International Journal of Differential Equations
Volume 2017 (2017), Article ID 1467049, 17 pages
https://doi.org/10.1155/2017/1467049

Research Article

Identifying Initial Condition in Degenerate Parabolic Equation with Singular Potential

K. Atifi, Y. Balouki, El-H. Essoufi, and B. Khouiti

Laboratoire de Mathématiques, Informatique et Sciences de l’Ingénieur (MISI), Université Hassan 1er, 26000 Settat, Morocco

Correspondence should be addressed to B. Khouiti; khouiti_b@yahoo.fr

Received 28 September 2016; Revised 13 March 2017; Accepted 19 March 2017; Published 14 June 2017

Academic Editor: Jen-Chih Yao

Copyright © 2017 K. Atifi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A hybrid algorithm and regularization method are proposed, for the first time, to solve the one-dimensional degenerate inverse heat conduction problem to estimate the initial temperature distribution from point measurements. The evolution of the heat is given by a degenerate parabolic equation with singular potential. This problem can be formulated in a least-squares framework, an iterative procedure which minimizes the difference between the given measurements and the value at sensor locations of a reconstructed field. The mathematical model leads to a nonconvex minimization problem. To solve it, we prove the existence of at least one solution of problem and we propose two approaches: the first is based on a Tikhonov regularization, while the second approach is based on a hybrid genetic algorithm (married genetic with descent method type gradient). Some numerical experiments are given.

1. Introduction

The inverse problem is expressed when the PDE solution is measured or specified, and we are interested in determining some properties: coefficients, forcing term, boundary, or initial condition from the partial knowledge of the system in a limited time interval (see [1, 2]).

In the last recent years, an increasing interest has been devoted to degenerate parabolic equations. Indeed, many problems coming from physics (boundary layer models in [3], models of Kolmogorov type in [4], etc.), biology (Wright-Fisher models in [5] and Fleming-Viot models in [6]), and economics (Black-Merton-Scholes equations in [7]) are described by degenerate parabolic equations [8].

The identification of the initial state of nondegenerate parabolic problems is well studied in the literature (see [9–11]). However, as far as we know, the degenerate case has not been analysed in the literature.

In this paper, we are interested in estimating the initial condition by the variational method in data assimilation of degenerate/singular parabolic equation:

where , , , , and . With initial and boundary conditions

The mathematical model leads to a nonconvex minimization problem

where the functional is defined as follows:

subject to being the weak solution of the parabolic problem (1) with initial state , an observation of in , and the observation operator. The space is the set of admissible initial states.

Problem (3) is ill-posed in the sense of Hadamard. To solve this problem, we propose two approaches.

The first approach is based on regularization, for the first time, applied to solve a degenerate inverse problem. The problem thus consists of minimizing a functional of the form

Here, the last term in (5) stands for the so-called Tikhonov-type regularization ([12, 13]), being a small regularizing coefficient that provides extra convexity to the functional and a priori (background) knowledge of the true state (the state to estimate). We consider that the values of are given in each point of analysis grid-points.

The second approach is applied when there is a partial knowledge of values of (example 20%); the regularization parameter is very difficult to determine. To overcome this problem, we propose a new approach, based on a hybrid genetic algorithm (married genetic with descent method gradient type). Finally, we make a comparison between the two mentioned approaches (with 20% of ).

First of all, we prove that problem (3) has at least one solution. The gradient of the functional is calculated with the adjoint method. Numerical experiments are presented to show the performance of our approaches.

2. Problem Statement and Main Result

Consider the following problem:

where , ,and is the operator defined as

with , , and .

We want to estimate thanks to an observation of in . The minimization problem associated with this problem is

where the functional is as follows:

subject to being the weak solution of the parabolic problem (6) with initial state , the background state, and the observation operator. The space is the set of admissible initial states (will be defined later).

We now specify some notations we shall use. Let us introduce the following functional spaces (see [14–16]):

with

We recall that (see [16]) is an Hilbert space and it is the closure of for the norm . If then the injections

are compacts.

Firstly, we prove that problem (6) is well-posed, the functional is continuous, and is -derivable in .

