15
Brain Activity Reconstruction by Finding a Source Parameter in an Inverse Problem

Amir H. Hadian‐Rasanan1 and Jamal Amani Rad2

1Institute for Cognitive and Brain Sciences, Shahid Beheshti University, Evin, Tehran, Iran

2Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, Tehran, Iran

15.1 Introduction

Brain activity reconstruction is the field of studying and reconstructing human brain activities from imaging data such as functional magnetic resonance imaging (fMRI), electroencephalography (EEG), magnetoencephalography (MEG), and so on. This field of neuroscience has attracted a lot of attention recently and different tools and algorithms have been developed for this purpose. On the other hand, because deep learning methods obtained outperform results in different problems of science, some researchers employed these methods for their problems and found out that a good tool for reconstruction of the brain activities are deep learning methods. Despite the deep learning methods have good accuracy and are so useful, all of them have a major problem, which is considering the learning algorithm as a black box. Moreover, there is no meaningful way that can explain “Why do the employed deep learning methods get good results?”. Fortunately, this major problem can be solved by using a suitable mathematical model. There are various mathematical tools for modeling this phenomenon. One of the most useful mathematical models for this purpose is inverse model. As a definition of an inverse problem, we can remark that this is a mathematical framework that is used to extract information about a (unknown) system from the observed treatments of the system. In fact, the problem of discovering information about source of the electrical currents from EEG data can be treated as inverse problem. Figure 15.1 shows that how the brain activities can be considered as an inverse model.

Illustration depicting the brain activities as an inverse model, a mathematical framework that is used to extract information about a (unknown) system and discovering information about source of the electrical currents from EEG data.

Figure 15.1 The graph of problem modeling procedure for the brain activities.

The procedure of measuring electrical currents and applying some preprocessing algorithm on this raw data to prepare these data to monitoring is called forward problem. On the other hand, the procedure of estimating neural activity sources corresponding to a set of measured data is called inverse problem. It is worth to mention that in the field of brain activity reconstruction of solving both forward and inverse problem is very related to each other because if the forward problem does not solve accurately, the measured data does not have enough accuracy and we are not able to solve the inverse problem accurately by using this noisy data.

In addition to the brain activity reconstruction, inverse problems and especially determining a source parameter in an inverse problem from additional information have many applications in different fields of science and engineering. Generally, source parameter evaluating in a parabolic partial differential equations from additional information has been employed to specify the unknown attributes of a special region by evaluating the properties of its boundary or specific area of the domain. These unknown attributes contain valuable information about the (unknown) system, but measuring them directly is expensive and impossible [1].

Despite of the applicability of the inverse problems and source parameter evaluation, these problems have some challenging aspects including solution existence, solution uniqueness, and instability of the solution [2]. In the following sentences we explain each of these major problems briefly:

  1. Existence of the solution: Since the measured data contain noise and moreover in the process of modeling of the phenomena, some simplifying and approximation is added to the problem, so it is possible that the model does not fit to the data.
  2. Uniqueness of the solution: If the model fits to the data, the existence of the solution is satisfied, but the second major is uniqueness of the solution, which means there may be more than one solution for the model. This problem usually occurs in the potential field models.
  3. Ill‐posedness of the solution: The nature of the inverse problems is ill‐posed, which means that small changes in the measured data can cause big changes in the results. In order to overcome this major problem, some regularization algorithms can be employed to prevent obtaining an ill‐conditioned system of linear algebraic equations from the inverse problem.

In Section 15.1.1, we illustrate the problem that is considered to be solved in this chapter.

15.1.1 Statement of the Problem

In this chapter, we study the following nonlinear inverse problem of simultaneously finding unknown function images and unknown source parameter images from the following parabolic equation:

with initial condition

(15.2)equation

and boundary condition

(15.3)equation

subject to an over specialization at a point in the spatial domain

In order to overcome to the problem of nonlinearity of the problem and finding two unknown functions simultaneously, a transformation is used. By employing a pair of transformations images and images, Eqs. 15.115.4 are written in the following form:

with initial condition

(15.6)equation

and boundary condition

(15.7)equation

subject to

which yields

(15.9)equation

As you can see with this transformation, the nonlinear equation 15.1 is transformed to linear equation 15.5. Moreover, the role of images is transferred to images. As mentioned in Section 15.1,, this problem has special importance in different fields of science. Therefore, many researchers developed different numerical algorithms to solve this problem. Some of these efforts are presented in Section 15.1.2.

15.1.2 Brief Review of Other Methods Existing in the Literature

Since finding a source parameter has a significant role in different fields of science, this problem has been attracted over the past decade. Many researchers analyze different aspects of this problem. For example, the existence and uniqueness of the solutions to these problems and also some more applications are discussed in [39]. Some authors developed different numerical algorithms for simulating this problem. These numerical methods include various classes of basic numerical algorithms such as finite difference, meshless methods, or spectral method. We summarized some numerical schemes that are used to solve this problem as follows:

