Predictability in Spatially Extended Systems with Model Uncertainty.
Macroscopic models for spatially extended systems under random influences are often described by stochastic partial differential equations. Some techniques for understanding solutions of such equations, such as estimating correlations, Liapunov exponents and impact of noises, are discussed. They are...
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| Опубліковано в: : | Электронное моделирование |
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| Дата: | 2009 |
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Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
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| Цитувати: | Predictability in Spatially Extended Systems with Model Uncertainty. I / J. Duan // Электронное моделирование. — 2009. — Т. 31, № 2. — С. 17-32. — Бібліогр.: 35 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859726919942012928 |
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| author | Duan, J. |
| author_facet | Duan, J. |
| citation_txt | Predictability in Spatially Extended Systems with Model Uncertainty. I / J. Duan // Электронное моделирование. — 2009. — Т. 31, № 2. — С. 17-32. — Бібліогр.: 35 назв. — англ. |
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| description | Macroscopic models for spatially extended systems under random influences are often described by stochastic partial differential equations. Some techniques for understanding solutions of such equations, such as estimating correlations, Liapunov exponents and impact of noises, are discussed. They are relevant for understanding predictability in spatially extended systems with model uncertainty, for example, in physics, geophysics and biological sciences. The presentation is for a wide audience.
Рассмотрены некоторые методы представления решений стохастических дифференциальных уравнений в частных производных, в частности в задачах корреляции оценки, экспоненты Ляпунова и воздействие шумов. Методы пригодны для понимания предсказуемости в пространственно распределенных системах с неопределенностью модели, например, в физике, геофизике и биологических науках.
Розглянуто деякі методи представлення розв'язків стохастичних диференціальних рівнянь у частинних похідних, зокрема у задачах кореляції оцінки, експоненти Ляпунова та впливу шумів. Методи придатні для розуміння передбачуваності у просторово розподілених системах з невизначеністю моделі, наприклад, у фізиці, геофізиці та біологічних науках.
|
| first_indexed | 2025-12-01T11:23:10Z |
| format | Article |
| fulltext |
J. Duan
Department of Applied Mathematics
Illinois Institute of Technology
(Chicago, IL 60616, USA, E-mail: duan@iit.edu)
Predictability in Spatially Extended Systems
with Model Uncertainty *. I
Macroscopic models for spatially extended systems under random influences are often described
by stochastic partial differential equations. Some techniques for understanding solutions of such
equations, such as estimating correlations, Liapunov exponents and impact of noises, are dis-
cussed. They are relevant for understanding predictability in spatially extended systems with
model uncertainty, for example, in physics, geophysics and biological sciences. The presentation
is for a wide audience.
Ðàññìîòðåíû íåêîòîðûå ìåòîäû ïðåäñòàâëåíèÿ ðåøåíèé ñòîõàñòè÷åñêèõ äèôôåðåíöèàëüíûõ
óðàâíåíèé â ÷àñòíûõ ïðîèçâîäíûõ, â ÷àñòíîñòè â çàäà÷àõ êîððåëÿöèè îöåíêè, ýêñïîíåíòû
Ëÿïóíîâà è âîçäåéñòâèå øóìîâ. Ìåòîäû ïðèãîäíû äëÿ ïîíèìàíèÿ ïðåäñêàçóåìîñòè â ïðîñò-
ðàíñòâåííî ðàñïðåäåëåííûõ ñèñòåìàõ ñ íåîïðåäåëåííîñòüþ ìîäåëè, íàïðèìåð, â ôèçèêå,
ãåîôèçèêå è áèîëîãè÷åñêèõ íàóêàõ.
K e y w o r d s: stochastic partial differential equations, correlation, Liapunov exponents, pre-
dictability, uncertainty, invariant manifolds, impact of noise
1. Motivation. Scientific and engineering systems are often subject to uncer-
tainty or random influence. Randomness can have delicate impact on the overall
evolution of such systems, for example, stochastic bifurcation [1], stochastic
resonance [2], and noise-induced pattern formation [3]. Taking stochastic effects
into account is of central importance for the development of mathematical models of
complex phenomena in engineering and science.
