =
q2
isi
s2
i
,if q2
i> si,
0,if q2
isi
(25)
Based on the value of σ2
ci, the basis function φiis added to or removed from the
basis matrix Φ. Note that the Fast RVM yields msignificant basis functions
φito be accounted for in Eq. (18) compared to the nfunctions φiused in the225
standard RVM. A more detailed presentation of the scheme can be found in
[32].
Next, to assess the performance of the two RVM schemes, an experiment
is conducted involving reconstruction of 100 synthetically generated coefficient
vectors c. In Fig. 2, the average relative errors of the RVM and the Fast RVM230
schemes are shown. Note that the error is defined as ||ˆ
cc||2/||ˆ
c||2, where ˆ
c
11
Figure 2: Average relative error of coefficient vector cusing RVM (top) and Fast RVM
(bottom). The white line indicates 5% relative error.
is the reference coefficient vector to be reconstructed and cis the mean of the
Bayesian posterior. Comparisons are made for various values of samples mand
sparsity k, for a given number of coefficients n. Also, the basis matrix Φis
randomly generated and a zero-mean Gaussian noise with standard deviation235
σe= 0.005 is added to each of the msamples.
Remarkably, for an indicative sparsity ratio of k/n = 0.2 (a value commonly
used in problems considered herein), utilizing a relatively low sampling ratio of
m/n 0.6 yields a reconstruction error smaller than 5% for both schemes.
3.4. Optimal selection of sample points240
According to Section 2.2, the samples vector yin Eq. (9) can be obtained
based on a random selection of points over the effective domain of the system
12
response log-PDF. However, such an approach can be further optimized based
on a judicious selection of the samples.
In this regard, a scheme is proposed in this section for optimal selection of245
points by minimizing the differential entropy of the total samples vector y0[19].
Specifically, the differential entropy of y0when adding a new point in ytakes
the form
hnew(y0) = h(y0)1
2log(1 + (φT
m+1Σφm+1 )2
e) (26)
where φm+1 is the new row added to the matrix Φof Eq. (9). The minimization
of Eq. (26) is equivalent to maximizing the variance of the new sample vector250
ym+1 based on equation
φT
m+1Σφm+1 =φT
m+1Cov(c)φm+1 =V ar(ym+1) (27)
In practice, this is achieved by adding to the previous sample vector ymthe
point of the log-PDF y0with the largest variance, or equivalently, the point of
the PDF with the largest relative variance (σ/µ)2. Indeed, considering Eqs.
(20) and (21), as well as Eq. (19), yields255
V ar{p(¯
x,˙
¯
x)}
E{p(¯
x,˙
¯
x)}2=exp(diag(BΣBT)) 1 = exp(V ar(y0)) 1 (28)
The performance of the proposed scheme for optimal selection of sample
points is assessed in the numerical examples of Section 4, where it is shown that
the same relative error of the reconstructed coefficient vector ccan be obtained
by using fewer sample points compared to the standard implementation.
3.5. Mechanization of the technique260
The mechanization of the developed WPI technique based on Bayesian CS
comprises the following steps:
(a) The expansion basis matrix Bin Eq. (8) is constructed, which is associ-
ated with an n-dimensional coefficient vector cto be determined.
(b) The number m<nof joint response PDF sample points to be used265
in the underdetermined linear system of Eq. (9) is selected. To this aim, the
13
results shown in Fig. 2 and pertaining to the RVM schemes presented in section
3.3 can serve as a guide.
(c) The locations of the msamples in Eq. (9) over the effective domain of the
system response log-PDF are selected either randomly, or based on the optimal270
sampling scheme proposed in section 3.4.
(d) The posterior distribution of the coefficient vector cis Gaussian with a
mean vector and covariance matrix given by Eq. (18). These are determined by
employing, for instance, the RVM schemes presented in section 3.3 in conjunc-
tion with the optimal sampling scheme in section 3.4.275
(e) The distribution of the WPI-based estimate p(x,˙
x) at an arbitrary point
(x,˙
x) is log-normal with a mean and a variance given by Eqs. (20) and (21),
respectively.
