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# NUMERICAL SOLUTIONS FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS VIA ACCELERATED GENETIC ALGORITHM

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This paper introduced a new accelerated Genetic Algorithms (GAs) method to find a numerical solutions of stochastic Partial differential equations driven by space-time white nose wiener process . The numerical scheme is based on a representation of the solution of the equation involving a stochastic part arising from the noise and a deterministic partial differential equation . By using Doss-Sussmann transformation that enables us to work with a partial differential equation instead of the stochastic partial differential equation. Then compare these solutions obtained by our method with saul'yev method and deterministic solution.
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IJESRT
INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH
TECHNOLOGY
NUMERICAL SOLUTIONS FOR STOCHASTIC PARTIAL DIFFERENTIAL
EQUATIONS VIA ACCELERATED GENETIC ALGORITHM
Dr. Eman A. Hussain*, Dr. Yaseen M. Alrajhi
* Al-Mustansiriyah University, College of Science , Department of Mathematics, Iraq
Al-muthanna University, College of Science , Department of Mathematics, Iraq
ABSTRACT
This paper introduced a new accelerated Genetic Algorithms (GAs) method to find a numerical solutions of stochastic
Partial differential equations driven by space-time white nose wiener process . The numerical scheme is based on a
representation of the solution of the equation involving a stochastic part arising from the noise and a deterministic
partial differential equation . By using Doss-Sussmann transformation that enables us to work with a partial
differential equation instead of the stochastic partial differential equation. Then compare these solutions obtained by
our method with saul'yev method and deterministic solution.
KEYWORDS: SPDS, Accelerated Genetic Algorithm method, Numerical solution of stochastic partial differential
equations.
INTRODUCTION
In this paper we want to take a quicker look at the numerical solutions for stochastic partial differential equations
(SPDEs). Working on the numerical solutions for SPDEs we face many difficulties. On the one hand we have to
consider problems known from numerically solving deterministic partial differential equations. On the other hand we
are faced with problems triggered by numerically solving stochastic ordinary differential equations (SODEs). And
additionally new issues arise resulting from the infinite dimensional nature of the underlying noise processes ,[1].
Stochastic partial differential equations (SPDEs) are used as a model in many applications. This area of mathematics
is especially motivated by the need to describe random phenomena studied in natural sciences like physics, chemistry,
biology, and in control theory, [2]. So, we can define SPDEs, by combine deterministic partial differential equations
with some kind of noise.
Consider the SPDE with space-time white noise ,[4].

with , and 
Then , two different ways of writing this equation (1) are :




or 
 
Such that , three kinds of space-time white noise as in [4] are :
Brownian Sheet 
Cylindrical Brownian motion family of Gaussian random variables
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 a Hilbert space, s.t.

Space-time white noise 

 , where  is assumed a Basis of the
Hilbert space we’re in , if  is a complete orthonormal system, then  independent
standard Brownian motion.
The connection between the three kinds: If , then


and  
 
where we assume that

Then we get equations of the form :

 
Or in integral form



 
For . The process  is an n-dimensional Brownian motion . The operator is defined
as : 


 

 
 
Where the diffusion matrix  and the drift coefficient  . the initial condition , the
functions  are suppose to be smooth functions of the space variable ,  are bounded and holder
continuous of order 1/2. Thus the equation (5) has a unique regular strong solution.
In this paper, we focus on the stochastic heat equation. Thus, we simplify the above equation to :

where is a multiplication operator of the form

Taking a closer look at the noise in this equation we see that we can split it into two types, additive and
multiplicative noise. We speak of additive noise if the operator is a constant operator and of multiplicative noise if
is not constant.
The objective of our work is to develop a numerical scheme for the random field . The problem of numerical
solutions of (5) has been studied by many authors with different approaches. The ideas that lead us to propose a new
scheme are twofold.
On the one hand we wish to propose a numerical scheme that separates the noise from the second order
operator . This idea has been used in [2] ,[5] in a filtering context in which the authors’ scheme first
performs off-line a wide number of solutions of partial differential equations by the finite element method.
The stochastic part of the simulation is done after this first step.
On the other hand we want to use the accelerated genetic algorithm method to find numerical solution for
the partial differential equations that may appear in our scheme.
In order to implement the above ideas, we need to introduce the d-dimensional Markov process  whose
infinitesimal generator is given by the second order operator in equation (7) .
The Markov process is governed by infinitesimal generator of the stochastic differential equation is :
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