Macroscopic models for systems with spatial dependence («spatially ex-
tended») are often in the form of partial differential equations (PDEs). Random-
ness appears in these models as stochastic forcing, uncertain parameters, random
sources or inputs, and random boundary conditions (BCs). These models are
usually called stochastic partial differential equations (SPDEs). Note that
SPDEs may also serve as intermediate «mesoscopic» models in some multiscale
systems. Although we may think that SPDEs could be reduced to large systems
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 2 17
������� ����
��
����������
����
��������� ��
* This work was partly supported by the NSF Grants 0542450 and 0620539.
of stochastic ordinary differential equations (SODEs) in numerical approaches
[4, 5], it is beneficial to work on SPDEs directly when dealing with some
dynamical issues [6—18].
There is a growing recognition of a role for the inclusion of stochastic terms
in the modeling of complex systems. For example, there has been increasing in-
terest in mathematical modeling via SPDEs, for the climate system, condensed
matter physics, materials sciences, mechanical and electrical engineering, and
finance, to name just a few. The inclusion of stochastic effects has led to interest-
ing new mathematical problems at the interface of dynamical systems, partial
differential equations, scientific computing, and probability theory. Problems
arising in the context of stochastic dynamical modeling have inspired interesting
research topics about, for example, the interaction between noise, nonlinearity
and multiple scales, and about efficient numerical methods for simulating ran-
dom phenomena.
There has been some promising new development in understanding dynamics
of SPDEs via invariant manifolds [19—22] and stochastic homogenization [23,
24]. But we will not discuss these issues in this paper. For general background on
SPDEs, see [25—29].
Although some progress has been made in SPDEs in the past decade, many
challenges remain and new problems arise in modeling basic mechanisms in
complex systems under uncertainty. These challenging problems include overall
impact of noise, stochastic bifurcation, ergodic theory, invariant manifolds, and
predictability of dynamical behavior, to name just a few. Solutions for these
problems will greatly enhance our ability in understanding, quantifying, and
managing uncertainty and predictability in engineering and science. Break-
throughs in solving these challenging problems are expected to emerge.
This paper is organized as follows. After reviewing some basic concepts on
probability in Hilbert space in part 2, we discuss stochastic analysis and SPDEs
in part 3. Then we derive correlations of some linear SPDEs, Lyapunov expo-
nents, and the impact of uncertainty in parts 4, 5 and 6, respectively.
2. Stochastic Tools in Hilbert Space. Hilbert space. Recall that the Eucli-
dean space �
n
is equipped with the usual metric or distance
d x y x yj j
j
n
( , ) ( )� �
�
�
2
1
,
norm or length
x x j
j
n
�
�
�
2
1
,
J. Duan
18 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 2
and the usual scalar product
x y x y x yj j
j
n
� �� � �
�
�,
1
.
The Borel�-field of �
n
, i. e.,B ( )�
n
is generated by all open balls in �
n
.
Hilbert space H is a set with three mathematical operations: scalar multipli-
cation, addition and scalar product � �, , satisfying the usual properties as we are
familiar with in elementary mathematics. The scalar product induces a natural
norm u u u� , . The Borel �-field of H, i. e., B ( )H is generated by all open
balls in H.
Probability in Hilbert space. Given a probability space ( , , ) F � with sam-
ple space ,�-fieldF and probability measure �. Consider a random variable in
Hilbert space H (i.e., taking values in H):
X H:
.
Its mean or mathematical expectation is defined in terms of the integral with re-
spect to the probability measure �:
� �( ) ( ) ( )X X d�
�
� �
.
Its variance is:
Var X X X X X X X X X( ) ( ), ( ) ( ) ( ) .� � � � � � �� � � � � � �
2 2 2
Especially, if � ( )X �0, then Var (X) = � X
2
. Covariance operator of X is de-
fined as
Cov X X X X X( ) [( ( )) ( ( ))]� �
�� � � ,
where for any a, b �H, we denote a b
the linear operator in H defined by
a b H H
: , ( ) ,a b h a b h
� , h H� .
Let X and Y be two random variables taking values in Hilbert space H. The
correlation operator of X and Y is defined by
Cor X Y X X Y Y( , ) [( ( )) ( ( ))]� �
�� � � .
Remark 1. Cov (X) is a symmetric positive and trace-class linear operator
with trace
Tr Cov X X X X X X X( ) ( ), ( ) ( )� � � � �� � � � �
2
.
Moreover, Tr Cor X Y X X Y Y( , ) ( ), ( )� � �� � � .
