Predictability in Spatially Extended Systems with Model Uncertainty. II
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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| Цитувати: | Predictability in Spatially Extended Systems with Model Uncertainty. II / J. Duan // Электронное моделирование. — 2009. — Т. 31, № 3. — С. 21-35. — Бібліогр.: 7 назв. — рос. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860261432772263936 |
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| author | Duan, J. |
| author_facet | Duan, J. |
| citation_txt | Predictability in Spatially Extended Systems with Model Uncertainty. II / J. Duan // Электронное моделирование. — 2009. — Т. 31, № 3. — С. 21-35. — Бібліогр.: 7 назв. — рос. |
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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 | 2025-12-07T18:55:44Z |
| 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 *. II
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
4. Correlation. In this section, we discuss correlation of solutions, at different
time instants, of some linear SPDEs. We first recall some information about
Fourier series in Hilbert space.
Hilbert-Schmidt theory and Fourier series in Hilbert space. A separable
Hilbert space H has a countable orthonormal basis{ }en n�
�
1. Namely, e em n mn, �� ,
where �mn is the Kronecker delta function. Moreover, for any h �H, we have
Fourier series expansion
h h e en n
n
�
�
�
� ,
1
.
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 3 21
* J. Duan. Predictability in Spatially Extended Systems with Model Uncertainty. I. — Ýëåêòðîí-
íîå ìîäåëèðîâàíèå, ¹ 2, 2009. This work was partly supported by the NSF Grants 0542450
and 0620539.
In the context of solving stochastic PDEs, we may choose to work on a
Hilbert space with an appropriate orthonormal basis. This is naturally possible
with the help of the Hilbert-Schmidt theory [1, p. 232].
The Hilbert-Schmidt theorem [1, p. 232] says that a linear compact symmet-
ric operator A on a separable Hilbert space H has a set of eigenvectors that form a
complete orthonormal basis for H. Moreover, all the eigenvalues of A are real,
each non-zero eigenvalue has finite multiplicity, and two eigenvectors that cor-
respond to different eigenvalues are orthogonal.
This theorem applies to a strong (self-adjoined) elliptic differential operator B
Bu D a x D u
m
� �
� �
� ( ) ( ( ) )
,
1
0
�
�
�
�
, x D n
� � ,
where the domain of definition of B is an appropriate dense subspace of
H L D� 2( ), depending on the boundary condition specified for u (x).
In this case, A := B–1 is a linear symmetric compact operator in a Hilbert
space, e. g., H L D� 2( ). We may consider A := (B + aI)–1 for some real number a.
This may be necessary in order for the operator to be invertible, i.e., no zero
eigenvalue, such as in the case of Laplace operator with zero Neumann boundary
condition.
By the Hilbert-Schmidt theorem, eigenvectors (also called eigenfunctions in
this context) of A = B–1 form an orthonormal basis for H L D� 2( ). Note that A
and B share the same set of eigenfunctions. So we can claim that the strong ellip-
tic operator B’s eigenfunctions form an orthonormal basis for H L D� 2( ).
In the case of one spatial variable, the elliptic differential operator is the so
called Sturm-Liouville operator [1, p. 245]. For example Bu pu qu� � � �
( ) ,
x l
( , )0 where p x( ), �p x( )and q x( )are continuous on (0, l). This operator arises
in the method of separating variables for solving linear (deterministic) partial
differential equations in the next section. By the Hilbert-Schmidt theorem,
eigenfunctions of the Sturm-Liouville operator form an orthonormal basis for
H L l� 2 0( , ).
The wave equation with additive noise. Consider the stochastic wave equa-
tion with additive noise
u c u Wtt xx t�
2
º , 0 0� � �x l t, ,
u t u l t( , ) ( , )0 0� � ,
u x f x( , ) ( )0 � , u x g xt ( , ) ( )0 � ,
where º is a real parameter modeling the noise intensity, c > 0 is a constant (wave
speed), and Wt is a Brownian motion taking values in Hilbert space H L l� 2 0( , ).