4. Numerical examples
4.1. Duffing nonlinear oscillator280
Consider a stochastically excited single-DOF Duffing nonlinear oscillator,
whose equation of motion is given by
m¨x+c˙x+kx(1 + x2) = w(t) (29)
where m= 1, k= 1, c= 0.1, = 0.1, and w(t) is a white noise excitation with
a constant power spectrum value S0= 0.0637. Note that the exact stationary
joint response log-PDF for this oscillator has the closed-form expression (e.g.,285
[34])
yexact
0=log(p(x, ˙x)) = c
πS0m(kx2
2m+kx4
4m+˙x2
2) + C (30)
Clearly, the expression in Eq. (30) represents a 4th order polynomial with 3
non-zero coefficients, and C is a normalization constant. In this regard, selecting
a 4th order polynomial (p= 4) as an approximating basis in Eq. (8) yields
n= 15, whereas based on the exact solution of Eq. (30), the sparsity in the290
stationary phase is k= 3.
14
Figure 3: Assessing the performance of various RVM schemes for determining the joint re-
sponse PDF of a Duffing nonlinear oscillator at an arbitrarily chosen time instant t= 20s.
Average relative error (top) and average variance (bottom) of the Bayesian estimates for
various sample ratios m/n.
Next, the two RVM schemes discussed in section 3.3, as well as their im-
plementations based on the optimal selection of points presented in section 3.4,
are compared in conjunction with the Duffing nonlinear oscillator of Eq. (29).
Specifically, the relative error ||ˆ
y0y0||2/||ˆ
y0||2and the variance of y0at an295
arbitrarily chosen time instant t= 20s(both averaged over 1000 trials) are
plotted in Fig. 3. ˆ
y0denotes the estimate obtained by the standard brute-force
WPI formulation presented in section 2.1, whereas y0denotes the mean of the
Bayesian estimate in Eq. (19). It is readily seen that the Fast RVM coupled with
the optimal sampling scheme of section 3.4 yields the smallest error and exhibits300
the lowest uncertainty degree compared to the other alternatives, particularly
15
Figure 4: Uncertainty quantification of Duffing nonlinear oscillator joint response PDF ex-
pansion coefficient vectors at t= 1sand t= 20s. Comparisons between the standard WPI
technique with a polynomial PDF approximation (deterministic estimates) and the Bayesian
WPI formulation (estimates of the coefficient vector distribution based on Eq. (18)). The
exact coefficient values based on Eq. (30) are also included for the stationary phase (t= 20s).
for smaller values of the sampling ratio m/n.
In the following, the Fast RVM with the optimal sampling scheme is em-
ployed and the value m/n = 0.6 is utilized. This translates into using m= 9
points for computing the n= 15-dimensional coefficient vector c, and thus, for305
determining the log-PDF y0. In this regard, the computed coefficient vector
values ci, with i= 1, ..., 15, are shown in Fig. 4. Specifically, the determinis-
tic estimates obtained by the standard WPI technique with a polynomial PDF
approximation are compared with the estimated distributions obtained by the
herein developed Bayesian framework. The target (exact) expansion coefficient310
16
Figure 5: Joint response PDF of a Duffing nonlinear oscillator at t= 1s(left) and t= 20s
(right); standard brute-force WPI formulation (top), and Bayesian formulation - mean values
estimates (bottom).
values based on Eq. (30) are also included for the stationary phase (t= 20s). In
general, it is seen that the mean values of the Bayesian estimates compare well
with the respective deterministic estimates, both for the non-stationary (t= 1s)
and the stationary (t= 20s) phases.
Further, the joint response PDFs at t= 1sand t= 20sobtained by a stan-315
dard brute-force implementation of the WPI are shown in Fig. 5 (top). This
refers to a discretized PDF domain of N2= 2601 points. Furthermore, the log-
17
Figure 6: Uncertainty quantification of the Duffing nonlinear oscillator joint response PDF
estimates based on the proposed Bayesian WPI formulation for t= 1s(left) and t= 20s
(right).
normal distribution of the joint response PDF estimate at an arbitrary point is
obtained by utilizing Eqs. (20) and (21). In Fig. 5 (bottom), the mean values
of the log-normal distribution are plotted demonstrating excellent agreement320
with the deterministic estimates in Fig. 5 (top). Moreover, Fig. 6 shows the
log-normal distributions corresponding to each and every point of the joint re-
sponse PDF domain. It becomes clear that a significant advantage of the herein
developed technique relates to its capability of quantifying the uncertainty of
the response PDF estimates.325
Next, the response PDF estimates at two indicative time instants t= 1s
and t= 20s(stationary phase), obtained by the standard WPI technique, are
shown in Fig. 7 for the displacement xand in Fig. 8 for the velocity ˙x. These
are compared both with pertinent MCS-based estimates (20,000 realizations)
and with the closed-form exact stationary PDF of Eq. (30) for t= 20s. Also,330
the mean values of the Bayesian estimates are included as well. Clearly, these
are in good agreement with the respective deterministic WPI-based estimates.
