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Physical systems with random uncertainties: Chaos
representations with arbitrary probability measure
Christian Soize, R. Ghanem
To cite this version:
Christian Soize, R. Ghanem. Physical systems with random uncertainties: Chaos representations
with arbitrary probability measure. SIAM Journal on Scientic Computing, Society for Industrial
and Applied Mathematics, 2004, 26 (2), pp.395-410. �10.1137/S1064827503424505�. �hal-00686211�
PHYSICAL SYSTEMS WITH RANDOM UNCERTAINTIES: CHAOS
REPRESENTATIONS WITH ARBITRARY PROBABILITY
MEASURE
∗
CHRISTIAN SOIZE
†
AND ROGER GHANEM
‡
Abstract. The basic random variables on which random uncertainties can in a given model
depend can be viewed as defining a measure space wit h respect to which the solution to the mathe-
matical problem can be defined. This measure space is defined on a product measure associated with
the collection of basic random variables. This paper clarifies the mathematical structure of this space
and its relationship to the underlying spaces associated with each of the random variables. Cases of
both dependent and independent basic random variables are addressed. Bases on the product space
are developed that can be viewed as generalizations of the standard polynomial chaos approximation.
Moreover, two numerical constructions of approximations in this space are presented along with the
associated convergence analysis.
Key words. stochastic analysis, uncertainty quantification, stochastic representations
AMS subject classifications. 60H35, 60H15, 60H25, 60H40, 65C50
DOI. 10.1137/S1064827503424505
1. Introduction. Characterizing the membership of a mathematical function
in the most suitable functional space is a critical step toward analyzing it and iden-
tifying sequences of efficient approximants to it. In most cases encountered in scien-
tific computing, many of the relevant functional spaces are asso ciated with the same
Lebesgue measure which is often omitted from the analysis. However, in the context
of modeling physical systems that exhibit uncertainty either in their behavior or in
their environment, probability theory is often used as a framework for modeling the
uncertainty [14, 10, 2 0, 21]. In these cases, the functional spaces over which the var -
ious quantities of interest are defined are associated with different measures. These
are typically mixtures of probability measures, each tagging the probabilistic content
of some related function [3]. This paper describes the most general mathematical
setting for characterizing such problems in the case where r andom uncertainties are
defined by a finite number of basic vector-valued random variables with arbitrary
probability distributions. This arbitrariness is manifested by the possibility of a mul-
tidimensional, non-Gaussian probability measure for each basic vector-valued random
variable, whose comp onents are a set of generally dependent random variables.
In the paper, a general stochastic physical system is thought of as a nonlinear
transformation of a finite set of basic random variables defined over a suitable product
space. For clarity of presentation, and without loss of generality, the pap er deals with
transformations taking values in a finite-dimensional vector space. The extension
to transformations with values in a Hilbert space ca n be readily obtained, with the
present case being a finite-dimensional approximation, such as obtained via finite
∗
Received by the editors March 24, 2003; accepted for publication (in revised form) March 4,
2004; published electronically December 22, 2004. The authors acknowledge the financial support of
Sandia National Laboratories through contract 14652 and the Office of Naval Research under grant
N000149910900.
http://www.siam.org/journals/sisc/26-2/42450.html
†
Laboratoire de Mecanique, Universit´e Marne La-Vall´ee, 5, Boulevard Descartes, 77454 Marne-
la-Vall´ee cedex 2, France (soize@univ-mlv.fr).
‡
201 Latrobe Hall, The Johns Hopkins University, Baltimore, MD 21218 (ghanem@jhu.edu).
1
2 CHRISTIAN SOIZE AND ROGER GHANEM
element formalisms.
Clearly, the finite-dimensional assumption on the basic random variables corre-
sponds to the situation where the uncerta inty in the problem is inher ently associated
with a finite number of random variables. This assumption can also be justified for
situations where the uncertainty derives from infinite-dimensional stochastic processes
or fields that have been reduced through adapted techniques such as the Karhunen–
Lo eve expansion or the polynomial chaos decomp osition [22, 17, 10].