J. Duan
22 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 3
Method of eigenfunction expansion:
u u t e xn
n
n�
�
�
�
1
( ) ( ), W q W t e xt n n n
n
�
�
�
� ( ) ( )
1
,
where
e x l
n x
l
n ( ) sin� 2
�
, � �n n� ( )2, n �1 2, , ... .
The final solution
u x t A
l
cn
q
cn
l
sdW sn n
t
n( , ) sin ( ) cos� �
�
�
�
�
�
�
�
�
�
�
�
��
�º
�
�
0
cn
l
t
n
�
�
�
�
1
�
�
�
�
�
�
�
�
�
�
�
��
�B
l
cn
q
cn
l
sdW s
cn
l
t en n
t
nº
�
� �
0
cos ( ) sin n x( ) ,
where
A f en n� , , B
l
cn
g en n�
�
, .
When the noise is at one mode, say at the first mode e x1( ) (i.e., q1 > 0 but qn =
= 0, n = 2, 3, ...), we see that the solution contains randomness only at that mode.
So for the linear stochastic diffusion system, there is no interactions between
modes. In other words, if we randomly force a few fast modes, then there is no
impact on slow modes.
Mean value for the solution:
�u x t A
cn t
l
B
cn t
l
n n
n
( , ) cos sin�
!
"
#
$
%
!
"
#
$
%
�
��
�
�
�
� � �
1 �
� e xn ( ).
Covariance for the solution: now we calculate the covariance of solution u at
different time instants t and s, i. e. � � �� � � �u x t u x t u x s u x s( , ) ( , ), ( , ) ( , ) . Us-
ing the Ito’s isometry, we get
� � �� � � � �u x t u x t u x s u x s( , ) ( , ), ( , ) ( , )
�
�
�
�
�
&
�
º
2 2
2 2 2
0
2l q
c n
cn r
l
dr
cn t
l
cn s
l
n
t s
�
� � �
sin cos cos
n�
�
�
1
�
&
�
0
2
t s
cn r
l
dr
cn t
l
cn s
l
cos sin sin
� � �
Predictability in Spatially Extended Systems with Model Uncertainty. II
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 3 23
�
&
�
0
t s
cn r
l
cn r
l
dr
cn t
l
cn s
l
cn s
l
sin cos cos sin cos s
� � � � �
in
cn t
l
�
!
"
#
$
%
�
�
�
�
.
After integrations, we get the covariance as
Cov u x t u x s u x t u x t u x s u x s( ( , ), ( , )) ( , ) ( , ), ( , ) ( , )� � � �� � � � �
� &
��
��
�
�
�
�
º
2 2
2 2 2
1 2 2
2l q
c n
t s
cn t s
l
l
cn
cn
n �
�
�
( ) cos
( )
sin
n t s
l
cn t s
l
� �( )
cos
( )&
&
�
�l
cn
cn t s
l
cn t s
l
l
cn
cn t s
l2
2
2�
� �
�
�
cos
( )
sin
( )
sin
( )
��
�
� &
��
��
�
�
�
º
2 2
2 2 2
1 2 2
l q
c n
t s
cn t s
l
l
cn
cnn
n �
�
�
( ) cos
( )
sin
�( ( ))t s t s
l
� &
�
2
�
�
��
l
cn
cn t s
l2 �
�
sin
( )
.
In particular, for t = s we get the variance.
Variance for the solution:
Var u x t
l
c n
q t
l
cn
cn
l
tn( ( , )) sin� �
��
�
!
"
#
$
º
2 2
2 2 2
1
2 4
2
� �
�
%
�
���
�
�
n 1
.
Energy evolution for the solution:
E t u c u dxt x
l
( ) [ ]�
�
1
2
2 2 2
0
.
Taking time derivative,
� ( ) [ ] ( , ) � ( )E t u u c u dx u x t W x dxt tt xx
l
t t
l
� � �� �
2
0 0
º .
Or in integral form,
E t E u x t dW x dx
t
s s
l
( ) ( ) ( , ) ( )�
� �0
0 0
º .
J. Duan
24 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 3
It can be shown that
�E t E( ) ( )� 0 , Var E t u x s dW dx
l
t s
t
( ( )) ( , )�
!