Interestingly, it is observed that the variance, and thus the uncertainty of the
joint response PDF estimates, is larger for t= 20scompared to t= 1s. This can
18
Figure 7: WPI-based estimates and uncertainty quantification of a Duffing nonlinear oscillator
response displacement PDF at t= 1s(top) and at t= 20s- stationary phase (bottom).
Comparisons with Monte Carlo simulations (20,000 realizations) and the exact solution for
the stationary phase.
be attributed, at least partly, to the larger time interval considered in the BVP335
of Eq. (4). In other words, it is anticipated that the uncertainty of the PDF
estimate increases for larger values of the final time instant tf, or equivalently,
for a final configuration less correlated with the known deterministic initial state
(xi,˙xi) at ti. Further, the uncertainty of the estimates depends, obviously, also
on the amount of missing data. This is seen in Fig. 9, where the relative standard340
deviation σ/µ of the estimates obtained by the Bayesian WPI decreases with
increasing sample ratio m/n. Furthermore, the superior performance of the Fast
RVM coupled with the optimal sampling scheme of section 3.4 compared to the
standard Fast RVM is demonstrated in Fig. 9. In fact, the former appears to
19
Figure 8: WPI-based estimates and uncertainty quantification of a Duffing nonlinear oscil-
lator response velocity PDF at t= 1s(top) and at t= 20s- stationary phase (bottom).
Comparisons with Monte Carlo simulations (20,000 realizations) and the exact solution for
the stationary phase.
converge faster to smaller values of relative variance than the latter for increasing345
m/n. Note that this is in agreement with similar findings observed in Fig. 3.
Moreover, in Table 1, the performance of the various WPI technique formu-
lations is assessed in terms of accuracy based on comparisons with MCS data
(20,000 realizations). In this regard, l2-norm errors of the estimated joint re-
sponse PDFs are reported. Specifically, it is seen that the Bayesian formulation350
with m/n = 1 exhibits practically the same high accuracy degree as the brute-
force implementation with N2= 2601 points. Further, the Bayesian formulation
with m/n = 0.6 exhibits a robust behavior with relatively constant error val-
ues with increasing time instants. The largest error values are reported for the
20
standard WPI formulation coupled with a polynomial PDF approximation. This355
can be attributed, at least partly, to the fact that the resulting linear system of
equations becomes ill-conditioned in many cases, particularly for larger time in-
stants. Notably, the Bayesian WPI formulation appears capable of meliorating
this effect to a certain extent.
Figure 9: Reduction of the relative variance σ/µ of the Bayesian WPI estimates with increasing
sample ratio m/n. The results refer to the joint response PDF of a Duffing nonlinear oscillator
at t= 20s. Comparisons between the standard Fast RVM (top) and the Fast RVM coupled
with the proposed optimal sampling scheme of section 3.4 (bottom).
21
Relative error (%) of WPI
compared to MCS t = 1s t = 10s t = 15s t = 20s
Brute-force implementation
(N2points) 5.3 6.9 6.9 7.6
Polynomial approximation
(Bayesian CS with m/n = 0.6) 8.4 8.4 8.6 8.8
Polynomial approximation
(Bayesian CS with m/n = 1) 5.5 6.6 6.5 7.3
Polynomial approximation
(Standard linear system of equations
with m/n = 0.6) 5.4 11.5 12.3 11.3
Table 1: L2-norm errors of Duffing nonlinear oscillator joint response PDF estimates compared
to MCS data (20,000 realizations). Results refer to various WPI technique formulations and
correspond to indicative time instants.