The finite-dimensional character of the basic random variables permits the natu-
ral extension of the standard Wiener chaos decomposition, well defined for Gaussian
basic random va riables, to the case of second-order random variables with arbitrary
probability measure. The Hilbert space to which the random solution of the math-
ematical problem belongs can be constructed as a tensor product of Hilbert spaces
associated with the basic variables. Each of these spaces is itself written as a tensor
product of Hilbert spaces.
In this paper, the finite-dimensional chaos decomposition is constructed as a
Hilbertian basis of the Hilbert space of the solution, taking into considera tion the
tensorized structure of this vector space. This Hilbertian basis is thus obtained as the
tensor product of Hilbertian bases associated with the basic random variables. This
construction differs from two standard constructions. The first one deals with the ca se
of Gaussian infinite-dimensional basic variables for which mathematical methods used
for the constructio n of Fock spaces are applicable [12, 19, 11]. The second standard
method deals with multidimensional polynomial approximations over product vector
spaces. The standard approach to this problem coincides with the stochastic problem
in the case where each basic random variable consists of mutually independent com-
ponents. The present work, therefore, can be viewed as an extension of the second
construction to relax the independence assumption.
It should be noted that the mathematical tools used in this paper consist of
standard results in Hilbert spaces, specifically, the Hilbertian basis theorem and the
orthogonal projection theorem [13]. It is shown that these simple tools can be used
to construct a complete mathematical framework in which efficient solutions can be
developed to physical systems with general random uncertainties modeled by basic
vector-valued random variables which are not necessarily Gaussian, and for which the
components ar e not necessarily independent. The resulting mathematical framework
is very well suited for the a nalysis, within a computational context, of these systems.
Chaos decomposition techniques have indeed been recently applied to a wide range
of pro blems in scientific computing relating to uncertain systems [8, 9 , 7, 18, 6, 5 ,
2, 16, 4, 23]. Attempts at developing chaos decompositions that are adapted to non-
Gaussian basic variables have also been presented in the literatur e [23], and the present
work can be viewed as delineating the correct mathematical framework in which these
extensions to these efforts should be described.
The paper is self-contained in that quantities are defined when they first appear
and enough detail is provided to assist the reader in implementing the framework. In
the first part of the paper (sections 2–5), the mathematical construction is carried
out for the finite-dimensional chaos representation for vector-valued random variables
with arbitrary probability measure. In the second part (section 6), the implementation
of the chaos decomposition is demonstrated through its application to the model of
a physical system with random uncertainties. Construction and convergence issues
are also addressed. Finally, in the third part (the appendix), data is provided for the
construction of orthogonal p o lynomials with respect to the most common probability
CHAOS WITH ARBITRARY PROBABILITY MEASURE 3
measures.
2. Defining vector-valued random variables. Consider a physical system
featuring random uncertainties in some of the parameters of its mathematical model.
The random uncertainties are identified with the p basic vector-valued random vari-
ables Z
1
,...,Z
p
. The solution describing the behavior of the physical system is a
vector-valued random variable Y =
f(Z
1
,...,Z
p
)inwhich
f is a nonlinear mapping .
Consider an R
m
j
-val ued random variable Z
j
defined on a probability space
(A, T ,P) with mean m
Z
j
and a positive-definite covariance matrix C
Z
j
admitting
the Cholesky facto rization
(1) C
Z
j
= L
T
Z
j
L
Z
j
.
Then, Z
j
can be normalized:
(2) Z
j
= m
Z
j
+ L
T
Z
j
X
j
,
where X
j
is an R
m
j
-valued random variable with mean zero and covariance matrix
equal to the identity. Consequently, random variable Y =
f (Z
1
,...,Z
p
)canbe
rewritten as Y = f (X
1
,...,X
p
) in which the no nlinear mapping f is such that
f(X
1
,...,X
p
)=
f (m
Z
1
+ L
T
Z
1
X
1
,...,m
Z
p
+ L
T
Z
p
X
p
). Thus, without any loss of
generality, the subsequent analysis will be carried out for normalized random vectors.