  1. Finite difference methods: Dehghan is a pioneer in using various finite difference schemes to solve this problem. About 2000, he developed different finite difference schemes and discussed about the stability and convergence of these schemes in [1013]. He used these schemes for solving one or multidimensional problems. After one decade, Mohebbi and Dehghan developed some high‐order schemes. They presented an algorithm based on hybrid of boundary value method and a high‐order finite difference scheme in [14]. After that, in 2015, Mohebbi and Abbasi developed a compact finite difference scheme for this problem [1]. In addition, some other finite difference methods were presented by some other researchers [1517]. Recently, the authors have developed a multigrid compact finite difference method to solve such a problem from this family in [18].
  2. Meshless methods: Meshless methods are a powerful class of numerical algorithms that are used to solve different problems. One of the advantages of these algorithms is their ability of handling nonrectangular domains easily. Dehghan and Tatari used radial basis function method to solve one‐dimensional version of this problem [19,20]. Cheng used a moving least‐square method for solving this problem [21]. Parzlivand and Shahrezaee developed a meshless algorithm based on inverse quadratic radial basis functions to solve this problem [22]. Recently, some new improvements on this problem have accomplished using meshless algorithms. For example: Shivanian and Jafarabadi developed a spectral meshless radial point algorithm to solve the multidimensional version of this problem. They also applied their algorithm for solving the problem on nonrectangular domains [23]. Also, a greedy version of meshless local Petrov‐Galerkin (MLPG) method is employed to solve this problem by Takhtabnoos and Shirzadi [24].
  3. Spectral and pseudo‐spectral methods: The spectral methods are very famous for their stability and high convergence rate. These methods have been used frequently for solving different problems in science and engineering. In [25], a Tau method based on shifted Legendre polynomials has been employed to solve one‐dimensional version of this problem. Moreover, some other pseudo‐spectral methods based on Legendre polynomials are used to solve one‐dimensional [26] and multidimensional [27] version of the problem. Ritz Galerkin method [28] and Ritz least‐squares method [29] are used for this problem by some other researchers. Additionally, Chebyshev cardinal functions [30], Cardinal Hermite collocation method [31], sinc‐collocation method [32,33], Bernstein Galerkin method [34], Cubic B‐spline scaling functions [35], and some other spectral methods were developed by mathematicians. Recently, in 2019, Gholampoor and Kajani have developed a direct numerical method for solving this problem based on block‐pulse/Legendre hybrid functions [36].
  4. Analytical and semianalytical methods: These methods have used for solving different problems. Tatari and Dehghan developed Adomian decomposition method [37] for solving this problem in 2007. They also presented variational iteration method [38] for the same purpose in that year. Moreover, some other analytical and semianalytical methods have been employed for this problem including reproducing kernel space [39,40], homotopy perturbation method, and homotopy analysis method [41,42].

In addition to the above‐mentioned methods, some other researchers used different methods for solving the inverse control parameter such as method of lines [43] and predictor‐corrector method [44].

Weighted residual methods have attracted much attention in recent years. There are various types of the weighted residual methods such as Tau, Galerkin, Petrov–Galerkin, and collocation. Collocation algorithm is a powerful method based on the expansion of an unknown function by basis functions. Using different basis functions can help us to solve many different problems that have different dynamics [4548]. In addition, Chebyshev polynomials and their various types such as rational and fractional are a suitable choice as basis functions. Many researchers have been using these polynomials in their approaches [4956]. In this chapter, for spatial approximation, the Chebyshev collocation method is used, which yields a linear algebraic system of equations. On the other hand, in order to temporal approximation, images‐weighted finite difference scheme is used. The rest of this chapter is organized as follows. In Section 15.2, we introduce different parts of proposed method, which includes weighted residual method, collocation method, and function approximation using Chebyshev polynomials. In Section 15.3, we apply the proposed method on Eqs. 15.515.8, and in Section 15.4, we report the numerical experiments of solving Eqs. 15.115.4. Finally, a conclusion is given in Section 15.5.

15.2 Methodology

Before we explain how to solve the model in the form of Eqs. 15.515.8, we focus on the numerical tools that are used in this chapter. At first, the weighted residual method and collocation algorithm as a special case of the weighted residual method are introduced; afterward, the properties of Chebyshev polynomials and function approximation by using these polynomials are expressed.

15.2.1 Weighted Residual Methods and Collocation Algorithm

The spectral and pseudo‐spectral methods are used frequently to solve many problems in different fields such as computational neuroscience [57], cognitive science [58], biology [59], fluid dynamics [60], engineering [61], and so on. There is no obvious historical start point for these methods but there may be the main idea that these methods are inspired by Fourier analysis method. In 1820, Navier employed an approach based on expanding the unknown function by sine functions for solving the Laplace equation. He expands the two value unknown function in the following form:

(15.10)equation

in which images can be obtained using the orthogonality of the sine functions. Because the computational cost of this method was high and does not have acceptable accuracy, this method was not used frequently in that time. By growing the available computational resources, these methods have been attracted very much and some researchers developed these methods in different aspects such as error estimation analysis, convergence and stability analysis, parallelism, etc. The weighted residual methods are powerful approach to approximate the solution of differential equations, which are the general form of the spectral methods. Before we explain the procedure and the basics of the weighted residual methods, we should remark the definition of the weight function. The definition of the weight function is presented in Definition 15.1.

In order to illustrate the weighted residual method, consider a multidimensional (nonlinear) partial differential equation as follows:

with the boundary condition

where images is a (nonlinear) partial differential operator of the equation, images operates on the boundary conditions, images is the domain of the equation, and images is the boundary of the domain. In order to approximate the solution of the above equation, we select images as basis functions on the space of images and assumed the solution as a finite summation of multiplication of basis functions. Therefore, we obtain images as follows:

By substituting Eq. 15.13 in Eqs. 15.11 and 15.12, the following equations are obtained, which are named residual functions:

(15.14)equation

because the approximated solution and the exact solution are not equal, so the residuals are not equal to zero, but the weighted residual method will force the residual to minimize as much as possibly closer to zero.

To obtain unknown coefficients images weighted inner product of Eqs. 15.15 and 15.16 which is used by a set of test functions images, images, …, images. So we have

(15.17)equation

where images are positive weight functions. The choice of test functions determines the type of spectral methods. Here, a brief review on the different types of the weighted residual method is presented by considering images as a residual function:

  1. Collocation method: In the collocation method, the test function is Dirac delta function, and because of some properties in this test function, it yields that the residual function should be forced to zero at some points of the domain.
    (15.18)equation
  2. Subdomain method: In this type of weighted residual method, the domain is split into some subdomain parts and the test function in the each subdomain is considered equal to 1.
    (15.19)equation

    Therefore, this method forces the residual function to be zero over the all subdomains.

    (15.20)equation
  3. Least‐squares method: The least‐squares method is the oldest of all the methods of weighted residuals. The history of using least‐squares method returns to 1775 when Gauss introduced this method. In this method, the residual function is considered as test function, so the problem is reduced to minimizing the following expression:
    (15.21)equation
  4. Moment method: In the moment method, the test functions are considered equal to the family of single‐term polynomials. Therefore, in this method, images are considered as test function. This method has some major problems, for example, it is ill‐conditioned.
  5. Galerkin method: The Galerkin method is a modification of the least‐squares method. In this method, the basis functions are considered as test function. In 1915, Galerkin introduced this method for the first time [62].