Predictability in Spatially Extended Systems with Model Uncertainty. I
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 2 19
Gaussian random variables. Recall that a random variable taking values in �
n
X n
:
�
is called Gaussian, if for any a a an
n
� �( ,..., )
1
� , X a a X a Xn n� � � �
1 1
... is a
scalar Gaussian random variable. A Gaussian random variable in �
n
is denoted
as X m Q~ ( , )� , with mean vector m and covariance matrix Q. The covariance
matrix Q is symmetric and non-negative (i. e., eigenvalue� j � 0, j n�1,..., ). The
trace of Q is written asTr Q n( ) ...� � �� �
1
. The covariance matrix is defined as
Q Q X m X mij i i j j� � � �( ) ( [( ( ])� ) ) .
We use the notations E (X) = m and Cov (X) = Q. The probability density func-
tion for this Gaussian random variable X in �
n
is
f x f x x
A
en n
x m a x mj j jk k
( ) ( ,..., )
det ( )
( )
/
( ) (
� �
� � �
1 2
1
2
2�
k
j k
n
)
, �
�
1
,
where A Q a jk� �
�1
( ).
The probability distribution function of X is
F x X x f x dx
x
( ) ( : ( ) ) ( )� � �
��
�
� � � .
The probability distribution measure� (or law L X ) of X is:
� ( ) ( )B f x dx
B
�
�
, B n
�B ( ).�
Here are some observations. For a b n
, �� ,
� � �X a a X a X a m m ai i
i=
n
i i
i=
n
i i
i=
n
, ( ) ,� � � �� � �
1 1 1
� �( , , ) ( ) ( )X m a X m b a X m b X mi i i
i
j j j
j
� � � � �
�
�
�
�
�
�
�
�
�� �
� � � � �� �a b X m X m a b Q Qa bi j i i
i j
j j i j ij
i j
� [( )( )] ,
, ,
.
In particular, Qa a X m a, ,� � ��
2
0, which confirms that Q is nonnegative.
Also, Qa b a Qb, ,� , which implies that Q is symmetric.
J. Duan
20 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 2
Definition 1. A random variable X H:
in Hilbert space H is called a
Gaussian random variable and denoted as X m Q~ ( , )� , if for every a in H, the
real random variable X a, is a scalar Gaussian random variable (i. e., taking
values in �
1
).
Remark 2. If X is a Gaussian random variable taking values in Hilbert space
H, then for all a, b � H,
(i) Mean vector � �( ) : , ,X m X a m a� � ;
(ii) Covariance operator Cov X Q X m a X m b Qa b( ) : ( , , ) ,� � � ��
Remark 3. The Borel probability measure � on ( , ( ))H HB induced by a
Gaussian random variable X taking values in Hilbert space H, is called a Gaussi-
an measure. If � is a Gaussian measure in H, then there exist an element m � H
and a non-negative symmetric continuous linear operator X H:
such that:
For all h, h1, h2 � H,
(i) Mean vector m: h x d x m h
H
, ( ) ,� �
�
;
(ii) Covariance operator Q: h x h x d x m h m h Qh h
H
1 2 1 2 1 2
, , ( ) , , ,� � �
�
.
Since the covariance operator Q is non-negative and symmetric, the eigen-
values of Q are non-negative and the eigenvectors en’s form an orthonormal ba-
sis for Hilbert space H: Qe q en n n� , n = 1, 2, ... . Moreover, trace Tr Q qn
n
( ) �
�
�
�
1
.
Note that
X m X en n� ��
with coefficients X X m en n� � , ,
� �X X m e X m e Qe e q e e qn n n n n n n n n
2
� � � � � �( , , ) , , .
Therefore,
X m X n� ��
2 2
,
� �X m X q Tr Qn n� � � �� �
2 2
( ) .
We use L H2
( , ) , or just L2
( ) , to denote the (new) Hilbert space of
square-integrable random variables x :
H. In Hilbert space L H2
( , ) , the
scalar product is � � � � �x y x y, ( ), ( )� � � , where � denotes the mathematical
expectation (or mean) with respect to probability �. This scalar product induces
the usual mean square norm x x: ( )� � �
2
, which provides an appropriate
convergence concept.