"
"
#
$
%
%� �' (2
0 0
2
� ,
where Wt is in the following form
W W t q W t e xt n n n
n
� �
�
�
�( ) ( ) ( )
1
,
and ( t u can be written in the following form :
(
� � � �
t n nu A
cn
l
cn t
l
B
cn
l
cn t
l
� ���
�
!
"
#
$
%
� sin cos
�
�
�
�
�'
� �
q
cn s
l
dW s
cn t
l
n
t
n
0
sin ( ) sin
�
��
�
�
��
0
t
n n
cn s
l
dW s
cn t
l
e xcos ( ) cos ( )
� �
.
Set cn l n� )/ � and rewrite
( ' ) ) ) )t n n
t
n n n n nu F t q s t s t dW s�
�
�( ) (sin sin cos cos ) ( )
0�
�
�
�
�
�
�
�
�
�
��
�
�
�
��
�� e xn ( )
�
�
�
�
�
��
�
�
�
��
�� F t q t s dW s e xn n
t
n n n( ) cos ( ) ( ) ( )' )
0
,
where F t A s t B s tn n n n n n n( ) : sin cos� �
) ) ) ) , n = 1, 2, ... . For the simplicity of
notations, set
G t F t q t s dW sn n n
t
n n( ) : ( ) cos ( ) ( )�
��' )
0
, n = 1, 2, ... ,
then we have ( t n nu G t e x�� ( ) ( ). Thus
� �
0 0
2
0
l
t s
t l
n n s nu x s dW dx q e x u dW� � �
!
"
"
#
$
%
% �
�
�
�
�
( ( , ) ( ) ( )s dx
t
n 0
2
1
��
#
$
%
%
�
�
�
�
!
"
" �
�
�
�
�
�
�
�
�
�
�
�
����
�
�
� q u e x dW s dxn s n n
tl
n
( ) ( )
00
2
1
Predictability in Spatially Extended Systems with Model Uncertainty. II
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 3 25
�� q e x G s e x dx dWn
n
t
n j j
j
l
�
�
�
�
� � ��
�
�
�
!
"
"
#
$
%
%
1 0 10
( ) ( ) ( ) n s( )
�
�
�
�
�
2
�
�
�
�
#
$
%
%
�
�
�
�
� � � �� q G s e x e x dx dW sn
n
t
j
j
n j
l
n
1 0 1 0
( ) ( ) ( ) ( )
�
�
�
�
!
"
" �
2
�
�
�
�
�
�
�
�
�
�
�
�
�
� � � �� �q G s dW s q G s dn
n
t
n n n
n
t
n
1 0
2
1 0
2( ) ( ) ( ) s �
�
�
�
�
�
�
�
�
�
��
�
� � �q F s q s r dW r dn
n
t
n n n
s
n
1 0 0
2
� ( ) cos ( ) ( )' ) s �� q F s dsn
n
t
n
�
�
� �
1 0
2( )
�
�
�
�
�
�
�
�
�
�
�
�
� ��� ' )*q s r dr dsn
n
s
n
t
2
1 0
2
0
cos ( )
�
!
"
�
��
�
#
$
%
�
�
� q A
t
t B
t
n
n
n n
n
n n n
1
2 2 2 2
2
1
4
2
2
1
4
)
)
) )
)
sin
n
n tsin2)
!
"
#
$
%�
� � �
��
�
�
�
�1
2
1 2
4
1
8
1 22
1
2
2
A B t q
t
n n n n n
n n
) ) '
)
*( cos ) ( cos )n t)
�
�
�
�
�
� .
Therefore,
Var E t q A
t
t Bn
n
n n
n
n( ( )) sin� �
!
"
#
$
%
�
�
�
�
�
� ' )
)
)*
1
2 2
2
1
4
2 n n
n
n
t
t2 2
2
1
4
2)
)
)
!
"
#
$
%�sin
� � �
��
�
�
�
�
�
�
�1
2
1 2
4
1
8
12
1
2
2
A B t q
t
n n n n n
n n
) ) '
)
+( cos ) ( cos )2)n t
�
�
�
is obtained.