where the initial condition in .
TYPES OF SOLUTIONS OF SPDES
Stochastic partial differential equations of the form

have different notions of solutions. As in [1] we find :
Definition 1: -valued predictable process  is called an analytical strong solution of the problem
(6) if 


In particular ,the integral in the right-hand side have to be well-define ,[1],[4].
Definition 2: H-valued predictable process  is called an analytical weak solution of the problem (6)
if 


For each  , in particular ,the integral in the right-hand side have to be well-define.
Definition 3: H-valued predictable process  is called an mild solution of the problem (6) if



In particular ,the integral in the right-hand side have to be well-define,[1],[4].
STOCHASTIC INTEGRAL WITH RESPECT TO CYLINDRICAL WIENER PROCESS
We denote by
 the space of Hilbert-Schmidt operators acting from into Y , and by  , we
denote the space of linear bounded operators from U into Y ,[1],[4].
Let us consider the norm of the operator
:
  
  
 


Where and are eigenvalues and eigenfunctions of the operator ,  and are
orthonormal bases of spaces , and , respectively. The space
is a separable Hilbert space with the norm

In particular
1.When then and the space
becomes  .
2. When is a nuclear operator, that is , then . For, assume that ,
that is is linear bounded operator from the space into .
Let us consider the operator  , that is the restriction of operator to the space , where

. Because is nuclear operator, then
is Hilbert- Schmidt operator.
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Proposition.1 the formula 
 
defines Wiener process in with covariance operator such that  .
Proposition.2 For any  the process

 
is real-valued Wiener process and

and 

.
In the case when is nuclear operator,
is Hilbert-Schmidt operator. Taking , the process ,
defined by (17) is the classical Wiener process introduced.
Definition.4 The process  , defined in (17), is called cylindrical Wiener process in when .
As shown in Fig.1 below .
Fig.1 Cylindrical White noise with its distribution and spectral density
The stochastic integral with respect to cylindrical Wiener process is defined as follows. As we have already written
above, the process  defined by (10) is a Wiener process in the space with the covariance operator
such that .
Then the stochastic integral , 

where , with respect to the Wiener process is well defined on .
We denote by  the space of all stochastic processes

Such that  


and for all  , is a Y-valued stochastic process measurable with respect to the
filtration .
The stochastic integral 
with respect to cylindrical Wiener process, given by (10) for any process  can be defined as the limit
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
 

 
In  sense .
MATHEMATICAL SETTING AND ASSUMPTIONS ,[1]
Let and let  be a probability space with a normal filtration  . in addition let 
be a separable Hilbert space with norm denoted by .We will interpret the SPDE (1) in such a space . The objects
 , here are specified through the following assumptions.
Assumption 1: linear operator . There exist sequence of real eigenvalues and eigenfunctions
 of such that the linear operator  is given by :

 
For all 

Let  denote the interpolation space of the operator  ,[8].
Assumption 2: Cylindrical Brownian motion . there exist a sequence of  , of positive real numbers
 such that 
 
And independent real valued  , i.e. each is -adapted and the
increments 
 , are independent of . Then the cylindrical Brownian motion is given by :

 
Remark 1. The above series may not converge in , but in some space into which can be embedded, ([7] and
[8]). In our example with the Laplace operator in one dimension, we will have  and 
. This is the important case of spacetime white noise.
Assumption 3: nonlinearity . The nonlinearity  is two times continuously differentiable, it and its
derivatives satisfy 
For all 
and they satisfy



Remark 2. The function is usually given as a real-valued function of a real variable, but in the SPDE (1) it is
considered as a function defined on and taking values in some function space such as a subspace of .
Assumption 4: initial value . The initial value is a  valued random variable, which satisfies


where is given in assumption 2.2.
With the above assumptions we get by [JK11] that

