Noniterative Localized and Space-Time Localized RBF Meshless Method to Solve the Ill-Posed and Inverse Problem

In many references, both the ill-posed and inverse boundary value problems are solved iteratively. The iterative procedures are based on firstly converting the problem into a well-posed one by assuming the missing boundary values. Then, the problem is solved by using either a developed numerical algorithm or a conventional optimization scheme. The convergence of the technique is achieved when the approximated solution is well compared to the unused data. In the present paper, we present a different way to solve an ill-posed problem by applying the localized and space-time localized radial basis function collocation method depending on the problem considered and avoiding the iterative procedure. We demonstrate that the solution of certain ill-posed and inverse problems can be accomplished without iterations. Three different problems have been investigated: problems with missing boundary condition and internal data, problems with overspecified boundary condition, and backward heat conduction problem (BHCP). It has been demonstrated that the presented method is efficient and accurate and overcomes the stability analysis that is required in iterative techniques.


Introduction
In contrast to the stationary and nonstationary direct boundary value problems, ill-posed problems are characterized by unknown boundary conditions on a part of the boundary. An example is the problem of determining the temperature and the heat flux on the whole boundary or on its part, where the temperature and the heat flux are prescribed in selected points located inside the domain of the considered problem. Another statement of ill-posed problems is the one referred to as the final boundary value problem or backward heat conduction problem (BHCP). The problem is characterized by the unknown initial condition value. The temperature distribution and the heat flux are investigated from the known data which can be the temperature distribution at particular time t = t f > 0. From this data, the question arises as to whether the temperature distribution at any earlier time t < t f can be retrieved.
Since the solution of the BHCPs does not continuously depend on the given final data, it shows some difficulties to be solved using classical methods. So, many iterative schemes have been developed during the last decade. Some of them have been proposed by Kozlov and Maz'ya [1], Mera et al. [2], and Jourhmane et al. [3]. Some other methods based on the BEM, regularization techniques, and FSM cited in [4][5][6][7][8][9] are applied. For recently developed methods, we can mention the work developed by Ma et al. [10]. They transform the problem into an optimization one and use a conjugate gradient method to solve the inverse problem.
The investigations of the meshless method based on radial basis functions (RBFs) have seen many developments. For BHCP, we can mention the meshless method developed by Li et al. [11] based on the RBF method for the nonhomogeneous backward heat conduction problem. Beside the first work done by Cheng and Cabral [12] using global RBF to solve Poisson problems, we can cite the recent work published by Li et al. [13] in which they presented a stable local meshless collocation method based on CS-RBFs for solving certain inverse problems. Gu et al. [14] have also proposed a meshless singular boundary method for three-dimensional inverse heat conduction problems in general anisotropic media. Wang et al. introduced a stable and accurate meshless method based on collocation and radial basis functions to solve the inverse wave problem [15]. Compared to other methods, no iterative algorithm is needed in their new developed method. Furthermore, they addressed their noniterative method to identify the initial conditions [16] and boundary condition [17] arising in the inverse wave problem.
In this paper, a local mesh-free method based on RBFs for solving ill-posed problems was presented. For the first case, we solve a Poisson problem in the two-dimensional domain with missing boundary conditions in a part of the boundary. The two types of examples considered are a problem with missing boundary condition and internal data and a problem with overspecified boundary condition. And for the second case, we solve a nonstationary backward heat conduction problem (BHCP) characterized by the final condition using the space-time localized RBF collocation method. The technique is based on transforming the parabolic system from a d-dimensional problem into a ðd + 1Þ-dimensional one without distinguishing between the space and time variables. The collocation points have both the space and time coordinates. The (BHCP) parabolic equation is then solved by using the governing domain equation as condition on the boundary of missing condition, characterized by ft = 0g for BHCP. An advanced feature of our approach is that we solve the problem in one step for the tree treated examples without any iterative scheme. Another novelty is to use the spacetime approach for (BHCP)example and no time integration method is used. The present method is expected to be free of disadvantages related to the loss of stability of solutions due to the iteration schemes. In our approach, the problem is considered a well-posed one and the algebraic system solved is square for all cases. Results of numerical simulations given in the present paper show that the method is stable and efficient. Note that the two-dimensional nonstationary problems can be solved using the same approach.