Predictability in Spatially Extended Systems with Model Uncertainty. I
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 2 21
Brownian motion. Recall that a Brownian motion (or Wiener process) W (t),
also denoted as Wt, in R n
is a Gaussian stochastic process on a underlying proba-
bility space( , , ) F � , where is a sample space,F is a�-field composed of mea-
surable subsets of (called «events»), and � is a probability (also called proba-
bility measure). Being a special Gaussian process, Wt is characterized by its
mean vector (taking to be the zero vector) and its covariance operator, a n � n
symmetric positive definite matrix (taking to be the identity matrix). More spe-
cifically, Wt satisfies the following conditions [30]:
(a) W(0) = 0 a.s.,
(b) W has continuous paths or trajectories a.s.,
(c) W has independent increments,
(d) W(t) – W(s) ~ N t s I( ,( ) )0 � , t and s > 0 and t � s � 0, where I is the n � n
identity matrix. The Brownian motion in R1
is called a scalar Brownian motion.
Remark 4. (i) The covariance operator here is a constant n � n identity mat-
rix I, i. e., Q = I and Tr (Q) = n.
(i) W (t) ~ N tI( , )0 , i.e. W(t) has probability density function
p x
t
et n
x x
t
n
( )
( )
/
...
�
�
� �
1
2
2
2
1
2 2
�
.
(ii) For every ��
�
�
�
�
�
�0
1
2
, , for a.e.�� there exists C( )� such that
W t W s C t s( , ) ( , ) ( )� � �
�
� � � ,
namely, Brownian paths are H��older continuous with exponent less than one half.
Note that the generalized time derivative of Brownian motion Wt is a mathe-
matical model for white noise [31].
Now we define Wiener process, or Brownian motion, in Hilbert space U.
We consider a symmetric nonnegative linear operator Q in U. If the trace
Tr Q( )� ��, we say Q is a trace class (or nuclear) operator. Then there exist a
complete orthonormal system (eigenfunctions) { }ek in U, and a (bounded) se-
quence of nonnegative real numbers (eigenvalues) qk such that Qe q ek k k� , k =
= 1, 2, ... . A stochastic process W (t), or Wt, taking values in U for � 0 , is called a
Wiener process with covariance operator Q if :
(a) W(0) = 0 a.s.,
(b) W has continuous trajectories a.s.,
(c) W has independent increments,
(d) W (t) – W (s) ~ N (0, (t – s) Q), t � s:
Hence, �W( )0 0� and Cov (W (t)) = tQ.
J. Duan
22 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 2
We can think the covariance matrix Q as a ��� diagonal matrix, with diago-
nal elements q1, q2, …, qn, … . For any a � H,
a a e en n
n
� � �� , ,
Qa a e Qe q a e en n
n
n n n
n
� � � � � �� �, , .
We define, for � > 0, especially for ��� (0; 1),
Q a q a e en n n
n
� �
� � �� , ,
when the right hand side is defined.
R e p r e s e n t a i n s o f B r o w n i a n m o t i o n i n H i l b e r t s p a c e.
It is known that Wt has an infinite series representation [25]:
W q W t et n n n
n
( ) ( )� �
�
�
�
1
,
where
W t
W t e
q
q
q
n
n
n
n
n
( ) :
( ),
, ,
, .
�
�
�
!
"
#
"
0
0 0
are the standard scalar independent Brownian motions. Namely,W t tn ( ) ~ ( , )� 0 ,
�W tn ( ) �0, �W t tn ( )
2
� and �W t W s t sn n( ) ( ) min ( , )� . This infinite series con-
verges in L2
( ) , as long as Tr Q qn( ) � � �� .
Remark 5. For example in H L�
2
0 1( , ), we have an orthonormal basis
e n xn �sin ( )� . In the above infinite series representation, taking derivative with
respect to x, we get
$ � � �x t n n
n
W n q W t n x( ) ( ) ( ) cos ( )�
�
�
� 2
1
.
In order for this series to converge, we need 2( )n qn� converges to zero
sufficiently fast as n
�. So qn being small helps. In this sense, the trace
Tr Q qn( ) �� may be seen as a measurement for spatial regularity of white noise
�Wt the smaller the trace Tr (Q), the more regular of the noise.