The diffusion equation with multiplicative noise. Consider the stochastic
diffusion equations with zero Dirichlet boundary condition
u u uwt xx t�
º � , 0 1� �x ,
u x f x( , ) ( )0 � ,
(2)
J. Duan
26 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 3
where wt is a scalar Brownian motion. We take Hilbert space H L� 2 0 1( , )
with an orthonormal basis e n xn � 2sin ( )� . We use the method of eigenfunction
expansion:
u x t u t e xn n( , ) ( ) ( )�� ,
u u t e x u t n e xxx n n n n� � �� �( ) �� ( ) ( )( ) ( )� 2 .
Putting these into the above SPDE (2), with � �n n� ( )2, we get
� ( ) ( ) ( )( ) ( ) ( ) � .u t e x u t e u t e x wn n n n n n n t� � �� �
� º
We further obtain the following system of SODEs:
du t u t u t dw tn n n n( ) ( ) ( ) ( )� �
� º , n = 1, 2, 3, ... .
Thus
u t u t w tn n n( ) ( ) exp ( )� � �
!
"
#
$
%
!
"
#
$
%0
1
2
2� º º ,
where u x f x f x e x e x u e xn n n n( , ) ( ) ( ), ( ) ( ) ( ) ( )0 0� ��� � �� . Therefore, the
final solution is:
u x t a e x b t wn n n t( , ) ( ) exp( ),�
� º
with a f x e xn n�� �( ), ( ) and bn n� � �
!
"
#
$
%�
1
2
2
º .
Note that � �exp( ) exp( ) exp( ) exp( ) expb t w b t w b t tn t n t n
� �
!
º º º
1
2
2
"
#
$
% �
� �exp( )� n t . Therefore, we can find out the mean, variance, covariance and cor-
relation of the solution:
E u x t a e x tn n n( ( , )) ( ) exp( )� �� � ,
Var u x t u x t E u x t u x t E u x t( ( , )) ( , ) ( ( , )), ( , )) ( ( , ))� � � ��
� � ��a t tn n
2 22 1exp( )[exp( ) ]� º .
For , � t we have
� �exp{ ( )} exp{ ( ) }º º ºw w w w wt t
� �
�, , ,2
� � �� �exp{ ( )} exp{ }º ºw w wt , ,2
� ��
�
�
�
�
�
�
&exp ( ) exp{ } exp [( ) ( )]
1
2
2
1
2
22 2 2
º º ºt t t, , , ,�
�
�
�
�
�
.
Predictability in Spatially Extended Systems with Model Uncertainty. II
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 3 27
Therefore, by direct calculation, we can get
Cov u x t u x a b t tn n( ( , ), ( , )) exp( ( ) (( ), , ,�
�
�
�
� 2 21
2
2º ( )))t &
,
�
� �
!
"
#
$
%� �exp( ( )) exp exp� , � ,
�
,n n n n nt b t t t b
1
2
2
º
!
"
#
$
%
�
�
�
�
1
2
2
º ,
� �
& ��a t tn n
2 2 1exp{ ( )}[exp{ ( )} ]� , ,º
and
Corr u x t u x
Cov u x t u x
Var u x t V
( ( , ), ( , ))
( ( , ), ( , ))
( ( , ))
,
,
�
ar u x( ( , )),
�
�
�
& �
�
�a t t
a t
n n
n n
2 2
2
1
2
exp{ ( )}[exp{ ( )} ]
exp( )[ex
� , ,
�
º
p{ ) ] exp( )[exp{ ) ]� �� � �º º
2 2 21 2 1t an n� , ,
.
5. Lyapunov Exponents. Lyapunov exponents are tools for quantifying
growth or decay of linear systems (e. g., PDEs or SPDEs). The following discus-
sions are from [2, 3].