The weak formulation of problem (6) is

Let

We discuss the following cases.

( 1) Noncoercive Case (see [14], ). In this case, the bilinear form becomes

We have at , from where the bilinear form will be noncoercive.

Let

We recall the following theorem.

Theorem 1 (see [14, 17, 18]). For all and , there exists a unique weak solution which solves problem (6) such that

and there is a constant such that for any solution of (6)

if more then

and there is a constant such that

Theorem 2. Let be the weak solution of (6) with initial state . In noncoercive case, the function

is continuous, and the functional has at least one minimum in .

Theorem 3. Let be the weak solution of (6) with initial state . The function

is -derivable in .

( 2) Subcritical Potential Case (see [19, 20], ). Then the bilinear form becomes

Since at and , the bilinear form is noncoercive and is noncontinuous at .

Consider the not bounded operator where

Let

We recall the following theorem.

Theorem 4 (see [15, 19]). If then, for all , problem (6) has a unique weak solution

if more then

if then, for all , problem (6) has a unique solution

Theorem 5. Let be the weak solution of (6) with initial state . In subcritical potential case, the function

is continuous, and the functional is continuous in .

Theorem 6. Let be the weak solution of (6) with initial state . The function

is -derivable in .

3. Proof

Proof of Theorem 2. Let be a small variation such that .
Consider , with being the weak solution of (6) with initial state and is the weak solution of (6) with initial state
Consequently, is the solution of the variational problem:

Hence, is the weak solution of (6) with . We apply the estimate in Theorem 1 with . This gives the following.
There is a constant such that

therefore

And from (32) we have

Hence,

In addition, from (32) we have

Equations (34), (36), and (41) imply the continuity of the function

Hence, the functional is continuous in

We have , where , which gives Since the set is bounded in , then is a compact in Therefore, has at least one minimum in .

Proof of Theorem 3. Let and such that ; we define the function

where is the solution of the variational problem

and we pose

We want to show that

We easily verify that the function is solution of the following variational problem:

By the same way as that used in the proof of continuity, we deduce

Hence, the function is -derivable in and we deduce the existence of the gradient of the functional .

Proof of Theorem 5. Let be a small variation such that
Consider , with being the weak solution of (6) with initial state , and is the weak solution of (6) with initial state .
Consequently, is the solution of variational problem

Take ; this gives

since is independent of , which gives

By integrating between and with we obtain

and since and , , we obtain

this gives

From where

Which gives the continuity of the function

Hence, the functional is continuous in

Proof of Theorem 6. Let and such that ; we define the function

where is the solution of the variational problem

and we pose

We want to show that

We easily verify that the function is the solution of the following variational problem:

By the same way as that used in the proof of continuity, we deduce

Hence, in all cases, the function is -derivable in and we deduce the existence of the gradient of the functional .

Now, we are going to compute the gradient of with the adjoint state method.

4. Gradient of

We define the Gâteaux derivative of at in the direction , by

where is the weak solution of (6) with initial state , and is the weak solution of (6) with initial state .

We compute the Gâteaux (directional) derivative of (6) at in some direction , and we get the so-called tangent linear model:

We introduce the adjoint variable , and we integrate

Let us take ; then we may write .

And with we may now rewrite (69) as

this gives

The Gâteaux derivative of the functional

at in the direction is given by

After some calculations, we arrive at

The adjoint model is

Problem (75) is retrograde; we make the change of variable , which gives

with .

From (71), (74), and (75) the gradient of is given by

With the change of variable , the gradient becomes

To calculate a gradient of , we solve two problems: (6) and (76). The result solution of (6) is used in the second member of problem (76).

5. Discretization of Problem

Step 1 (full discretization). To resolve problem (6) and (76), we use the method -schema in time. This method is unconditionally stable for
Let be the steps in space and the steps in time.
Let

we put

Let

Therefore,

is approximated by

Let us define

Letting , finally we get

where

Step 2 (discretization of the functional one has).