We do some calculations. For a, b � H, we have the following identities:
� � �W W W q W t e q W t et t t n n n
n
n n n
n
, ( ) , ( )� � �
�
�
�
�
� �
2
1 1
Predictability in Spatially Extended Systems with Model Uncertainty. I
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 2 23
� � �
�
�
�
�
� �q W t W t t q tTr Qn n n
n
n
n
� ( ), ( ) ( )
1 1
,
� W a at , ,� �0 0 ,
� �( , , ) ( ) , ( ) ,W a W b q W t e a q W t e bt t n n n
n
n n n
n
�
%
&
'
(
)�
�
�
�
� �
1 1
*
�
� � �� � ��� q q W t W t e a e bm n m n
m n
m n( ) ( ) , ,
,
� � �� � � � �� � �� �tq e a e b t e a q e bn
n
n n
n
n n n, , , ,
� � �� �� � � � � �� �t e a Qe b t Q e a e b
n
n n
n
n n, , , ,
� � � � �� � ��t Q e a e b t Qa b
n
n n, , , ,
where we have used the fact that a e a e
n
n n� � �� , in the final step. In particular,
taking a = b, we obtain
� W a t Qa at , ,
2
� , Var W a t Qa at( , ) ,� .
More generally, � ( , , ) min ( , ) ,W a W b t s Qa bt s � � �. Moreover,
� �[ ( ) ( )] ( ) ( ) ( ) ( )W x W y q W t e x q W s e yt s n n n
n
m m m
m
�
�
�
�
�
� �
1 1
!
#
+
,
-
�
� �
�
�
� q q W t W s e x e yn m n m
n m
n m� [ ( ) ( )] ( ) ( )
, 1
� �
�
�
�min ( , ) ( ) ( ) min ( , ) ( , )t s q e x e y t s q x yn
n
n n
1
,
where
q x y q e x e yn
n
n n( , ) ( ) ( )�
�
�
�
1
.
On the other hand, the covariance operator may be represented in terms of
q x y( , ):
Qa Q e a e e a Qe e a q e
n
n n
n
n n
n
n n n� � � � � � � � � �� � �, , ,
J. Duan
24 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 2
� �� � �
n
n n na y e y dyq e x q x y a y dy( ) ( ) ( ) ( , ) ( )
0
1
0
1
.
Sometimes we call the kernel function q x y( , ) the spatial correlation. The
smoothness of q x y( , ) depends on the decaying property of qn ’s.
3. Stochastic Partial Differential Equations. Stochastic calculus in Hil-
bert space. We define the Ito stochastic integral:
. ( , )s dWs
T
�
0
�
.
Note that since Wt takes values in Hilbert space U. The integrand. ( , )t � is usu-
ally a linear operator from U to H (for each time t and each sample �):
. :U H
.
It is also possible to take Wt as a scalar, real-valued Brownian motion. For
example, in u s dWs
T
( )
0
�
if Wt is a scalar Brownian motion, we can interpret the
integrand u as a multiplication operator. For Brownian motionWt in U
W q W t et n n n
n
( ) ( )� �
�
�
�
1
,
we define
. .( , ) ( ) ( , ) ( )s dW q s e dW ss
T
n n n
T
n
� � �
0 01
� ���
�
�
.
A property of Ito integrals:
� . ( , ) ( )s dWs
T
� �
0
0
�
� .
Deterministic calculus in Hilbert space. In order to discuss more tools to
handle stochastic calculus in Hilbert space, we need to recall some concepts of
deterministic calculus. For calculus in Euclidean space �
n
, we have concepts
derivative and directional derivative. In Hilbert space, we have the correspond-
ing Frechet derivative and Gateaux derivative [32, 33].
Let H and �H be two Hilbert spaces, and F U H H: �
/
be a map, whose do-
main of definition U is an open subset of H. Let L H H( , � ) be the set of all
bounded linear operators A H H: �
. In particular, L H L H H( ) : ( , )� . We can
Predictability in Spatially Extended Systems with Model Uncertainty. I
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 2 25
also introduce a multilinear operator A H H H
1
: �
�
. The space of all these
multilinear operators is denoted as L H H H( , � )� .
Definition 2. The map F is Frechet difierentiable at u U
0
� if there is a linear
bounded operator A H H: �
such that
lim
( ) ( )
h
F u h F u Ah
h
� � �
�
0
0 0
0 ,
i. e. F u h F u Ah o h( ) ( ) ( )
0 0
� � � � , where � denotes norms in H or �H as ap-
propriate. The linear bounded operator A is called the Frechet derivative of F at
u0, and is denoted as F uu ( )
0
, or sometimes 0F u( )
0
.
If F is linear, its Frechet derivative is itself.