A deterministic PDE system. Let us first look at the following deterministic
PDE:
(
(
�
u
t
u uxx�
, (3)
u x f x( , ) ( )0 � , (4)
u x t( , ) �0, x D
( , (5)
where D x x� � �{ : }0 1 and the function f L
2 0 1( , ). An orthonormal basis for
L2 0 1( , ) is { ( )}e xn , n = 0, 1, 2, ..., ( �xx j j je e� � . Note that 0� -�� j . We then
can write:
f f ej j
j
�
�
�
�
0
,
(6)
where f f ej j� , . By using the method of eigenfunction expansion, it is known
that the unique solution to the problem is given below:
u x t t f e xj j j
j
( , ) exp( ( )) ( )� �
�
�
� � �
0
, t ./01 (7)
J. Duan
28 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 3
Theorem 2. Let us fix a non-zero initial condition f. Let j0 be the smallest
integer j . 0 in the expansion (6) of f such that f j0
02 . Then the Lyapunov expo-
nent of the system (3)—(5) exists as a limit and is given by � � �u
jf( ) � �
0
.
P r o o f. For a class of initial conditions f we calculate the Lyapunov expo-
nents, which are defined as
�u
t
L
f
t
u t( ) lim sup log�
3�
1
2( ) .
By applying (7), we obtain the Lyapunov exponents regarding to PDE sys-
tem (3)—(5),
� � �u
t
j j j
j
f
t
t f e x( ) lim sup log exp( ( )) ( )� �
3� �
�
�1
0
.
On the one hand,
1
0t
t f e xj j j
j
log exp( ( )) ( )�
�
�
�
� � �
� �
!
"
"
#
$
%
% � �
�
�
�1 1
0
0
0
2
1 2
t
t f
t
j j
j j
jlog exp( ( ))
/
� � � �
log f .
On the other hand,
1
0t
t f e xj j j
j
log exp( ( )) ( )�
.
�
�
� � �
. �
� �
�
1 1
0 0 0 0t
t f
t
fj j j jlog exp( ( )) log� � � �
.
A SPDE system. We now consider the following SPDE
dv v v dt vdwxx t�
( ) 4 , (8)
v x f x( , ) ( )0 � , x D
, (9)
v x t( , ) �0 , x D
( , (10)
where wt is a scalar Brownian motion. The conditions (9) and (10) hold for
a.a. 5
6.
We seek the solution with expansion with respect to the basis { }e j (see the
last subsection)
v x t y t e xj j
j
( , ) ( ) ( )�
�
�
�
0
,
(11)
Predictability in Spatially Extended Systems with Model Uncertainty. II
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 3 29
where y tj ( ), for j = 0, 1, 2, … satisfy the following stochastic ordinary differen-
tial equations:
dy t y t dt y t dwj j j j t( ) ( ) ( ) ( )� �
� 4 ,
y fj j( )0 � .
So
y t w t fj t j j( ) exp( ) exp� �
�
!
"
#
$
%
!
"
#
$
%4 � 4
1
2
2 .
Thus from (11), we obtain,
v x t w t f et j j j
j
( , ) exp( ) exp� �
�
!
"
#
$
%
!
"
#
$
%
�
�
� 4 � 4
1
2
2
0
.
Observe that
v x t w t u t xt( , ) exp( ) exp ( ( , )�
!
"
#
$
%
!
"
#
$
%4 ��7 4
1
2
2 , (12)
where u (t, x) is the solution to the above deterministic PDE (3)—(5).
By (12), we can calculate the Lyapunov exponent of the stochastic system
(8)—(10) as a function of the Lyapunov exponent of the deterministic system
(3)—(5) as follows:
�v
t
f
t
v t( ) lim sup log� �
3�
1
( )
� �
!
"
#
$
%
!
"
#
$
%
3�
lim sup log exp( ) exp (
t
t
t
w t u
1 1
2
24 ��7 4 ( )t �
�
�� �7 �
8
*
4*u f( ) ( , a.s.
by the strong law of large number.
Let us state the result in the following theorem.
Theorem 3. Let f 2 0. Then the Lyapunov exponent of the SPDE
(8)—(10) almost surely exists as a limit, is non-random and is given in the fol-
lowing formula:
� � �7 �
8
*
4*v uf f( ) ( ) (�
� , a. s.