In the above‐mentioned methods recently, some new pseudo‐spectral methods based on a new family of Lagrange functions have been introduced [63,64]. In order to solve our model, we employed the collocation algorithm. In the following, this algorithm is explained as a special case of weighted residual method in details.

The collocation algorithm is the simplest and most common weighted residual method that is used to solve many problems in different areas of science. In 1934, Salter and Kantrovic employed this method for the first time. Afterward, about three years later, Frazer, Jones, and Skan used this algorithm for approximating the solution of ordinary differential equations. Lanczos illustrated the significance of choosing basis functions and collocation points in 1938. From 1957 till 1964, some studies were done by Clenshaw, Norton, and Wright, which cased to revive this method. As mentioned, the collocation method is obtained by choosing Dirac delta function as the test function. Dirac delta function has some special properties, which should be remarked. The definition and some essential properties of Dirac delta function are presented.

This method is a member of weighted residual methods, which force the residual to zero at some points of the domain. As a result of choosing images in the role of collocation points and Dirac delta function as test functions, it yields that images. Therefore, if we take images, Eqs. 15.16 and 15.17 become as follows:

Now, if we select images equations from Eqs. 15.22 and 15.23, a images linear or nonlinear system of equations is obtained. The unknown coefficients are specified from solving of this system. The pseudo‐code of collocation algorithm is presented in Algorithm 15.1

15.2.2 Function Approximation Using Chebyshev Polynomials

Orthogonal polynomials have significant role in function approximation, spectral methods, and specially in the collocation algorithm. These functions have many applications in different fields of science and engineering. For example, in pattern recognition, many support vector machine algorithms have been developed based on orthogonal functions [6569]. Moreover, some algorithms for signal and image representation [70] use orthogonal functions. In the field of complex network, computing the reliability of the network has a special importance. Recently, some approaches based on orthogonal functions have employed to obtain the reliability of the network [71]. In the spectral methods, choosing suitable basis function has significant effect on convergence and accuracy of the algorithm. These polynomials have suitable properties to choose as a basis function. The general properties that the basis functions should have are as follows:

  • convenient in computations
  • completeness
  • high convergence rate

Some classes of the orthogonal functions have the above‐mentioned properties simultaneously. One of these suitable properties is completeness over the arbitrary close finite interval. In Theorem 15.1, we explain this property.

It is worth to mention that all classes of orthogonal functions are not complete and the mentioned theorem is only about the orthogonal functions that are the solution of a Sturm–Liouville differential equation. One of these orthogonal polynomials is Chebyshev polynomials. The Chebyshev polynomials have been used frequently in many areas of numerical analysis and scientific computing. The Chebyshev polynomials are a special case of Gegenbauer and Jacobi polynomials and have four kinds. The first kind of Chebyshev polynomials is solutions of the following Sturm–Liouville differential equation:

(15.24)equation

The solutions of this differential equation depends on images. The imagesth solution is images, where images and the range of images is images. Therefore, by denoting the imagesth solution by images, we have:

(15.25)equation

where images is the imagesth member of the first kind of Chebyshev polynomials. All classical orthogonal polynomials can be obtained by a recursive formula. The recursive formula for these polynomials is as follows:

(15.26)equation

In addition to the recursive formula, there is some direct formula to obtain the Chebyshev polynomials. For example, the following series give the imagesth Chebyshev polynomial:

(15.27)equation

Or

(15.28)equation

As mentioned above, the Chebyshev polynomials have the orthogonality property. These functions are orthogonal over the interval images by images weight functions as follows:

(15.29)equation

where images is the Kronecker delta function, and

(15.30)equation

Another important property of the Chebyshev polynomials is that the imagesth member of Chebyshev series has images real roots. The following theorem explain this fact formally.

Because these polynomials are orthogonal over the interval images and sometimes the problem is defined over the another interval, so we should use a transformation that shifts the problem to the interval images or the Chebyshev polynomials to the interval of the problem. If we want to shift these functions from images to images, we can use the following transformation:

(15.31)equation

where images and images. In this paper, we use shifted first kind of Chebyshev polynomials on images. The remaining thing, which is essential for the spectral method, is calculating the differentiation of these polynomials. In order to calculate the differentiation of the Chebyshev polynomials, we can use an operational matrix of differentiation. By denoting the operational matrix of first derivative of shifted Chebyshev polynomials by images, its elements can be obtained as follows:

(15.32)equation

in which

(15.33)equation

It is worth to mention that for obtaining a higher order of derivative of Chebyshev polynomials by using this operational matrix, we can use the powers of this operational matrix. For example, in order to obtain the second order of derivative of Chebyshev polynomials, it is just needed to compute images.

In order to approximate functions using shifted Chebyshev polynomials, some definition and theorem are needed, which expressed in the rest of this section.

It is known that any function images can be expanded as follows:

(15.34)equation

where images denotes shifted Chebyshev polynomials to images and

(15.35)equation

that is,

(15.36)equation

Now, let us assume that

equation

is a finite dimensional subspace, as mentioned in Theorem 15.1 images is a complete subspace of images [47,51]. Let us define the images‐orthogonal projection images, that for any function images:

(15.37)equation

It is clear that images is the best approximation of images in images and can be expanded as [72]:

(15.38)equation

15.3 Implementation

In this part, we demonstrate that how to apply numerical methods that are explained in Section 15.2. First, we discretize the temporal dimension of the problem using a finite difference scheme and then we apply the collocation algorithm for spatial dimensions. Many researchers have developed different finite difference schemes to approximate the fractional or integer order of derivatives of a (unknown) function. Moreover, some other researchers have used and combined these schemes with some other methods to solve various problems. The basic idea of the finite difference method is inspired from Taylor series. In this chapter, we use images‐weighted finite difference scheme for temporal approximation. The stability of the used scheme is a challenging problem in the numerical methods. Crank–Nicolson is the most famous images‐weighted finite difference scheme that is unconditionally stable.