With
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
 has unique mild solution

 .
A RELATED PARTIAL DIFFERENTIAL EQUATION
We present in this section a transformation that enables us to work with a partial differential equation instead of the
stochastic partial differential equation (5). This method is classical and it is known as the Doss-Sussmann transform
when one applies it to stochastic differential equation ([7] and [8]). It is a useful trick that permits to rewrite a large
class of one dimensional stochastic dynamic as a one dimensional random ordinary dynamic (by stochastic dynamic
we mean stochastic differential equation or stochastic partial differential equation). It has been successfully used in
[8] in which the authors have estimated the probability of finite-time blowup of positive solutions of stochastic partial
differential equations with Dirichlet boundary condition.
Doss-Susmann transform
The particular form of (5) will allow us to use a Doss-Susmann transform. We may write that ,[5].

 
with that solves the partial differential equation


 
As regard to the expression of the function , it is clear that can be simulated off-line. Indeed the coefficients in the
above partial differential equation are and the coefficients of the Markov process . They are all supposed
to be known. Consequently, we can perform a wide number of computations related to the partial differential equation
satisfied by . Then we shall come back to the simulation of itself and we use as much as we want the previous
computations. Thus we have split our scheme into a deterministic part (the approximation of ) and a stochastic part
(the immediate computation of when one simulates the Brownian motion ). The approximation of and the
Markov process will be achieved by a accelerated genetic algorithm . We have the following proposition.
Proposition 3. Let be the solution of (5). Then the function defined almost-surely by

 
is the unique strong solution of the following parabolic partial differential equation


 
, The above equation is understood trajectory wise since it is valid for almost-all .
Proof. We denote  the process defined by

 
It is a semi-martingale with the decomposition



 
In view of (1), for all ,  is a semi-martingale and we have

 
 
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Since does not depend on the space variable x, it holds that

and the integration by parts formula yields the result.
ACCELERATED GENETIC ALGORITHM
The principles of genetic algorithm are discussed in previous paper [9]. Where the components of the genetic
algorithm, ,[10],[11],[12]are :
1. Initialization
The value of mutation rate and selection rate are stated ,[9]. The initialization of every chromosome is
performed by randomly selecting an integer for every element of the corresponding vector.
2. Fitness-evaluation
Expressing the Partial differential equation in the following form:






The associated boundary conditions are expressed as:

The steps for the fitness evaluation of the population are the following:
1. Choose equidistant points in the box , equidistant points on the boundary at
 and at  , equidistant points on the boundary at  and at 
2. For every chromosome :
(i) Construct the corresponding model , expressed in the grammar described earlier.
(ii) Calculate the quantity





 
(iii) Calculate an associated penalty  . The penalty function depends on the boundary conditions and it has
the form:



 (39)




(iiii) Calculate the fitness value of the chromosome as:

3. Genetic operators
The genetic operators that are applied to the genetic population are the initialization, the crossover and the mutation.
A random integer of each chromosome was selected to be in the range [0..255] . The parents are selected via
tournament selection, i.e. :
- First, create a groups of  randomly selected individuals from the current population.
- The individuals with the best fitness in the group is selected, the others are discarded.
The final genetic operator used is the mutation, where for every element in a chromosome a random number in the
range  is chosen,[9].
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4. Termination control
Creating new generation required for application genetic operators to the population in order to find the best
chromosome having better fitness or whenever the maximum number of generations was obtained.
5. Technical of the Accelerated Method
To make the method is faster to arrived the exact solution of the partial differential equations by the following :
1- Insert the boundary conditions of the partial differential equation as a part of chromosomes in the our population
of the problem, the algorithm gives the exact solution or numerical approximation solution in a few generations.
2- Insert a part of exact solution ( or particular solution ) as a part of a chromosome in the population, find the
algorithm that gives an exact solution in a few generations.
3- Insert the vector of exact solution ( if exist ) as a chromosome in the our population of the problem, the algorithm
gives the exact solution in the first generation.
APPLICATION OF THE ACCELERATED GENETIC ALGORITHM
In this section we applied our algorithm on some SPDEs driven by cylindrical Brownian motion with additive and
multiplicative cases.
1. Stochastic Partial differential equations with additive noise.
We first look at SPDEs with additive noise to get a reference about how well the earlier presented method work. We
consider the stochastic heat equation with additive spacetime white noise on the one-dimensional domain  over
the time interval  with .
Consider the following SPDE