We recall that the method of Thomas Simpson to calculate an integral is

with , , , .
Let the functions

This gives

where

with , , , .
Therefore,

Step 3 (discretization of ). The adjoint problem (76) is discretized as (85), so,

6. Numerical Experiments and Results

In this section, we discuss two cases:

In case we have a priori knowledge of in each point of analysis grid-points, we apply the Tikhonov approach to solve the minimization problem (8). The data is assumed to be corrupted by measurement errors, which we will refer to as noise. In particular, we suppose that . Here, we study the impact of () on the construction of the solution.

In case we have a partial knowledge of values of (example 20%): firstly, we apply the hybrid approach to rebuild the initial state. Secondly, we make a comparison between both hybrid and Tikhonov approaches.

The tests have been performed in Matlab 2012A, on a Windows 7 platform.

6.1. Regularization Approach

The differentiability and continuity in of the functional,

is deduced from the differentiability and continuity of the functional , and we have

where is the solution of (76).

The main steps for descent method at each iteration are the following:

(i)Calculate solution of (6) with initial condition .(ii)Calculate solution of (76).(iii)Calculate the descent direction .(iv)Find (v)Update the variable .

The algorithm ends when , where is a given small precision.

is chosen by the inaccurate linear search by Rule Armijo-Goldstein as follows:

 let and  if   and stop if not .

We do all the tests on Pc with the following configurations: Intel Core i3 CPU 2.27 GHz; RAM = 4 GB (2.93 usable).

In all figures, the observed function is drawn in red and built function in blue.

Let be number of points in space and number of points in time.

6.1.1. The Noncoercive Case

Let and parameters , .

(i) Tests with . See Figures 1, 2, 3, and 4.

Figure 1: Initial temperature. This figure shows that we can rebuild the initial state.

Figure 2: Final temperature.

Figure 3: Graph of .

Figure 4: Norm of gradient of .

(ii) Tests with . In Figures 5, 6, 7, and 8, is drawn in red and (rebuilt initial condition) in blue.

Figure 5: Initial temperature in err = 2%   case. This figure shows that we can rebuild the true state.

Figure 6: Initial temperature in err = 5%   case. The reconstructed initial condition is not far from the true state.

Figure 7: Initial temperature in err = 10%   case. The reconstructed initial condition begins to move away from the true state.

Figure 8: Initial temperature in err = 20%   case. This figure shows that we cannot rebuild the true state.

6.1.2. Sub Critical Potential Case

Let , , and the parameters , .

(i) Tests with . See Figures 9, 10, 11, and 12.

Figure 9: Initial temperature. This figure shows that we can rebuild the initial state.

Figure 10: Final temperature.

Figure 11: Graph of .

Figure 12: Norm of gradient of .

(ii) Tests with . See Figures 13, 14, 15, and 16.

Figure 13: Initial temperature in err = 2%   case. This figure shows that we can rebuild the true state.

Figure 14: Initial temperature in err = 5%   case. The reconstructed initial condition is not far from the true state.

Figure 15: Initial temperature in err = 10%   case. The reconstructed initial condition began to move away from the true state.

Figure 16: Initial temperature in err = 20%   case. This figure shows that we cannot rebuild the true state.

6.2. Hybrid Algorithm

The genetic algorithms (GA) are adaptive search and optimization methods that are based on the genetic processes of biological organisms. Their principles have been first laid down by Holland. The aim of GA is to optimize a problem-defined function, called the fitness function. To do this, GA maintain a population of individuals (suitably represented candidate solutions) and evolve this population over time. At each iteration, called generation, the new population is created by the process of selecting individuals according to their level of fitness in the problem domain and breeding them together using operators borrowed from natural genetics, as, for instance, crossover and mutation. As the population evolves, the individuals in general tend toward the optimal solution [21–24]. The basic structure of a GA is the following:

  Initialize a population of individuals;  Evaluate each individual in the population;  While the stop criterion is not reached do   Select individuals for the next population;  Apply genetic operators (crossover, mutation) to produce new individuals;  Evaluate the new individuals;   return the best individual.

The hybrid methods combine principles from genetic algorithms and other optimization methods. In this approach, we will combine the genetic algorithm with method descent (steepest descent algorithm (FP)).