The domain of problem is images. In order to discretize temporal dimension, the interval images is divided into images steps with time step size images. Note that images and we denote images by images and images by images. By applying images‐weighted finite difference scheme on Eq. 15.5, we obtain the following equation:

(15.39)equation

in which images. If we choose images, the mentioned scheme is called explicit; if images, the scheme is called implicit; and if images, the Crank–Nicolson scheme is obtained. In the above equation, the goal is obtaining images, so by reordering it, we have:

(15.40)equation

In order to approximate images in different time steps, we expand it as follows:

or equivalently in the matrix form:

(15.42)equation

where

(15.43)equation

and

(15.44)equation

It is worth to mention that images is the shifted Chebyshev polynomial to interval images. Therefore, if we consider images as Eq. 15.41 for images, it is obtained:

(15.45)equation

or equivalently in the matrix form:

(15.46)equation

We construct the residual functions as follows:

(15.47)equation
(15.48)equation

By choosing images as collocation point and substituting Eqs. 15.4115.45 in residual functions and equal them to zero, then selecting images equations from them, we obtain images linear algebraic equations, which can be solved for the unknown coefficients images using any linear solver such as LU decomposition or successive over‐relaxation (SOR) in each time step.

The aim of this problem is finding unknown source parameter. For obtaining unknown source parameter, images from Eqs. 15.1 and 15.4 we have

(15.49)equation

Note that for images, we can write

(15.50)equation

15.4 Numerical Results and Discussion

In this section, we are going to test the presented algorithm in different aspects. In order to this purpose, five examples with fixed values of their parameters are provided to test the accuracy, efficiency, stability, and convergence of the proposed algorithm. Because usually the inverse problems are ill‐posed, the stability of the algorithm should be checked. In order to check the stability of the proposed algorithm, some noises are added to overspecified condition. We add 1%, 3%, and 5% of noise to images as follows:

(15.51)equation

in which images is a random number in interval images, where images and images denotes the noisy condition. On the other hand, the convergence and order of convergence of the algorithm are demonstrated by increasing the number of collocation points, which means it is demonstrated that by increasing the number of collocation points, the error of approximated solutions decreased. In order to compute the error of approximated solutions and also compare our results to each other and with other methods in literature, the following norms are used:

(15.52)equation
(15.53)equation

It is worth to mention that we performed all computations using Maple software. Moreover, in our computations, it is considered that images and images.

15.4.1 Test Problem 1

Let us consider the following inverse problem as the first test example

(15.54)equation

where images and the exact solution is given as follows:

(15.55)equation
(15.56)equation

This equation is solved in [1] by Mohebbi and Abbasi using a compact finite difference scheme. Here, we solve this equation by considering images in our computations for this example. In Table 15.1, the obtained absolute error of presented numerical algorithm for unknown function is compared with the compact finite difference scheme, which is presented in [1]. Additionally, the comparison between the absolute errors for source parameter is presented in Table 15.2. It is concluded from Tables 15.1 and 15.2 that the proposed method is more accurate than finite difference type solvers.

Table 15.1 A comparison between absolute error obtained by the methods of [1] and the presented method for images in test problem 1.

images Compact [1] Present method Present method
images = 100, images = 40 images = 10, images = 18 images = 10, images = 25
0.1 images images images
0.2 images images images
0.3 images images images
0.4 images images images
0.5 images images images
0.6 images images images
0.7 images images images
0.8 images images images
0.9 images images images

Table 15.2 A comparison between absolute error obtained by the methods of [1] and the presented method for images in test problem 1.

images Exact images Compact [1] Present method Present method
images = 100, images = 40 images = 10, images = 18 images = 10, images = 25
0.1 1.01 images images images
0.2 1.04 images images images
0.3 1.09 images images images
0.4 1.16 images images images
0.5 1.25 images images images
0.6 1.36 images images images
0.7 1.49 images images images
0.8 1.64 images images images
0.9 1.81 images images images
1.0 2.00 images images images

As can be seen in Tables 15.1 and 15.2, the obtained results for the proposed method are extremely more accurate than the finite difference method, which is presented in [1]. Another aspect of the proposed method, which should be checked, is stability. Figure 15.2 shows the stability of the presented algorithm. In Figure 15.2, the absolute errors of proposed method by using different numbers of collocation points for the first test problem with various percentage of noisy condition are presented, which show the stability of the mentioned method. As shown in Figure 15.2, the accuracy of the proposed method is about images for noisy data, which is acceptable accuracy for this problem. We can conclude from Figure 15.2 that the presented approach is stable for different numbers of collocation points and different temporal steps.

Image described by caption.

Figure 15.2 Obtaining errors in approximating images, for test problem 1 with different percents of noise when (a) images, (b) images, (c) images, (d) images, (e) images, and (f) images.

Moreover, convergence of the algorithm is very significant. Figure 15.3 is presented to show the convergence of the suggested numerical method. In Figure 15.3, we plot the residual error for this example by using different numbers of collocation points in Figure 15.3a, and on the other hand, we plot the 2 norm of error function for different values of collocation points in Figure 15.3b. As can be seen from Figure 15.3, the order of convergence of the algorithm is exponential.

Graph depicting the convergence of a suggested numerical method, plotting the residual error by using different
numbers of collocation points, exponentially.

Figure 15.3 The graph of (a) images and (b) images for the various numbers of collocation points for test problem 1.

15.4.2 Test Problem 2

In the second example, consider the following one‐dimensional inverse problem:

(15.57)equation

where images, and the exact solution is given as follows:

(15.58)equation
(15.59)equation

Similar to the previous example, this problem is solved in [1] by compact finite difference method. Here, by applying the presented numerical algorithm with images on this example, we obtain high‐accuracy results, which are presented in Tables 15.3 and 15.4. In Table 15.3, the obtained absolute error of the proposed method for an unknown function is compared with a compact finite difference results [1]; in addition, the comparison between the obtained absolute error of presented method and the method of [1] for evaluating the source parameter is provided in Table 15.4. As can be seen in Tables 15.3 and 15.4, we can obtain a more accurate approximated solution by using less spatial points and time steps.

