Content uploaded by Yaseen Merzah Hemzah

Author content

All content in this area was uploaded by Yaseen Merzah Hemzah on Apr 28, 2017

Content may be subject to copyright.

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[68]

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

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[69]

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 :

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[70]

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.

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[71]

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

Additionally,

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

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[72]

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 space–time 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

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[73]

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

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[74]

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].

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[75]

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 space–time 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

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[76]

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 :

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[77]

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 space–time 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:

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[78]

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

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[79]

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

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[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

[Hussain* et al., 5(8): August, 2016] ISSN: 2277-9655

IC™ Value: 3.00 Impact Factor: 4.116

http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology

[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

[1] Jentzen, P. E. Kloeden, "Taylor Approximations for Stochastic Partial Differential Equations", by the Society

for Industrial and Applied Mathematics , (2011).

[2] S. Lototsky, R. Mikulevicius & B. L. Rozovskii , "Nonlinear filtering revisited: a spectral approach" , SIAM

J. Control Optim. 35 (1997), no. 2, p. 435–461.

[3] S. V. Lototsky ,"Wiener chaos and nonlinear filtering" , Appl. Math. Optim. 54 (2006)

[4] J.B. Walsh ,"An introduction to stochastic partial differential equations", Lecture Notes in mathematics ,

Volume 1180, (1986) , pp 265-439

[5] B. Saussereau , "A new numerical scheme for stochastic partial differential equations with multiplicative

noise"', bruno.saussereau@univ-fcomte.fr December 19, (2012).

[6] M. Dozzi & J. A. Lopez-Mimbela,"Finite-time blowup and existence of global positive solutions of a semi-

linear SPDE", Stochastic Process. Appl. 120 (2010), no. 6, p. 767–776.

[7] E. Pardoux & S. Peng ,"Backward stochastic differential equations and quasilinear parabolic partial

differential equations, in Stochastic partial differential equations and their applications",(Charlotte, NC,

1991), Lecture Notes in Control and Inform. Sci., vol. 176, Springer, Berlin, 1992, p. 200–217.

[8] H. J. Sussmann ," On the gap between deterministic and stochastic ordinary differential equations", Ann.

Probability 6 (1978), no. 1, p. 19–41.

[9] E.A. Hussain and Y.M. Alrajhi , "Solution of partial differential equations using accelerated genetic

Algorithm", Int.J. of Mathematics and Statistics Studies, Vol. 2, No.1, pp. 55-69, March 2014.

[10] D.E. Goldberg, "Genetic algorithms in search, Optimization and Machine Learning", Addison Wesley, 1989.

[11] G. Tsoulos. I. E, "Solving differential equations with genetic programming", P.O. Box 1186, Ioannina

45110, 2003

[12] P. Naur, “Revised report on the algorithmic language ALGOL, 1963.

[13] A. R. SOHEILI , M. B. Niasar and M. Arezoomandan, "Approximation of stochastic parabolic differential

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