Mathematical Formulation of the Problems
In this section, we give a brief description of the models of the ill-posed boundary value problem considered in this work. The first treated problem, characterized by missing boundary condition and internal data, has the following form: The internal data for this problem is described by uðx, ℓ int Þ = g 1 ðxÞwith ℓ int < ℓ y and x ∈ 0, ℓ x ½. No boundary condition is taken on the fourth side fy = ℓ y ; 0 < x < ℓ x g. Figure 1 describes the domain considered with the boundary and internal conditions. The second problem is a stationary heat problem with missing boundary condition and overspecified boundary condition on one part of the boundary; this problem has the following form: Figure 2 describes the domain considered with the boundary conditions. For these two problems, g 1 ðxÞ, g 2 ðxÞ, h 1 ðxÞ, and h 2 ðxÞ are known functions.
The third treated problem is a typical example of an inverse and ill-posed problem of parabolic equation representing the heat phenomena. The problem is given by the following system: where f 0 , f 1 , and g 1 are functions that describe the boundary and initial conditions, respectively, and t f is a given positive value of the final time and a is a positive number. The boundary temperatures f 0 and f 1 and the final temperature g 1 are known while the initial temperature uðx, 0Þ is unknown and has to be determined. This is usually referred to as the final boundary value problem or the backward heat conduction problem (BHCP). This problem is easily transformed into an initial boundary value problem by a simple variable change. The domain considered with boundary conditions is illustrated in Figure 3.

Localized and Space-Time Localized RBF Collocation Method for the Inverse and Ill-Posed Problem
As one of the investigated inverse problems is the time evolutionary partial differential equation, a methodology based on space-time problem formulation is needed. In this section, we describe the space-time problem transformation and localized and space-time localized radial basis function (RBF) collocation methods. The mathematical formulation 2 Modelling and Simulation in Engineering of the two applied methods are the same with respect to the fact that the space-time localized RBF collocation method (ST-LRBFCM) is applied to the evolutionary problem in a space-time domain by combining a space variable and time variable in one vector variable, where the second one, localized radial basis function method (LRBFCM), is applied to the independent time problem defined in the space domain. It can be then remarked that the localized RBF method is just a variant of the space-time localized RBF method. Based on that, only the space-time localized RBF method is reviewed in this section.

Space-Time Localized RBF Method for the Well-Posed
Problem. In the presented formulation, the radial basis function is formulated by taking into account both the spatial and time variables to construct the center points. The d-dimensional space evolving problem is then transformed into a ðd + 1Þ-dimensional problem and solved in a space-time variable [18]. To apply the technique in the case of a given and Ω × ft = t f g, we require to formulate the boundary conditions of the new formulated system of equations: for the equation in the space-time domain Ω × 0, t f ½, and on ∂Ω × ½0, t f and Ω × ft = t f g, respectively. As the problem is still ill-posed for the space-time domain since it needs a boundary condition on Ω × ft = 0g, Equation (4) can be considered a boundary condition on Ω × ft = 0g: The new formulation leads to a complete problem in the space-time variable domain. Then, it can be solved by applying the localized RBFs collocation method described below, which gives us the approximate solution at any point ðx, tÞ (see [18] and the next section for more details).

Localized RBF Method for the Well-Posed
Problem. Let us consider the following boundary value problem: where   Modelling and Simulation in Engineering (3) and Ω ′ = Ω in the case of the stationary problem (1) or (2) and L and B are the given linear domain and boundary differential operators, respectively.
To recall the technique, let fx j g N j=1 ∈ Ω ′ ∪ ∂Ω ′ be center points; for any point x s ∈ Ω′ ∪ ∂Ω′, a localized influence domain Ω s is created (see Figure 4). It contains n s nodal points fx ½s k g n s k=1 , including x s . Following the method of particular solutions (MPS) [20,21], the solution uðx s Þ can be approximated in Ω s by a linear combination of n s radial basis functions in the following form: where fα k g n s k=1 are undetermined coefficients and k·k is the Euclidean norm. Using Equation (9) and collocating at all fx ½s k g n s k=1 ⊂ Ω s , we get the following system: where and α ½s = ½α 1 , α 2 , ⋯, α n s T .From Equation (10), α ½s can be written as follows: For x s ∈ Ω s , we apply the differential operator L to Equation (9) to obtain the following equation: where Ξ s = BΦ ½s ðΦ ½s Þ −1 and Ξ is the expansion of Ξ ½s by adding zeros. By substituting Equation (12) into Equation (7) for x s ∈ Ω ′ and Equation (13) into Equation (8) for x s ∈ ∂Ω ′ , we obtain the following equations: By collocating all the interpolation points fx j g N j=1 using Equation (14), we get the following sparse linear system: where  Note that the linear algebraic system is square since the number of unknowns (the values of the approximate function) and the collocation points are equal. In the next section, the application of such methods to the ill-posed and inverse problem will be demonstrated.