Definition 3. The directional derivative of F at u U
0
� in the direction h � H
is defined by the limit
1F u h
F u th F u
tt
( ; ) : lim
( ) ( )
0
0
0 0
�
� �
.
If this limit exists for every h � H, and 0 �F u h F u hG ( ) : ( ; )
0 0
1 is a linear map,
then we say that F is Gateaux differentiable at u0. The linear map 0F uG ( )
0
is called
the Gateaux derivative of F at u0.
In fact, if F is Frechet differentiable at u0, then it is also Gateaux differen-
tiable at u0 and they are equal [32, 33]: F u F uu G( ) ( )
0 0
� 0 . But the converse is not
usually true. It is true under suitable conditions [32, p. 68].
For any nonlinear map F :U H Y/
, its Frechet derivative 0F u( )
0
is a lin-
ear operator, i.e. 0 �F u L H Y( ) ( , )
0
. Similarly, we can define higher order Frechet
derivatives. Each of these derivatives is a multilinear operator. For example,
00 �
f u H H Y( ) :
0
,
( , ) ( ) ( , )h k f u h k
00
0
.
We denote
00 � 00f u h f u h h( ) : ( ) ( , )
0
2
0
,
000 � 000f u h f u h h h( ) : ( ) ( , , )
0
3
0
,
and similarly for higher order derivatives. Then we have the Taylor expansion in
Hilbert space
f u h f u f u h f u h
m
f u h Rm m
m( ) ( ) ( )
!
( ) ...
!
( )
( )
� � � 0 � 00 � � �
1
2
12
�1
( , )u h ,
where the remainder
R u h
m
s f u sh h dsm
m m m
�
� �
�
�
� �
�1
1 1
0
1
1
1
1( , )
( )!
( ) ( )
( )
.
J. Duan
26 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 2
Remark 6. It is interesting to relate these two concepts with the classical
concept of variational derivative (or functional derivative) that is used in the
context of calculus of variations. The variational derivative is usually considered
for functionals defined as spatial integrals, such as a Langrange functional in
mechanics. For example,
F u G u x u x dxx
l
( ) ( ( ), ( ))�
�
0
,
where u is defined on x l�[ , ]0 and satisfies zero Dirichlet boundary condition at
x = 0, l. Then it is known [34] that
F u h
F
u
h x dxu
l
( ) ( )�
�
1
1
0
, (1)
for h in the Hilbert space H l
0
1
0( , ). The quantity
1
1
F
u
is the classical variational de-
rivative of F. The equation (31) above gives the relation between Frechet deriva-
tive and variational derivative.
Ito’s formula in Hilbert space. We get back to stochastic calculus in Hilbert
space H. We first look at the Ito’s formula; see [25] or [29].
Theorem 1. Let u be the solution of the SPDE
du b u dt u dWt� �( ) ( ). , u u( )0
0
� .
Assume that F (t, u) be a given smooth (deterministic) function:
F : [ , )0 � �H
�
1
.
Then
(i) Ito’s Formula: Difierential form
dF t u t F t u t u t dW F u t F t uu t t u( , ( )) ( , ( )) ( ( ( )) ) { ( )) ( , (� � �. t b u t)) ( ( ( )))�
�
1
2
1
2
1
2Tr F t u t u t Q u t Q dtuu[ ( , ( ))( ( ( )) ) ( ( ( )) ) ]}
*
. . ,
where Fu and Fuu are Frechet derivatives, Ft is the usual partial derivative in
time, and *denotes adjoined operator. This formula is understood with the fol-
lowing symbolic operations in mind:
dt dW dt dWt t, ,� �0, dW dW Tr Q dtt t, ( )� .
(ii) Ito’s Formula: Integral form
F t u t F u F s u s u s dWu s
t
( , ( )) ( , ( )) ( , ( )) ( ( ( )) )� � �
�
0 0
0
.
Predictability in Spatially Extended Systems with Model Uncertainty. I
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 2 27
� � �
!
#
�
0
t
t uF s u s F s u s b u s( , ( )) ( , ( )) ( ( ( )))
�
%
&
'
'
(
)
*
*
+1
2
1
2
1
2Tr F s u s u s Q u s Quu ( , ( ))( ( ( )) ) ( ( ( )) )
*
. . ,
"
-
"
ds ,
where Fu and Fuu are Frechet derivatives and Ft is the usual partial derivative in
time. Moreover,
F s u s u s dW u s dWu s
t
s
t
( , ( )) ( ( ( )) )
~
( ( )). .