4.2. Oscillator with asymmetric nonlinearities360
Consider next a single-DOF oscillator with asymmetric nonlinearities, whose
equation of motion is given by
m¨x+c˙x+kx(1 + 0x) = w(t) (31)
where m= 1, k= 1, c= 0.2, 0= 0.5 and w(t) is a white noise excitation
with a constant power spectrum value S0= 0.0637. Further, it can be argued
that a polynomial expansion used for approximating the response PDF of the365
oscillator in Eq. (31) is anticipated to be less sparse than the respective one
used for the Duffing oscillator in section 4.1. This is due to the fact that the
form of the nonlinearity function in Eq. (31) produces a response PDF that is
asymmetric, and thus, an increased number of non-zero terms in the polynomial
expansion is required for approximating the PDF shape accurately. Based on370
this observation, higher-order polynomial bases are employed next. In particu-
lar, a 6th order polynomial (p= 6) with n= 28 is utilized in the following for
the arbitrarily selected time instants t= 1sand t= 2s, whereas a 12th order
22
Figure 10: Assessing the performance of various RVM schemes for determining the joint
response PDF of an oscillator with asymmetric nonlinearities at an arbitrarily chosen time
instant t= 2s. Average relative error (top) and average variance (bottom) of the Bayesian
estimates for various sample ratios m/n.
polynomial (p= 12) with n= 91 is used for t= 3s.
Next, the performance of the various RVM schemes described in sections375
3.3 and 3.4 is assessed in conjunction with the nonlinear oscillator of Eq. (31).
Specifically, the relative error ||ˆ
y0y0||2/||ˆ
y0||2and the variance of y0at an
indicative time instant t= 2s(both averaged over 1000 trials) are plotted in
Fig. 10. ˆ
y0denotes the estimate obtained by the standard brute-force WPI
technique, whereas y0represents the mean of the Bayesian estimate in Eq. (19).380
Obviously, the RVM schemes coupled with the optimal sampling scheme yield
smaller errors and exhibit a lower uncertainty degree compared to their standard
counterparts.
23
Figure 11: Uncertainty quantification of joint response PDF expansion coefficient vectors at
t= 1, 2 and 3sfor an oscillator with asymmetric nonlinearities. Comparisons between the
standard WPI technique in conjunction with a polynomial PDF approximation (deterministic
estimates) and the Bayesian WPI formulation (estimates of the coefficient vector distribution
based on Eq. (18)).
24
Figure 12: Joint response PDF of an oscillator with asymmetric nonlinearities at t= 1s(left),
t= 2s(middle), and t= 3s(right); standard brute-force WPI formulation (top), and Bayesian
formulation - mean values estimates (bottom).
In the ensuing analysis, the RVM with the optimal sampling scheme is used
with m/n = 0.6. This translates into utilizing m= 17 points for determining the385
n= 28-dimensional coefficient vector cfor t= 1sand t= 2s. Further, m= 55
points are used for determining the n= 91-dimensional coefficient vector cfor
t= 3s. In this regard, the coefficient vector values obtained by the standard
WPI technique are compared in Fig. 11 with the distributions of the estimates
25
Figure 13: Uncertainty quantification of joint response PDF estimates based on the developed
Bayesian WPI formulation and corresponding to an oscillator with asymmetric nonlinearities
for t= 1s(left), t= 2s(middle), t= 3s(right).
obtained by the herein developed Bayesian CS framework. In general, it is seen390
that the mean values of the Bayesian estimates agree well with the respective
deterministic coefficient vector estimates.
Further, the joint response PDFs at t= 1,2 and 3sobtained by a standard
brute-force implementation of the WPI technique with N2= 2601 points are
shown in Fig. 12 (top). Furthermore, the log-normal distribution of the joint395
response PDF estimate at an arbitrary point is obtained by utilizing Eqs. (20)
and (21). In Fig. 12 (bottom), the mean values of the log-normal distribution
are plotted demonstrating excellent agreement with the deterministic estimates
in Fig. 12 (top). Moreover, Fig. 13 shows the log-normal distributions corre-
sponding to each and every point of the joint response PDF domain.400
Next, the response displacement and velocity PDFs obtained by a standard
brute-force implementation of the WPI technique are plotted in Figs. 14 and 15,
respectively, for various indicative time instants. These are compared both with
the mean values of Bayesian estimates and with pertinent MCS data (20,000
realizations), demonstrating a high degree of agreement. Further, Fig. 16 shows405
the relative standard deviation σ/µ of the estimates obtained by the Bayesian
26
Figure 14: WPI-based estimates and uncertainty quantification of the response displacement
PDF of an oscillator with asymmetric nonlinearities corresponding to (top) t= 1s, (middle)
t= 2s, and (bottom) t= 3s. Comparisons with Monte Carlo simulations (20,000 realizations).