Localized RBF Method for the Ill-Posed and Inverse
Problem. Firstly, we start by explaining the application of the method to the ill-posed problems (1) and (2). As given in the section above, the most important part of these problems is defined by

Modelling and Simulation in Engineering
We can then remark that on the boundary part defined by 0, ℓ x ½× fℓ y g, no condition is defined. To solve the problem, more data or boundary condition is needed. For problem (1), potential values at M interior points uðx j , y j Þ, j = 1, ⋯, M, are given. They are defined by uðx, ℓ int Þ = g 1 ðxÞ in system (1). And, for problem (2), another prescribed boundary condition on the part of the boundary 0, ℓ x ½ × f0g is given and defined by Generally, these PDEs have the following form: The domain Ω is a subset of ℝ d with a boundary ∂Ω = Γ 1 ∪ Γ 3 , Γ 1 ∩ Γ 3 = ∅. No boundary condition is given on Γ 3 . Γ 2 can be a part of ∂Ω or even Γ 2 ⊂ _ Ω (interior data). Problem (19) is ill-posed when Γ 2 ⊂ Ω or Γ 2 ⊂ ∂Ω and B 1 ≠ B 2 . The operators B 1 and B 2 can be a Dirichlet, Neumann, or mixed boundary condition. B 2 can be the identity operator when h 2 is an interior data.
To solve numerically this problem, we choose N distinct nodes fx j g N j=1 ∈ Ω ∪ ∂Ω as centers with N = N d + N b . N d and N b are the numbers of centers in Ω and on the boundary ∂Ω, respectively. For the collocation nodes, we take fx j g N d j=1 ∈ Ω, fx j g N 1 j=N d +1 ∈ Γ 1 , and N 2 = N − N 1 nodes fx k g k ∈ Γ 2 . Then, we apply (14) to system (19) for each node in Ω or ∂Ω (depending on the operator used). The obtained algebraic system is square since card ðΓ 2 Þ = card ðΓ 3 Þ (the number of centers in Γ 3 is equal to the number of collocation nodes in Γ 2 ). Finally, we get an algebraic system similar to system (15). For the last problem (BHCP), it is solved exactly as shown in Sections 3.1 and 3.2.

Numerical Results and Discussions
In this work, we consider two kinds of ill-posed heat problem. The first one, described in Figures 1 and 2, is a two-dimensional stationary heat equation defined by where a condition on a part of the boundary is not known. Two examples of this first type are considered, one with missing boundary condition and internal data and the second with overspecified boundary condition on a part of the boundary. The second kind of problem is a one-dimensional nonstationary heat problem given by with unknown initial condition. Figure 3 describes the considered example. We should mention that the first problem is solved using the localized RBF method and the second one by the space-time localized RBF method. Throughout this section, n s denotes the number of neighboring points in an influence domain for the used localized collocation method. The numbers N x , N y , and N t are the numbers of partitions in each axis used to generate the total number of interpolation points N. The parameter ϵ is either the shape parameter of the well-known multiquadric radial basis function φðrÞ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + ε 2 r 2 p or the inverse multiquadric function φðrÞ = 1/ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + ε 2 r 2 p . The determination of the optimal value of ϵ is still an open subject.
To measure the numerical accuracy, we consider the maximum absolute error (MAE), the root mean squared error (RMSE), and the L 1 er relative error defined as follows: where uðx j Þ andûðx j Þ are the exact and approximate solutions at the nodex j , respectively. We considerx j = ðx j , y j Þ for the 2D space examples andx j = ðx j , t j Þ for the BHCP example.
The impact of a number of parameters on the accuracy of the numerical solutions, such as the number of local nodes, the shape parameter of RBFs, and the total number of collocation nodes being used, is also investigated. In the first two examples, we solve the inverse heat problem under the form Δu = 0 in the two-dimensional domain.
We solve the problem without using any boundary condition on 0, ℓ½ × fy = ℓg and even without collocating the governing equation on its nodal points as it has been done by Cheng and Cabral [12]. So, the algebraic matrix obtained is square. The performance and robustness of this technique will be investigated using the localized RBF collocating method with different radial basis functions such as MQ and IMQ.
To conduct numerical experiments, we take ℓ = 1 and n s = 9. Tables 1 and 2 show the numerical results obtained for some ϵ, N x , and N y using the MQ and IMQ radial basis functions. Figure 5 shows the absolute error of the problem using the MQ function with ϵ = 0:02 and the IMQ function with ϵ = 0:08 and taking N = N x × N y = 11 × 11. It can be remarked that accurate results are obtained in all different cases.