Remark 7. Let us consider a special case when � � . Then by the above
theorem, for a fixed initial condition f, the Lyapunov exponent of the stochastic
system (8)—(10) is
� � �
8
*
4*v uf f( ) ( )� ,
J. Duan
30 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 3
which obviously is smaller than the Lyapunov exponent of the corresponding
deterministic system (3)—(5). The result implies that this stochastically per-
turbed system is more stable than the original deterministic system.
6. Impact of Uncertainty. In this section, we first recall some inequalities
for estimating solutions of SPDEs, and then we estimate the impact of noises on
solutions of the nonlinear Burgers equation.
Differential and integral inequalities. G r o n w a l l i n e q u a l i t y: D i f f e -
r e n t i a l f o r m [4]. Assuming that y t( ) . 0, g (t) and h (t) are integrable, if
dy
dt
� g t y h t( ) ( )
for t t. 0, then
y t y t e h s e ds
g d g d
t
t
t
t
t
t
( ) ( ) ( )[ ]
( ) ( )
�
� �
�0
0 0
0
, , , ,
, t t. 0.
In particular, if
dy
dt
gy h�
for t t. 0 with g, h being constants and t0 = 0, we have
y t y e
h
g
egt gt( ) ( ) ( )� �0 1 , t . 0 .
Note that when constant g < 0, then lim ( )
t
y t
h
g3�
� � .
G r o n w a l l i n e q u a l i t y: I n t e g r a l f o r m [5, 6]. If u (t), v (t) and c (t)
are all non-negative, c (t) is differentiable, and v t c t u s v s ds
t
( ) ( ) ( ) ( )�
�
0
for
t t. 0, then
v t v t e c s e ds
u d u d
t
t
t
t
s
t
( ) ( ) ( )[ ]
( ) ( )
�
�
� �
�0
0
0
, , , ,
, t t. 0.
In particular, assuming that y (t) . 0 and is continuous and y t C K y s ds
t
( ) ( )�
�
0
,
with C, K being positive constants, for t > 0. Then y t CeKt( ) � , t . 0.
Sobolev inequalities. We first introduce some common Sobolev spaces.
For k = 1, 2, ..., we define H l f f f f L lk k( , ) : { : , , ..., ( , )}( )0 02� �
. Each of these
is a Hilbert space with the scalar product
u v uv u v u v dx
k
k k
l
, [ ... ]( ) ( )�
� �
�
0
,
Predictability in Spatially Extended Systems with Model Uncertainty. II
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 3 31
and the norm
u u u u u u dx
k k
k
l
� �
�
�, [ ( ) ... ( ) ]( )2 2 2
0
.
For k = 1, 2, ... and p . 1, we further define another class of Sobolev spaces
W D u u Du D u L D kk p p, ( ) { : , , ..., ( ), }�
�� � with norm u u
k p L
p
p,
�
!
"
�
#
$
%u u
L
p k
L
p
p
p p
... ( )
1
.
Moreover, H lk
0 0( , )denotes the closure ofC lc
� ( , )0 in H lk ( , )0 (i.e., under the
norm 9
k
). It is a sub-Hilbert space in H lk ( , )0 . Similarly,W lk p
0
0, ( , ) denotes the
closure of C lc
� ( , )0 in W lk p, ( , )0 (i.e., under the norm 9
k p,
). It is a sub-Hilbert
space inW lk p, ( , )0 .
Standard abbreviations L L D2 2� ( ), H H Dk k
0 0� ( ), k = 1, 2, ..., are used for the
common Sobolev spaces in fluid mechanics, with < 9 , 9 > and 9 denoting the
usual (spatial) scalar product and norm, respectively, in L D2( ):
� � � �f g fgdxdy
D
, : , f f f f x y dxdy
D
: , ( , )� � � � � .
C a u c h y - S c h w a r z i n e q u a l i t y. In the space L D2( ) of
square-integrable functions defined on a domain D n� � :
f x g x dx f x dx g x dx
D D D
( ) ( ) ( ) ( )� � �� 2 2 .
H ��o l d e r i n e q u a l i t y. In the space L Dr ( ) of functions defined on a do-
main D n� � :
f x g x dx f x dx g x dx
D
p
D
p
q
D
( ) ( ) ( ) ( )� � ��
!