Table 15.3 A comparison between absolute error obtained by the methods of [1] and the presented method for images in test problem 2.

images Compact [1] Present method Present method
images = 1000, images = 50 images = 10, images = 15 images = 10, images = 20
0.1 images images images
0.2 images images images
0.3 images images images
0.4 images images images
0.5 0 0 0
0.6 images images images
0.7 images images images
0.8 images images images
0.9 images images images

Table 15.4 A comparison between absolute error obtained by the methods of [1] and the presented method for images in test problem 2.

images Exact images Compact [1] Present method Present method
images = 1000, images = 50 images = 10, images = 15 images = 10, images = 25
0.1 images images images images
0.2 images images images images
0.3 images images images images
0.4 images images images images
0.5 images images images images
0.6 images images images images
0.7 images images images images
0.8 images images images images
0.9 images images images images

Similar to test problem 1, we test the stability of the proposed algorithm by adding some noise to the overspecified condition. For a different percent of noise, we apply the proposed method by various numbers of collocation points. The obtained results are presented in Figure 15.4, which shows the stability of the provided approach.

Image described by caption.

Figure 15.4 Obtaining errors in approximating images, for test problem 2 with a different percent of noise when (a) images, (b) images, (c) images, (d) images, (e) images, and (f) images.

Moreover, the convergence of provided method is shown in Figure 15.5. The process of decreasing the value of error and residual function is shown in Figure 15.5, which means, by increasing the number of collocation points, the residual and error of approximated solution decrease in the exponential sense.

Graph depicting the process of decreasing the value of
error and residual function; by increasing the number of collocation points, the residual and error of approximated solution decrease in the exponential sense.

Figure 15.5 The graph of (a) images and (b) images for the various numbers of collocation points for test problem 2.

15.4.3 Test Problem 3

In the third example, we consider the following two‐dimensional inverse problem:

(15.60)equation

in which images and its exact solution is :

(15.61)equation
(15.62)equation

Shivanian and Jafarabadi developed a spectral meshless method for solving this problem in [23]. In Table 15.5, the obtained results of the proposed method by using images are compared with the meshless method, which is presented in [23], in terms of images for unknown function and source parameter, in different numbers of collocation points and time steps. Table 15.5 shows that the presented algorithm is more accurate than meshless method, which is introduced in [23].

Table 15.5 A comparison between the presented method and the method of [23] in the sense of images for test problem 3.

Meshless [23] Present method Meshless [23] Present method
images images images images
images images images images images
images images images images images
images, images images images images images

As can be seen from Table 15.5, the presented method has the higher convergence rate in comparison with the method of [23]. The convergence of the presented method is illustrated in Figure 15.6. In addition to Table 15.5, for convergence examination of the suggested algorithm, the values of images and images, which are obtained by using different number collocation points, is shown in Figure 15.6.

Graph depicting the convergence of a method for a suggested algorithm 2, the values of which are obtained by using different numbers of  collocation points.

Figure 15.6 The graph of (a) images and (b) images for the various numbers of collocation points for test problem 3.

On the other hand, in order to show that this method is stable, the provided algorithm is tested by noisy data. The obtained results by adding different percent of noise to images is presented in Figure 15.7, in which the number of collocation points is images. Moreover, the obtained results by using 20 collocation points is presented in Figure 15.8. In both cases, the number of time steps is equal to 10.

Image described by caption.

Figure 15.7 Obtaining errors in approximating images, for test problem 3 with images and images by: (a) images noise, (b) images noise, and (c) images noise.

Image described by caption.

Figure 15.8 Obtaining errors in approximating images for test problem 3 with images and images by: (a) images noise, (b) images noise, and (c) images noise.

15.4.4 Test Problem 4

In the fourth example, we consider another two‐dimensional inverse problem as follows:

(15.63)equation

in which images and its exact solution is

(15.64)equation
(15.65)equation

This equation is solved by a splitting finite difference method in [16] and also by a greedy MLPG meshless method in [24]. The best obtained solution by the method of [16] is about images by using images time steps and 50 nodes for each spatial dimension. In Tables 15.6 and 15.7, the obtained results of proposed method by considering images is reported.

Table 15.6 Table of obtaining accuracy for unknown function by using images time steps in test problem 4.

images images images images images images
0.1 0.1 0.1 images images images
0.2 0.2 0.2 images images images
0.3 0.3 0.3 images images images
0.4 0.4 0.4 images images images
0.5 0.5 0.5 images images images
0.6 0.6 0.6 images images images
0.7 0.7 0.7 images images images
0.8 0.8 0.8 images images images
0.9 0.9 0.9 images images images

Table 15.7 Table of obtaining accuracy for discovering the source parameter by using images time steps in test problem 4.

images Exact images images = 5 images = 10 images = 15
0.1 2.894 829 081 924 352 375 188 292 173 51 images images images
0.2 2.778 597 241 839 830 166 078 928 005 36 images images images
0.3 2.650 141 192 423 996 896 016 255 686 67 images images images
0.4 2.508 175 302 358 729 682 175 147 047 16 images images images
0.5 2.351 278 729 299 871 853 151 349 212 19 images images images
0.6 2.177 881 199 609 491 025 124 632 331 84 images images images
0.7 1.986 247 292 529 523 478 375 450 611 42 images images images
0.8 1.774 459 071 507 532 395 420 462 468 60 images images images
0.9 1.540 396 888 843 050 336 199 873 436 40 images images images

In addition to high accuracy of the algorithm, the convergence of the presented method can be conclude from Tables 15.6 and 15.7. Moreover, in order to show the convergence of the presented algorithm, the values of images and images, which are obtained by using different numbers of collocation points, are shown in Figure 15.9.

Graph depicting the convergence of an algorithm, the values of which are obtained using different numbers of collocation points.

Figure 15.9 The graph of (a) images and (b) images for the various numbers of collocation points for test problem 4.

On the other hand, in order to show that the proposed method is stable, the provided algorithm is tested by noisy data. The obtained results by adding various percents of noise to images are presented in Figure 15.10, in which the number of collocation points is images. Moreover, the obtained results by using 20 collocation points are presented in Figure 15.11.