0 0
� �
�
and for all s, 2�H,�� the operator
~
( ( )). u s is defined by
~
( ( ))( ) : ( , ( )) ( ( ( )) ). .u s v F s u s u s vu� .
Also,
Tr F s u s u s Q u s Quu ( , ( )) ( ( ( )) ) ( ( ( )) )
*
. .
1
2
1
2
%
&
'
'
(
)
*
*
�
�
%
&
'
'
(
)
*
*
Tr u s Q F s u s u s Quu( ( ( )) ) ( , ( )) ( ( ( )) )
*
. .
1
2
1
2 .
Note that for the symmetric non-negative covariance operator Q with
eigenvalues qn � 0 and eigenvector en, n = 1, 2, … , we have
Qu q u e e Q u q u e en n n
n
n n n
n
� �� �, , ,
1
2
1
2 .
In fact, for a given function h : � �
, we define the operator h (Q) through the
following natural formula [35, p. 293—294],
h A u h q u e en n n
n
( ) ( ) ,�� ,
when the right hand side is defined.
Example 1. A typical application of Ito’s formula for SPDEs
du b u dt u dWt� �( ) ( ). , u u( )0
0
� .
Take Hilbert space H = L D2
( ), D n
/ � with the usual scalar product u v, �
�
�
uvdx
D
. Energy functional
F u u dx u
D
( ) � �
�
1
2
1
2
2 2
.
J. Duan
28 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 2
In this case,
F u h uhdxu
D
( )( ) �
�
,
and
F u h k h x k x dxuu
D
( )( , ) ( ) ( )�
�
,
1
2
1
2
2
1
2
1
2d u u b u Tr u Q u Q dx dt� �
!
#
+
,
"
-
"
, ( ) [( ( ) )( ( ) ) ]
*
. . �
�
u u dWt
D
, ( ). .
Integrating and taking mathematical expectation, we obtain
1
2
1
2
0
2 2
0
� � �u u u b u dt
t
� � �
�
( ) , ( )
�
%
&
'
'
(
)
*
*
� �
1
2
0
1
2
1
2� Tr u r Q u r Q dxdr
t
D
( ( ( )) )( ( ( )) )
*
. . .
Note that in this special case, Fu is a bounded operator in L H( , )� , which
can be identified with H itself due to the Riesz representation theorem.
Example 2. Energy functional
F u u dx u dx
p
D
p
D
( ) ( )� �
� �
2 2
for p� �[ , )1 . In this case,
F u h p u u hdxu
p
D
( )( )
0 0
2 2
0
2�
�
�
and
F u h k p u hkdx p p u u h x u kuu
p
D
p
( )( , ) ( ) ( )
0 0
2 2
0
2 4
0 0
2 4 1� � �
� �
�
( )x dx
D
�
.
Example 3 [28, p. 153]. Let H be a Hilbert space with scalar product� � ��,
and norm � �� � ��
2
, . Consider an energy functional F u u
p
( ) �
2
for p� �[ , )1 .
In this case,
F u h p u u hu
p
( )( ) ,
0 0
2 2
0
2� � �
�
and
F u h k p u h k p p u u h uuu
p p
( )( , ) , ( ) , ,
0 0
2 2
0
2 4
0 0
2 4 1� � � � � � ��
� �
k � �
� � � � � �
�
� �
2 4 1
0
2 2
0
2 4
0 0
p u h k p p u u u h k
p p
, ( ) ( ) , ,
where ( ) : ,a b h a b h
� � � (see part 2 or [25, p. 25]).
Predictability in Spatially Extended Systems with Model Uncertainty. I
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 2 29
S t o c h a s t i c p r o d u c t r u l e. Let u and v be solutions of two SPDEs.
Then
d uv udv du v dudv( ) ( )� � � .
I t o i s o m e t r y:
� �t dW Tr r Q r Qt
t t
. . .( , ( ) ( )�3
0
2
0
1
2
1
2
� �
�
�
�
�
�
�
�
�
�
%
&
'
'
�
�
�
�
�
�
�
�
(
)
*
*
*
dr.