"
"
#
$
%
%
!
"
"
#
$
%
%
1 1
q
.
M i n k o w s k i i n e q u a l i t y. In the space L Dp ( )of functions defined on a
domain D n� � :
f x g x dx f x dx g x
p
D
p
p
D
p
p
( ) ( ) ( ) ( ):
!
"
"
#
$
%
% �
!
"
"
#
$
%
%
� �
1 1
dx
D
p
�
!
"
"
#
$
%
%
1
.
J. Duan
32 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 3
P o i n c a r e i n e q u a l i t y [4]. For g H D
0
1( ),
g g x y dxdy
D
g dxdy
D
g
D D
2 2 2 2� � ; � ;� �( , )
� �
,
where D is the Lebesgue measure of the domain D.
For u W Dp
0
1, ( ),1� � �p and D n� � a bounded domain u C u
p p
� ; ,
where C is a positive constant depending only on the domain D.
Let u W Dp
1, ( ),1� � �p and D n� � a bounded convex domain. Let S � D
be a measurable subset, and define the spatial average of u over S by u
S
udxS
D
� �
1
(here S is the volume or Lebesgue measure of S). Then u u C uS p p
� � ; ,
where C is a positive constant depending only on the domain D and S.
A g m o n i n e q u a l i t y [4]. Let D n� � . There exists a constant C depend-
ing only on domain D such that
u C u u
L D
H D H D
n n� �
�
( )
( ) ( )
1
2
1
2
1
2
1
2 , for n odd,
u u u
L D
H D H D
n n� �
�
( )
( ) ( )
2
2
2
2
1
2
1
2 , for n even.
In particular, for n = 1 and u H l
1 0( , ),
u C u u
L l L l H l
� �
( , ) ( , ) ( , )0 0
1
2
0
1
2
2 1
.
Moreover, for n = 1 and u H l
0
1 0( , ),
u C u u
L l L l
x
L l
� �
( , ) ( , ) ( , )0 0
1
2
0
1
2
2 2
.
Stochastic Burgers equation.We now consider the Burgers equation with
additive noise forcing as in [7]:
( ( ( <t x x tu u u v u W
�
2
� ,
u t( , )0 0� , u l t( , ) �0, u x u x( , ) ( )0 0� ,
whereWt is a Brownian motion, with covariance Q, taking values in the Hilbert
space L2(0, l) with the usual scalar product 9 9, . We assume that the trace Tr (Q)
is finite. So �Wt is noise colored in space but white in time.
Predictability in Spatially Extended Systems with Model Uncertainty. II
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 3 33
Taking
F u u dx u u
l
( ) ,� ��
1
2
1
2
2
0
and applying the Ito’s formula, we obtain
1
2
1
2
2 2d u u dW u vu uu lTr Q dtt xx x�
�
�
��
�
��
, , ( )< < .
Thus
d
dt
u u vu uu lTr Q v u lTr Qxx x x�
2 2 2 22 2� �
� �
, ( ) ( )< < .
By the Poincare inequality u c ux
2 2� for some positive constant depending
only on the interval (0, l), we have
d
dt
u
v
c
u lTr Q�
2 2 22
� �
< ( ).
Then using the Gronwall inequality, we finally get
� �u u e c lTr Q e
v
c
t
v
c
t
2
0
2
2
2
2
1
2
1�
�
� �
< ( )[ ] .
Note that the first term in this estimate involves the initial data, and the second
term involves the noise intensity < as well as the trace of the noise covariance.
We finally consider the Burgers equation with multiplicative noise forcing
( ( ( <t x x tu u u v u uw
�
2
� , with the same boundary condition and initial condi-
tion as above, where wt is a scalar Brownian motion (e. g., with covariance Q = 1
and the trace Tr (Q) = 1). So �Wt is noise homogeneous in space but white in time.
By the Ito’s formula, we obtain
1
2
1
2
2 2 2
d u u udw u vu uu u dtt xx x�
�
�
��
�
��
, ,< < .
Thus
d
dt
u u vu uu u v u u
v
c
uxx x x�
2 2 2 2 2 2 2 2
2 2
2
� �
� �
� �
!