Image described by caption.

Figure 15.10 Obtaining errors in approximating images for test problem 4 with images and images by: (a) images noise, (b) images noise, and (c) images noise.

Image described by caption.

Figure 15.11 Obtaining errors in approximating images for test problem 4 with images and images by: (a) images noise, (b) images noise, and (c) images noise.

15.4.5 Test Problem 5

As the final example, the following three‐dimensional inverse problem is considered as fifth test example:

(15.66)equation

where images, and it has the following exact solution:

(15.67)equation
(15.68)equation

Dehgan solved this equation in [13] by a seven‐point finite difference scheme. The best accuracy, which is reported in [13], is images for images and images for images, while it is using images time steps and 50 nodes for each spatial dimension. In this example, the computations are done by considering images. The images is a measure of comparison between the presented algorithm and method of [13], which is discussed in Table 15.8.

Table 15.8 Table of obtaining results for different time steps and different numbers of collocation points for test problem 5 in the sense of images.

images images
images, images 0.3375 0.5656
images, images images images
images, images images images

As can be seen in Table 15.8, the presented method can obtain more accurate results by much less than time steps and collocation pints for spatial dimensions in comparison with finite difference method of [13]. Moreover, Table 15.8 yields the convergence of the algorithm because by increasing the time steps and collocation points, the infinite norm of error decreased. On the other hand, the stability of presented algorithm is tested by adding some percent of noise to overspecified condition. The results of adding the noise to overspecified condition are illustrated in Figures 15.1215.14.

Image described by caption.

Figure 15.12 Obtaining errors in approximating images for test problem 5 with images, images, and images noise when (a) images, (b) images, and (c) images.

Image described by caption.

Figure 15.13 Obtaining errors in approximating images, for test problem 5 with images, images, and images noise when (a) images, (b) images, and (c) images.

Image described by caption.

Figure 15.14 Obtaining errors in approximating images for test problem 5 with images, images, and images noise when (a) images, (b) images, and (c) images.

15.5 Conclusion

In this chapter, a multidimensional inverse problem of finding a source parameter is considered, and the connection between the brain activity reconstruction and evaluating a source parameter in an inverse problem is illustrated. On the other hand, in order to approximate the solution of this inverse problem, a finite difference/pseudo‐spectral method was developed. In the presented method, a finite difference scheme is used to discretize the temporal dimension, and a pseudo‐spectral scheme based on collocation algorithm is used to approximate the spatial dimensions. We choose a images scheme weighted for the finite different method. Moreover, in order to function approximation, the first kind of Chebyshev polynomials are selected as basis functions. The presented algorithm is tested for convergence and stability. In order to test the convergence of the algorithm, we show that by increasing the amount of basis functions and collocation points, the value of residual is decreased. On the other hand, in order to show the stability of the presented method, some noises are added to the overspecified condition and the obtained numerical results are reported. In addition, computational experiments confirmed the high accuracy of the proposed method in comparison with other method such as meshless methods, finite difference methods, and other relevant methods in the literature. Another important point of the presented algorithm is discovering both linear and nonlinear source parameters strongly.