WPI technique decreasing with increasing sample ratio m/n. Note that the
RVM coupled with the optimal sampling scheme of section 3.4 converges faster to
27
Figure 15: WPI-based estimates and uncertainty quantification of the response velocity PDF
of an oscillator with asymmetric nonlinearities corresponding to (top) t= 1s, (middle) t= 2s,
and (bottom) t= 3s. Comparisons with Monte Carlo simulations (20,000 realizations).
smaller values of σ/µ than the standard RVM scheme. This enhanced behavior
due to the optimal sampling scheme is anticipated taking also into account the410
28
Figure 16: Reduction of the relative variance σ/µ of the Bayesian WPI estimates with in-
creasing sample ratio m/n. The results refer to the joint response PDF of an oscillator with
asymmetric nonlinearities at t= 2s. Comparisons between the standard RVM (top) and the
RVM coupled with the proposed optimal sampling scheme of section 3.4 (bottom).
findings in Fig. 10.
In Table 2, the accuracy degree of the various WPI technique formulations
is assessed based on comparisons with MCS data (20,000 realizations). Specif-
ically, based on calculations of the l2-norm errors referring to joint response
PDF estimates, it is seen that the Bayesian approach with m/n = 1 yields the415
smallest error, approximately equal to that of the brute-force implementation
with N2= 2601 points. Further, the Bayesian WPI technique exhibits a robust
behavior, since even for m/n = 0.6 there is only a slight increase in the reported
error compared to the case with m/n = 1.
29
Relative error (%) of WPI
compared to MCS t = 1s t = 2s t = 3s
Brute-force implementation
(N2points) 5.1 5.5 10.8
Polynomial approximation
(Bayesian CS with m/n = 0.6) 5.4 5.7 13
Polynomial approximation
(Bayesian CS with m/n = 1) 5.1 5.4 11
Polynomial approximation
(Standard linear system of equations
with m/n = 0.6) 5.1 5.7 11.6
Table 2: L2-norm errors of joint response PDF estimates compared to MCS data (20,000
realizations) for an oscillator with asymmetric nonlinearities. Results refer to various WPI
technique formulations and correspond to indicative time instants.
5. Concluding remarks420
In this paper, the WPI technique for determining the stochastic response of
diverse nonlinear dynamical systems has been enhanced based on a Bayesian
CS treatment. Specifically, sparse expansions of the polynomial kind have been
utilized for representing the system response joint PDF. Next, obtaining PDF
values at specific points based on the WPI technique localization capabilities425
has led to an underdetermined linear system of equations for the expansion co-
efficients. Further, a solution treatment based on a Bayesian CS formulation
has yielded a posterior distribution for the expansion coefficient vector. Clearly,
a significant advantage of the herein developed methodology relates to its novel
aspect of quantifying the uncertainty of the response PDF estimates obtained430
by the WPI technique. Furthermore, an adaptive scheme has been proposed
based on the quantified uncertainty of the estimates for optimal selection of
PDF sample points. In this regard, the total number of BVPs to be solved
30
as part of the WPI technique is reduced, and thus, the associated computa-
tional cost decreases. Note that the developed technique can be construed as435
a generalization and enhancement of earlier efforts in the literature (e.g., [16])
that relied on standard CS tools and provided with deterministic estimates of
the response PDF coefficient vector. The efficiency and reliability of the tech-
nique have been demonstrated based on comparisons with pertinent MCS data.
This has been done in conjunction with two indicative numerical examples per-440
taining to a Duffing nonlinear oscillator and to an oscillator with asymmetric
nonlinearities.
Acknowledgment
I. A. Kougioumtzoglou gratefully acknowledges the support through his CA-
REER award by the CMMI Division of the National Science Foundation, USA445
(Award number: 1748537).
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35
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