Next, consider the measurable function (x
1
,...,x
p
) → f (x
1
,...,x
p
)fromR
m
1
×
···×R
m
p
into C
m
. Mo reover, let P
X
1
,...,X
p
(dx
1
,...,dx
p
) be the probability measure
of the random variable (X
1
,...,X
p
) with values in R
m
1
×···×R
m
p
and let Y =
f(X
1
,...,X
p
)beaC
m
-valued random variable. Furthermore, assume that Y is a
second-order random variable, that is,
(3) E
f(X
1
,...,X
p
)
2
< +∞,
in which E{.} denotes the mathematical expectation, and where . denotes the
Hermitian norm in C
m
associated with the inner pro duct f, g
C
m
=
m
j=1
f
j
¯g
j
,in
which an overbar denotes complex conjuga tion. This inner product reduces in an
obvious manner to the r e al case. It is assumed that the random vectors X
1
,...,X
p
are mutually independent, resulting in
(4) P
X
1
,...,X
p
= P
X
1
⊗···⊗P
X
p
,
where P
X
j
is the probability distribution of random variable X
j
.LetL
2
µ
(F, G)de-
note the space of µ-square-integrable functions from top ological vector space F into
topological vector space G,inwhichµ is a probability measure on F equipped with
its Borel field.
3. Hilbert spaces for finite-dimensional chaos representations. Amath-
ematical structure appropriate for the construction of the finite-dimensional chaos
representation requires the characterization of the complex Hilbert space H
(m)
to
which (x
1
,...,x
p
) → f (x
1
,...,x
p
) belongs. Given the multiparameter dependence
of Y = f(X
1
,...,X
p
) and the multidimensional nature of each of these param-
eters, real Hilbert spaces H
j
and H, associated with the measures P
X
j
(dx
j
)and
4 CHRISTIAN SOIZE AND ROGER GHANEM
P
X
1
,...,X
p
(dx
1
,...,dx
p
), must also be characterized:
H
j
= L
2
P
X
j
(R
m
j
, R),(5)
H
(m)
= L
2
P
X
1
,...,X
p
(R
m
1
×···×R
m
p
, C
m
)
≃
L
2
P
X
1
,...,X
p
(R
m
1
×···×R
m
p
, R)
⊗ C
m
=
⊗
p
j=1
H
j
⊗ C
m
.(6)
We can then write H
(m)
as
(7) H
(m)
= H ⊗ C
m
,
in which H is a real Hilb ert space defined by
(8) H = ⊗
p
j=1
H
j
.
Througho ut this paper, the tensor product H
1
⊗ H
2
of two real Hilb ert spaces
H
1
and H
2
is defined with respect to the universal pro perty of the tensor product.
In addition, in order to simplify the notation, the tensor product H
1
⊗ H
2
has to be
understood as the completion H
1
⊗H
2
of the space H
1
⊗ H
2
.
Real Hilbert space H
j
and complex Hilbert space H
(m)
are equipped with the
following inner products:
u, v
H
j
=
R
m
j
u(x
j
)v(x
j
)P
X
j
(dx
j
)
= E
u(X
j
)v(X
j
)
(9)
and
f, g
H
(m)
=
R
m
1
···
R
m
p
f(x
1
,...,x
p
), g(x
1
,...,x
p
)
C
m
P
X
1
,...,X
p
(dx
1
,...,dx
p
)
= E
f(X
1
,...,X
p
), g(X
1
,...,X
p
)
C
m
.(10)
4. Finite-dimensional chaos representation. The chaos representation of
random variable Y = f(X
1
,...,X
p
) is obtained by representing (x
1
,...,x
p
) →
f(x
1
,...,x
p
) on a Hilbertian basis (complete orthonormal family of functions) of
H
(m)
.SinceX =(X
1
,...,X
p
) i s a random variable with values in a finite-dimensional
vector space, the associated chaos representation of Y is said to be finite-dimensional.
4.1. Hilbertian basis for H
j
. Consider a Hilbertian basis of real Hilbert space
H
j
given by
φ
j
α
j
, α
j
=(α
j
1
,...,α
j
m
j
) ∈ N
m
j
;thus
φ
j
α
j
,φ
j
β
j
H
j
= E{φ
j
α
j
(X
j
) φ
j
β
j
(X
j
)}
= δ
α
j
β
j
.(11)
Therefore any function h ∈ H
j
canbeexpandedas
(12) h(x
j
)=
α
j
∈N
m
j
h
α
j
φ
j
α
j
(x
j
),