"
#
$
%, < < < .
Therefore,
� �u u e
v
c
t
2
0
2
22
�
�
!
" #
$
%<
.
J. Duan
34 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 3
Note here that the multiplicative noise affects the mean energy growth or de-
cay rate, while the additive noise affects the mean energy upper bound.
Likelihood for staying bounded. By the Chebyshev inequality, we can esti-
mate the likelihood of solution orbits staying inside or outside a bounded domain
in Hilbert space H L l� 2 0( , ). Taking the bounded domain as a ball centered at the
origin with radius � > 0. For example, for the above Burgers equation with
multiplicative noise, we have
� �
�
( : )5 �
� �* *
<
u u
u
e
v
c
t
. � �
�
!
" #
$
%1 2 0
2 22
and
� �
�
( : ) ( : )5 � 5 �
�*
<
u u
u
e
v
c
t
� � � . . �
�
!
" #
$
%
1 1 0
2 22
.
I would like to thank Hongbo Fu and Jiarui Yang for helpful comments.
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.
1. Zeidler E. Applied Functional Analysis: Applications to Mathematical Physics. — New
York: Springer, 1995.
2. Caraballo T., Langa J. A Comparison of the Longtime Behavior of Linear Ito and Stra-
tonovich Partial Differential Equations // Stochastic Anal. Appl. — 2001. — 19, ¹ 2. —
P. 183—195.
3. Kwiecifinska A. A. Stabilization of Evolution Equations by Noise//Proc. Amer. Math. Soc. —
2002. — 130, ¹ 10. — P. 3067—3074.
4. Temam R. Infinite-dimensional Dynamical Systems in Mechanics and Physics. — New
York: Springer-Verlag, Second Edition, 1997.
5. Coddington E. A., Levinson N. Theory of Ordinary Differential Equations. — New York
McGraw Hill, 1955.
6. Guckenheimer J., Holmes P. Nonlinear Oscillations, Dynamical Systems and Bifurcations
of Vector Fields. — New York: Springer-Verlag, 1983.
7. Blomker D., Duan J. Predictability of the Burgers Dynamics under Model Uncertainty. In
Boris Rozovsky 60th Birthday Volume Stochastic Differential Equations: Theory and Ap-
plications, P. Baxendale and S. Lototsky (Eds.) — New Jersey: World Scientific, 2007. —
P. 71—90
Ïîñòóïèëà 08.12.08
Predictability in Spatially Extended Systems with Model Uncertainty. II
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 3 35
|
| id | nasplib_isofts_kiev_ua-123456789-101491 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0204-3572 |
| language | English |
| last_indexed | 2025-12-07T18:55:44Z |
| publishDate | 2009 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Duan, J. 2016-06-03T19:00:19Z 2016-06-03T19:00:19Z 2009 Predictability in Spatially Extended Systems with Model Uncertainty. II / J. Duan // Электронное моделирование. — 2009. — Т. 31, № 3. — С. 21-35. — Бібліогр.: 7 назв. — рос. 0204-3572 https://nasplib.isofts.kiev.ua/handle/123456789/101491 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. I would like to thank Hongbo Fu and Jiarui Yang for helpful comments. en 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. II Article published earlier |
| spellingShingle | Predictability in Spatially Extended Systems with Model Uncertainty. II Duan, J. Математические методы и модели |
| title | Predictability in Spatially Extended Systems with Model Uncertainty. II |
| title_full | Predictability in Spatially Extended Systems with Model Uncertainty. II |
| title_fullStr | Predictability in Spatially Extended Systems with Model Uncertainty. II |
| title_full_unstemmed | Predictability in Spatially Extended Systems with Model Uncertainty. II |
| title_short | Predictability in Spatially Extended Systems with Model Uncertainty. II |
| title_sort | predictability in spatially extended systems with model uncertainty. ii |
| topic | Математические методы и модели |
| topic_facet | Математические методы и модели |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/101491 |
| work_keys_str_mv | AT duanj predictabilityinspatiallyextendedsystemswithmodeluncertaintyii |