with  .
and  ,  , where the noise  here is the space-time white noise wiener process
 
with for all in view of assumption 2.2. (The summation here is just
formal, it does not converge in .) Therefore, we have
,with an arbitrary small in our situation.
Then the SPDE 
has unique mild solution 
. where described in [Kru12] can be written as


 
where we use the eigenvalues 
and eigenvectors

for all of the operator .
Example 1 Let we try to find the numerical solution of the SPDE with additive noise.


With 
 where  is space-time white noise wiener process.
By using Doss-Susmann transform (30). We find
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


 
with 


 



 



Then  ,and the Markov process was governed by the operator of this stochastic partial
differential equation is:

where the initial condition in . if  and , then (48 ) became :




With 
Now find the numerical solution of the partial differential equation (PDE)(51) by using an accelerated genetic
algorithm. We found that 

And the solution of the stochastic ordinary differential equation (50) (Markov process) generated by the infinitesimal
generator by accelerated genetic algorithm is :

Then , the solution of the original equation (46) is obtained by substituting (52),(53) in equation (29):

 
 

Fig.1 shown this solution
Fig.1 solution of SPDE (46)
and then compared this solution by our method with the solution obtained by Saul'yev method ,[13]. And with its
corresponding deterministic solution. (In this problem and other test examples, by a deterministic solution we mean
the numerical solution of the unperturbed problems). This comparison shown in Fig. 2 below :
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Fig.2 comparison of solutions of SPDE (46)
2. Stochastic Partial differential equations with multiplicative noise.
Look at SPDEs with multiplicative noise to get a reference about how well the earlier presented method work. We
consider the stochastic heat equation with multiplicative spacetime white noise on the one-dimensional domain 
over the time interval  with .
Example 2 Let us try to find the numerical solution of the SPDE with multiplicative noise.
Consider the SPDE 
With 

 where  is space-time white noise wiener process . where is a small parameter, we
will have
 . By using Doss-Susmann transform (30) . We find



 
With 


 




 




Then   ,and the Markov process was governed by the infinitesimal generator of this
stochastic differential equation is:


where the initial condition in . if and , then (57) became :

 


With 

.
Now find the numerical solution of the partial differential equation (PDE)(60) by using accelerated genetic algorithm.
We found that at generation 26, the numerical solution is:

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and the solution of the stochastic ordinary differential equation (59) (Markov process) generated by the infinitesimal
generator by accelerated genetic algorithm is :

Then , the solution of the original equation (55) is obtained by substituting (62),(63) in equation (29) , we find :

Fig.3 show this solution
Fig.3 The solution of SPDE (55 )
and then compared this solution by our method with the solution obtained by Saul'yev method and with its
corresponding deterministic solution. This comparison shown in Fig. 4 below :
Fig.4 comparison of solutions of SPDE (55)
The comparisons of errors of these solutions was shown in table (4.1).
Table (4.1) Comparisons of the errors.
t
x
|saul'yev-Gp26|
|ditermenistic-Gp26|
0
0
0
0
0.1
0.1
0.01575
0.01653
0.2
0.2
0.04425
0.04100
0.3
0.3
0.06718
0.04256
0.4
0.4
0.09982
0.09902
0.5
0.5
0.12078
0.12041
0.6
0.6
0.14600
0.12727
0.7
0.7
0.15877
0.11709
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0.8
0.8
0.08448
0.16523
0.9
0.9
0.08089
0.08428
1
1
0
0
Example 3 Let we try to find the approximation solution of the SPDE with multiplicative noise.