References

  1. 1 Mohebbi, A. and Abbasi, M. (2015). A fourth‐order compact difference scheme for the parabolic inverse problem with an overspecification at a point. Inverse Problems in Science and Engineering 23 (3): 457–478.
  2. 2 Aster, R.C., Borchers, B., and Thurber, C.H. (2018). Parameter Estimation and Inverse Problems. Elsevier.
  3. 3 Cannon, J.R. and Lin, Y. (1990). An inverse problem of finding a parameter in a semi‐linear heat equation. Journal of Mathematical Analysis and Applications 145 (2): 470–484.
  4. 4 MacBain, J.A. and Bednar, J.B. (1986). Existence and uniqueness properties for the one‐dimensional magnetotellurics inversion problem. Journal of Mathematical Physics 27 (2): 645–649.
  5. 5 Deng, Z.C., Yang, L., Yu, J.N., and Luo, G.W. (2009). An inverse problem of identifying the coefficient in a nonlinear parabolic equation. Nonlinear Analysis: Theory Methods & Applications 71 (12): 6212–6221.
  6. 6 Cannon, J.R., Lin, Y., and Wang, S. (1992). Determination of source parameter in parabolic equations. Meccanica 27 (2): 85–94.
  7. 7 Cannon, J.R., Lin, Y., and Xu, S. (1994). Numerical procedures for the determination of an unknown coefficient in semi‐linear parabolic differential equations. Inverse Problems 10 (2): 227.
  8. 8 Liu, S. and Triggiani, R. (2011). Global uniqueness and stability in determining the damping and potential coefficients of an inverse hyperbolic problem. Nonlinear Analysis: Real World Applications 12 (3): 1562–1590.
  9. 9 Prilepko, A.I. and Solovev, V.V. (1987). Solvability of the inverse boundary‐value problem of finding a coefficient of a lower‐order derivative in a parabolic equation. Differential Equations 23 (1): 101–107.
  10. 10 Dehghan, M. (2000). Finite difference schemes for two‐dimensional parabolic inverse problem with temperature overspecification. International Journal of Computer Mathematics 75 (3): 339–349.
  11. 11 Dehghan, M. (2001). Implicit solution of a two‐dimensional parabolic inverse problem with temperature overspecification. Journal of Computational Analysis and Applications 3 (4): 383–398.
  12. 12 Dehghan, M. (2003). Identifying a control function in two‐dimensional parabolic inverse problems. Applied Mathematics and Computation 143 (2–3): 375–391.
  13. 13 Dehghan, M. (2003). Determination of a control function in three‐dimensional parabolic equations. Mathematics and Computers in Simulation 61 (2): 89–100.
  14. 14 Mohebbi, A. and Dehghan, M. (2010). High‐order scheme for determination of a control parameter in an inverse problem from the over‐specified data. Computer Physics Communications 181 (12): 1947–1954.
  15. 15 Dehghan, M. (2005). Parameter determination in a partial differential equation from the overspecified data. Mathematical and Computer Modelling 41 (2–3): 196–213.
  16. 16 Daoud, D.S. and Subasi, D. (2005). A splitting up algorithm for the determination of the control parameter in multi dimensional parabolic problem. Applied Mathematics and Computation 166 (3): 584–595.
  17. 17 Fatullayev, A.G. and Cula, S. (2009). An iterative procedure for determining an unknown spacewise‐dependent coefficient in a parabolic equation. Applied Mathematics Letters 22 (7): 1033–1037.
  18. 18 Hadian Rasanan, A.H. and Rad, J.A. (2019). A computational source modeling of brain activity: an inverse problem. Neurodevelopmental Cognition 1 (1): 69–78.
  19. 19 Dehghan, M. and Tatari, M. (2006). Determination of a control parameter in a one‐dimensional parabolic equation using the method of radial basis functions. Mathematical and Computer Modelling 44 (11–12): 1160–1168.
  20. 20 Dehghan, M. and Tatari, M. (2007). The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data. Numerical Methods for Partial Differential Equations: An International Journal 23 (5): 984–997.
  21. 21 Cheng, R. (2008). Determination of a control parameter in a one‐dimensional parabolic equation using the moving least‐square approximation. International Journal of Computer Mathematics 85 (9): 1363–1373.
  22. 22 Parzlivand, F. and Shahrezaee, A. (2016). The use of inverse quadratic radial basis functions for the solution of an inverse heat problem. Bulletin of the Iranian Mathematical Society 42 (5): 1127–1142.
  23. 23 Shivanian, E. and Jafarabadi, A. (2018). An inverse problem of identifying the control function in two and three‐dimensional parabolic equations through the spectral meshless radial point interpolation. Applied Mathematics and Computation 325: 82–101.
  24. 24 Takhtabnoos, F. and Shirzadi, A. (2018). A greedy MLPG method for identifying a control parameter in 2D parabolic PDEs. Inverse Problems in Science and Engineering 26 (11): 1676–1694.
  25. 25 Dehghan, M. and Saadatmandi, A. (2006). A tau method for the one‐dimensional parabolic inverse problem subject to temperature overspecification. Computers & Mathematics with Applications 52 (6–7): 933–940.
  26. 26 Abdelkawy, M.A., Ahmed, E.A., and Sanchez, P. (2015). A method based on legendre pseudo‐spectral approximations for solving inverse problems of parabolic types equations. Mathematical Sciences Letters 4 (1): 81–90.
  27. 27 Shamsi, M. and Dehghan, M. (2012). Determination of a control function in three‐dimensional parabolic equations by legendre pseudospectral method. Numerical Methods for Partial Differential Equations 28 (1): 74–93.
  28. 28 Dehghan, M., Yousefi, S.A., and Rashedi, K. (2013). Ritz–Galerkin method for solving an inverse heat conduction problem with a nonlinear source term via bernstein multi‐scaling functions and cubic B‐spline functions. Inverse Problems in Science and Engineering 21 (3): 500–523.
  29. 29 Khorshidi, M.N. and Yousefi, S.A. (2016). Ritz‐least squares method for finding a control parameter in a one‐dimensional parabolic inverse problem. Journal of Applied Analysis 22 (2): 169–179.
  30. 30 Lakestani, M. and Dehghan, M. (2010). The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time‐dependent coefficient subject to an extra measurement. Journal of Computational and Applied Mathematics 235 (3): 669–678.
  31. 31 Ashpazzadeh, E., Lakestani, M., and Razzaghi, M. (2017). Cardinal Hermite interpolant multiscaling functions for solving a parabolic inverse problem. Turkish Journal of Mathematics 41 (4): 1009–1026.
  32. 32 Molai, A.A. and Houlari, T. (2013). Resolution of an inverse parabolic problem using Sinc‐Galërkin method. TWMS Journal of Applied and Engineering Mathematics 3 (2): 160–181.
  33. 33 Zolfaghari, R. (2013). Parameter determination in a parabolic inverse problem in general dimensions. Computational Methods for Differential Equations 1 (1): 55–70.
  34. 34 Yousefi, S.A. (2009). Finding a control parameter in a one‐dimensional parabolic inverse problem by using the Bernstein Galerkin method. Inverse Problems in Science and Engineering 17 (6): 821–828.
  35. 35 Lakestani, M. and Dehghan, M. (2008). A new technique for solution of a parabolic inverse problem. Kybernetes 37 (2): 352–364.
  36. 36 Gholampoor, I. and Kajani, M.T. (2019). A direct numerical method for approximate solution of inverse reaction diffusion equation via two‐dimensional Legendre hybrid functions. Numerical Algorithms 1–18. https://doi.org/10.1007/s11075-019-00691-0.