G e n e r a l i z e d I t o i s o m e t r y:
� �F t dW G t dW Tr G r Q
a
t
b
t
t
a b
0 0 0
1
2
� � �
�
�
�
4
( , , ( , ( ,�3 �3 �3
�
�
�
�
�
�
%
&
'
'
�
�
�
�
�
�
�
�
(
)
*
*
F r Q dr( ,
*
�3
1
2 ,
where a b a b4 �min ( , ).
Stochastic partial differential equations. A general class of SPDEs may be
written as du Au f u dt G u dWt t� � �[ ( )] ( ) , where Au is the linear part, f (u) is
the nonlinear part, G (u) the noise intensity (usually an operator), and Wt a
Brownian motion. When G depends on u, G (u) dWt is called a multiplicative
noise, otherwise it is an additive noise.
For general background on SPDEs, such as wellposedness and basic proper-
ties of solutions, see [25, 26, 29].
(The completion of the paper see in the next issue)
Ðîçãëÿíóòî äåÿê³ ìåòîäè ïðåäñòàâëåííÿ ðîçâ'ÿçê³â ñòîõàñòè÷íèõ äèôåðåíö³àëüíèõ ð³âíÿíü ó
÷àñòèííèõ ïîõ³äíèõ, çîêðåìà ó çàäà÷àõ êîðåëÿö³¿ îö³íêè, åêñïîíåíòè Ëÿïóíîâà òà âïëèâó
øóì³â. Ìåòîäè ïðèäàòí³ äëÿ ðîçóì³ííÿ ïåðåäáà÷óâàíîñò³ ó ïðîñòîðîâî ðîçïîä³ëåíèõ ñèñòå-
ìàõ ç íåâèçíà÷åí³ñòþ ìîäåë³, íàïðèêëàä, ó ô³çèö³, ãåîô³çèö³ òà á³îëîã³÷íèõ íàóêàõ.
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32 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 2
|
| id | nasplib_isofts_kiev_ua-123456789-101439 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0204-3572 |
| language | Ukrainian |
| last_indexed | 2025-12-01T11:23:10Z |
| publishDate | 2009 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Duan, J. 2016-06-03T16:16:29Z 2016-06-03T16:16:29Z 2009 Predictability in Spatially Extended Systems with Model Uncertainty. I / J. Duan // Электронное моделирование. — 2009. — Т. 31, № 2. — С. 17-32. — Бібліогр.: 35 назв. — англ. 0204-3572 https://nasplib.isofts.kiev.ua/handle/123456789/101439 Macroscopic models for spatially extended systems under random influences are often described by stochastic partial differential equations. Some techniques for understanding solutions of such equations, such as estimating correlations, Liapunov exponents and impact of noises, are discussed. They are relevant for understanding predictability in spatially extended systems with model uncertainty, for example, in physics, geophysics and biological sciences. The presentation is for a wide audience. Рассмотрены некоторые методы представления решений стохастических дифференциальных уравнений в частных производных, в частности в задачах корреляции оценки, экспоненты Ляпунова и воздействие шумов. Методы пригодны для понимания предсказуемости в пространственно распределенных системах с неопределенностью модели, например, в физике, геофизике и биологических науках. Розглянуто деякі методи представлення розв'язків стохастичних диференціальних рівнянь у частинних похідних, зокрема у задачах кореляції оцінки, експоненти Ляпунова та впливу шумів. Методи придатні для розуміння передбачуваності у просторово розподілених системах з невизначеністю моделі, наприклад, у фізиці, геофізиці та біологічних науках. This work was partly supported by the NSF Grants 0542450 and 0620539. uk Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Электронное моделирование Математические методы и модели Predictability in Spatially Extended Systems with Model Uncertainty. Article published earlier |
| spellingShingle | Predictability in Spatially Extended Systems with Model Uncertainty. Duan, J. Математические методы и модели |
| title | Predictability in Spatially Extended Systems with Model Uncertainty. |
| title_full | Predictability in Spatially Extended Systems with Model Uncertainty. |
| title_fullStr | Predictability in Spatially Extended Systems with Model Uncertainty. |
| title_full_unstemmed | Predictability in Spatially Extended Systems with Model Uncertainty. |
| title_short | Predictability in Spatially Extended Systems with Model Uncertainty. |
| title_sort | predictability in spatially extended systems with model uncertainty. |
| topic | Математические методы и модели |
| topic_facet | Математические методы и модели |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/101439 |
| work_keys_str_mv | AT duanj predictabilityinspatiallyextendedsystemswithmodeluncertainty |