Problems with Overspecified Boundary Condition.
This second ill-posed problem of steady-state heat conduction described below is solved in a square domain Ω = ½0, 1 2 as it was considered by Lesnic [23]. The schematic diagram of the computational square domain is shown in Figure 2. The anticipated exact solution of the problem is uðx, yÞ = cos ðxÞ cosh ðyÞ + sin ðxÞ sinh ðyÞ. In our situation, we assume that the boundary condition is missing on one side of the domain and the condition of the temperature is prescribed as Dirichlet condition on the three sides of the boundary as uðx, 0Þ = cos ðxÞ, uð0, yÞ = cosh ðyÞ, and uð1, yÞ = cos ð1Þ cosh ðyÞ + sin ð1Þ sinh ðyÞ. To formulate the problem as an ill-posed one, we assume that no boundary condition is given on the fourth part of the boundary described by {y = 1 ; 0 < x < 1} and an overspecified Neumann condition is given on the side {y = 0 ; 0 < x < 1}. For numerical tests, we take n s = 9 and chose different values of ϵ and various numbers of N x and N y . Tables 3 and 4 show that accurate results are obtained for different radial basis functions used. Figure 6 depicts the absolute errors in the entire domain of the problem using the MQ with ϵ = 0:44 and the IMQ with ϵ = 0:36 and taking N = 60 × 60 and N = 50 × 50, respectively.

Backward Heat Conduction Problem (BHCP).
In this section, we solve the backward heat condition problem given by Equation (3). This is an example of an ill-posed problem which is difficult to solve using classical numerical methods. This problem has been intensively discussed by Mera et al. [2] and by Jourhmane and Mera [24]. It has been shown in [2] that the matrix of the algebraic system obtained using BEM is severely ill-conditioned. Jourhmane and Mera have then developed an iterative scheme to deal with the illconditioning. Their technique is based on a sequence of solutions of well-posed forward heat conduction problems. To discuss the feasibility of our proposed technique, we consider   the example treated in [2,23,24] for which the analytical solution is given by uðx, tÞ = sin ðπxÞe −π 2 t . Following by setting the data f 0 and f 1 to be zero, it can be remarked that for any large t f , the information given by gðxÞ = sin ðπxÞe −π 2 t f are very weak since g approach zero and become smaller than the desired initial condition u 0 ðxÞ = sin ðπxÞ. Then, the problem shows some difficulties to be solved using some comment techniques. In our case, we show that this problem will be overtaken by using the space-time localized RBF method to solve the problem in the domain ½0, 1 × ½0, t f . For this numerical simulation, the obtained results are presented for the MQ-RBF and IMQ-RBF functions and n s = 13. The number of nodes on the x-axis and t-axis are chosen N x = N t = 40. In Tables 5 and 6  We remark that even for the big values of t f such as 0.5, 0.75, and 1, the RMSE is of the order 10 −4 . It has also been shown that for small values of t f , the same accurate results are obtained using less than 40 nodes on the t-axis.

Backward Heat Conduction Problem with Noisy Data.
Following Jourhmane and Mera [24], we furthermore investigate the sensitivity of the numerical solution with respect to the noisy boundary data. For that, we assume that the given function g is perturbed by small α and replaced by g + α, where α is a Gaussian random variable with mean zero and standard deviation σ = max jgjðs/100Þ. s is the percentage of additive noise included in the input data g. Figure 8 shows

Conclusion
In this paper, we presented a localized and space-time localized RBF collocation meshless method to solve the ill-posed and inverse problems in the same way that the well-posed problem is solved and without any iteration method. For the nonstationary problem, we adopted the new space-time localized collocation approach and the problem is solved by the same way as for the localized collocation approach for the stationary case. The results presented show that the method is efficient and gives an alternative of the iteration methods without losing the stability due to iterations. We note that the global RBF method was already used to solve this kind of problem resulting in a rectangular algebraic system [12]. As a further work, we extend the application of the local and space-time local methods for solving the ill-posed problems in higher dimensions and to the nonlinear problems. The numerical algorithms to determinate a good shape parameter will be also investigated.