We assume that we have a partial knowledge of background state at certain points , .

We assume the individual is a vector ; the population is a set of individuals.

The initialization of individual is as follows:

Starting by initial population, we apply genetic operators (crossover, mutation) to produce a new population in which each individual is an initial point for the descent method (FP). When a specific number of generations is reached without improvement of the best individual, only the fittest individuals (e.g., the first fittest individuals in the population) survive. The remaining die and their place is occupied by new individuals with new genetic ( are chosen randomly; the other are chosen as (96)). At each generation we keep the best. The algorithm ends when or , where is a given precision (see Figure 17).

Figure 17: Hybrid algorithm.

The main steps for descent method (FP) at each iteration are the following:

(i)Calculate solution of (6) with initial condition .(ii)Calculate solution of (76).(iii)Calculate the descent direction .(iv)Find .(v)Update the variable .

The algorithm ends when , where is a given small precision.

is chosen by the inaccurate linear search by Rule Armijo-Goldstein as follows:

 let and . if   and stop if not .

Consider we know of values of background state (), in this test we try to build the solution with the hybrid method.

In the figures below, the observed function is drawn in red and built function in blue.

Let be number of points in space and number of points in time.

6.2.1. The Noncoercive Case

Let and parameters , , number of individuals , and number of generations .

The test with simple descent gives Figure 18.

Figure 18: This figure shows that we cannot rebuild the solution.

The test with genetic algorithm gives Figure 19.

Figure 19: This figure shows that we cannot rebuild the solution.

Now we turn the hybrid algorithm. This gives Figure 20.

Figure 20: This figure shows that we can rebuild the solution.

6.2.2. Subcritical Potential Case

Let , , and parameters , , number of individuals = 30, and number of generations = 2000.

The test with simple descent gives Figure 21.

Figure 21: This figure shows that we cannot rebuild the solution.

The test with genetic algorithm gives Figure 22.

Figure 22: This figure shows that we cannot rebuild the solution.

Now we turn the hybrid algorithm. This gives Figure 23.

Figure 23: This figure shows that we can rebuild the solution.

6.3. Comparison between Hybrid Approach and Tikhonov Approach

Here, we assume that we know of values of background state ().

(i) Noncoercive Case. see Tables 1 and 2.

Table 1: Results on the Tikhonov approach. Comparison between different values of regularizing coefficient . The smallest value of is reached when

Minimum value of
Elapsed time (seconds)

8.38

24.47

23.11

21.50

22.64

18.95

15.01

17.33

6.04

Table 2: Results on the hybrid method.

Minimum value of
Elapsed time

1 min

2 min

7 min

11 min

23 min

6 h 43 min

11 h 20 min

The minimum value of reached by the Tikhonov algorithm was , whereas with the hybrid algorithm it was possible to reach the value in 11 h and 20 min with knowledge of 20% of ; if we take more than 20%, we got less than elapsed time.

(ii) Subcritical Potential Case. see Tables 3 and 4.

Table 3: Results on the Tikhonov aporoach. Comparison between different values of regularizing coefficient . The smallest value of is reached when

Minimum value of
Elapsed time (seconds)

10.91

8.68

8.75

7.91

11.17

11.61

10.20

9.57

9.66

Table 4: Results on the hybrid method.

Minimum value of
Elapsed time

1 min

2 min

4 min

43 min

1 h 56 min

6 h 43 min

13 h 40 min

The minimum value of reached by the Tikhonov algorithm was , whereas with the hybrid algorithm it was possible to reach the value in 13 h and 40 min with knowledge of 20% of ; if we take more than 20%, we got less than elapsed time.

7. Conclusion

In this paper, we have presented the regularization method and the hybrid method which are applied to determine an initial state from the point of measurements of parabolic degenerate/singular problem. These methods have proven efficiency to rebuild the solution. The proposed reconstruction algorithms are easily implanted.

The elapsed time of the hybrid method is long enough. To reduce it, in our coming work we will use the multiprogramming to run two approaches of parallelism.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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