With 
 ,where  is space-time white noise wiener process, and where is a small parameter,
we will have
 .
By using Doss-Susmann transform (30). We find



 
With 


 

  


 




Then   ,and the Markov process was governed by the infinitesimal generator of
this stochastic differential equation is:


where the initial condition in . if and then (67) becomes :

 


With 

Now find the numerical solution of the partial differential equation (PDE)(65) by using accelerated genetic algorithm.
We found at generation 10 that :

And the solution of the stochastic ordinary differential equation (69) (Markov process) generated by the infinitesimal
generator by accelerated genetic algorithm is :

Then , the solution of the original equation (65) is obtained by substituting (72),(73) in equation (29):

Fig.5 show this solution
[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655
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[80]
Fig.5 solution of SPDE (65 )
And then compared this solution by our method with the solution obtained by Saul'yev method and with its
corresponding deterministic solution. This comparison shown in
Fig. 6 below :
Fig.6 comparison of solutions of SPDE (65)
The comparisons of errors of these solutions was shown in table (4.2).
Table (4.2) Comparisons of the errors.
t
x
|saul'yev-Gp10|
|ditermenistic-Gp10|
0
0
0
0
0.1
0.1
0.00061
0.00131
0.2
0.2
0.00249
0.01378
0.3
0.3
0.04720
0.04396
0.4
0.4
0.09117
0.08608
0.5
0.5
0.14238
0.13393
0.6
0.6
0.03907
0.17049
0.7
0.7
0.22878
0.20517
0.8
0.8
0.22330
0.19364
0.9
0.9
0.15338
0.12285
1
1
0
0
CONCLUSIONS
Application of a new technique for solving stochastic partial differential equations. Such as applied of accelerated
genetic algorithm (AGA) to find the numerical solutions of stochastic partial differential equations with additive and
multiplicative cylindrical Brownian motion ( or space-time white noise ) , using Doss-Susmann transformation , to
transform these equation into partial differential equations and stochastic ordinary differential equation , then applied
the AGA to find the numerical solutions of transformed equations and then the solution of original equations. We
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[81]
noted that this method has general utility for applications , and we found that insertion of boundary condition as a
chromosomes in the population quick the algorithm to approximate the numerical solutions.
In order to compare the results that have been obtained by using accelerated genetic algorithm , validating, it has
comparison with some numerical methods (such as finite difference method and the saul'yev method), where these
methods are used to solve this kind of stochastic partial differential equations and it's always convergence. It turns out
that the results that have been obtained by using accelerated genetic algorithm are good results and convergence with
these methods.
The main problem that we faced during the application of the (AGA) to find numerical solutions of stochastic
differential equations , are noise-generating process, such as (Brownian motion or cylindrical Brownian motion ).
Where the values of the noise must be normally distributed with zero mean and variance equal to  i.e. . To
achieve this value of  must be very small change so that we get the largest number of values within the specified
interval , these issues that affect on the shape and distribution of the noise and shows its influence is clear in the final
solutions.
REFERENCES
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[3] S. V. Lototsky ,"Wiener chaos and nonlinear filtering" , Appl. Math. Optim. 54 (2006)
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noise"', bruno.saussereau@univ-fcomte.fr December 19, (2012).
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linear SPDE", Stochastic Process. Appl. 120 (2010), no. 6, p. 767776.
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Probability 6 (1978), no. 1, p. 1941.
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Algorithm", Int.J. of Mathematics and Statistics Studies, Vol. 2, No.1, pp. 55-69, March 2014.
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equations with two different finite difference schemes" , Special Issue of the Bulletin of the Iranian
Mathematical Society Vol. 37 No. 2 Part 1 (2011), pp 61-83
... The principles of genetic algorithm are discussed in previous paper [15]. Where The components of the genetic algorithm, [11,12] are: ...
... To produce fitness function we used Backus-Naur grammar form (BNF), [13,14,15]. ...
... The value of mutation rate and selection rate are stated, [13,15]. The initialization of every chromosome is performed by randomly selecting an integer for every element of the corresponding vector. ...
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