  37. 37 Tatari, M. and Dehghan, M. (2007). Identifying a control function in parabolic partial differential equations from overspecified boundary data. Computers & Mathematics with Applications 53 (12): 1933–1942.
  38. 38 Tatari, M. and Dehghan, M. (2007). He's variational iteration method for computing a control parameter in a semi‐linear inverse parabolic equation. Chaos, Solitons & Fractals 33 (2): 671–677.
  39. 39 Wang, W., Han, B., and Yamamoto, M. (2013). Inverse heat problem of determining time‐dependent source parameter in reproducing kernel space. Nonlinear Analysis: Real World Applications 14 (1): 875–887.
  40. 40 Mohammadi, M., Mokhtari, R., and Isfahani, F.T. (2014). Solving an inverse problem for a parabolic equation with a nonlocal boundary condition in the reproducing kernel space. Iranian Journal of Numerical Analysis and Optimization 4 (1): 57–76.
  41. 41 Khader, M.M. (2017). Analytical solution for determination the control parameter in the inverse parabolic equation using HAM. Applications and Applied Mathematics: An International Journal 12 (2): 1072–1087.
  42. 42 Rostamian, M. and Shahrezaee, A. (2016). Numerical solution of a semi‐linear inverse parabolic problem via HPM. International Journal of Computing Science and Mathematics 7 (4): 330–339.
  43. 43 Dehghan, M. and Shakeri, F. (2009). Method of lines solutions of the parabolic inverse problem with an overspecification at a point. Numerical Algorithms 50 (4): 417–437.
  44. 44 Deng, Z.C. and Yang, L. (2011). An inverse problem of identifying the coefficient of first‐order in a degenerate parabolic equation. Journal of Computational and Applied Mathematics 235 (15): 4404–4417.
  45. 45 Parand, K., Rezaei, A.R., and Taghavi, A. (2010). Numerical approximations for population growth model by rational Chebyshev and Hermite functions collocation approach: a comparison. Mathematical Methods in the Applied Sciences 33 (17): 2076–2086.
  46. 46 Parand, K. and Rad, J.A. (2012). Numerical solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via collocation method based on radial basis functions. Applied Mathematics and Computation 218 (9): 5292–5309.
  47. 47 Parand, K. and Delkhosh, M. (2017). Accurate solution of the Thomas–Fermi equation using the fractional order of rational Cchebyshev functions. Journal of Computational and Applied Mathematics 317: 624–642.
  48. 48 Parand, K., Lotfi, Y., and Rad, J.A. (2017). An accurate numerical analysis of the laminar two‐dimensional flow of an incompressible Eyring‐Powell fluid over a linear stretching sheet. The European Physical Journal Plus 132 (9): 397.
  49. 49 Baseri, A., Abbasbandy, S., and Babolian, E. (2018). A collocation method for fractional diffusion equation in a long time with Chebyshev functions. Applied Mathematics and Computation 322: 55–65.
  50. 50 Agarwal, P. and El‐Sayed, A.A. (2018). Non‐standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation. Physica A: Statistical Mechanics and Its Applications 500: 40–49.
  51. 51 Boyd, J.P. (2001). Chebyshev and Fourier Spectral Methods. Courier Corporation.
  52. 52 Boyd, J.P. (2011). Chebyshev spectral methods and the Lane‐Emden problem. Numerical Mathematics: Theory Methods and Applications 4 (2): 142–157.
  53. 53 Babolian, E., Fattahzadeh, F., and Raboky, E.G. (2007). A Chebyshev approximation for solving nonlinear integral equations of Hammerstein type. Applied Mathematics and Computation 189 (1): 641–646.
  54. 54 Parand, K. and Razzaghi, M. (2004). Rational Chebyshev tau method for solving Volterra's population model. Applied Mathematics and Computation 149 (3): 893–900.
  55. 55 Parand, K., Shahini, M., and Taghavi, A. (2009). Generalized Laguerre polynomials and rational Chebyshev collocation method for solving unsteady gas equation. International Journal of Contemporary Mathematical Sciences 4 (21): 1005–1011.
  56. 56 Parand, K. and Shahini, M. (2010). Rational Chebyshev collocation method for solving nonlinear ordinary differential equations of Lane‐Emden type. International Journal of Information and Systems Sciences 6: 72–83.
  57. 57 Olmos, D. and Shizgal, B.D. (2009). Pseudospectral method of solution of the Fitzhugh–Nagumo equation. Mathematics and Computers in Simulation 79 (7): 2258–2278.
  58. 58 Parand, K., Moayeri, M.M., Latifi, S., and Rad, J.A. (2019). Numerical study of a multidimensional dynamic quantum model arising in cognitive psychology especially in decision making. The European Physical Journal Plus 134 (3): 109–126.
  59. 59 Yüzbaşı, Ş. (2012). A numerical approach to solve the model for HIV infection of CD4T cells. Applied Mathematical Modelling 36 (12): 5876–5890.
  60. 60 Parand, K., Moayeri, M.M., Latifi, S., and Delkhosh, M. (2017). A numerical investigation of the boundary layer flow of an eyring‐powell fluid over a stretching sheet via rational Chebyshev functions. The European Physical Journal Plus 132 (7): 325–336.
  61. 61 Parand, K., Latifi, S., Moayeri, M.M., and Delkhosh, M. (2018). Generalized lagrange Jacobi Gauss‐Lobatto (GLJGL) collocation method for solving linear and nonlinear Fokker‐Planck equations. Communications in Theoretical Physics 69 (5): 519–531.
  62. 62 Galerkin, B.G. (1915). Vestnik inzhenerov i tekhnikovi. Tech 19: 897–908.
  63. 63 Delkhosh, M. and Parand, K. (2019). Generalized pseudospectral method: theory and applications. Journal of Computational Science 34: 11–32.
  64. 64 Delkhosh, M., Parand, K., and Hadian‐Rasanan, A.H. (2019). A development of lagrange interpolation, Part I: Theory. arXiv preprint arXiv:1904.12145.
  65. 65 Ozer, S., Chen, C.H., and Cirpan, H.A. (2011). A set of new Chebyshev kernel functions for support vector machine pattern classification. Pattern Recognition 44 (7): 1435–1447.
  66. 66 Parodi, M. and Gómez, J.C. (2014). Legendre polynomials based feature extraction for online signature verification. Consistency analysis of feature combinations. Pattern Recognition 47 (1): 128–140.
  67. 67 Benouini, R., Batioua, I., Zenkouar, K. et al. (2019). New set of generalized legendre moment invariants for pattern recognition. Pattern Recognition Letters 123: 39–46.
  68. 68 Moghaddam, V.H. and Hamidzadeh, J. (2016). New Hermite orthogonal polynomial kernel and combined kernels in support vector machine classifier. Pattern Recognition 60: 921–935.
  69. 69 Padierna, L.C., Carpio, M., Rojas‐Domínguez, A. et al. (2018). A novel formulation of orthogonal polynomial kernel functions for SVM classifiers: the Gegenbauer family. Pattern Recognition 84: 211–225.
  70. 70 Benouini, R., Batioua, I., Zenkouar, K. et al. (2019). Fractional‐order orthogonal Chebyshev moments and moment invariants for image representation and pattern recognition. Pattern Recognition 86: 332–343.
  71. 71 Robledo, F., Romero, P., and Sartor, P. (2013). A novel interpolation technique to address the edge‐reliability problem. In: 2013 5th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 187–192. IEEE.
  72. 72 Ben‐Yu, G. (1998). Spectral Methods and Their Applications. World Scientific.
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