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

Authors: Mohammed Hamaidi; Ahmed Naji; Fatima Ghafrani; Mostafa Jourhmane
Journal: Modelling and Simulation in Engineering
 (2020)

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## Abstract

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.


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## Body

## 1. 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 timet=tf>0. From this data, the question arises as to whether the temperature distribution at any earlier time t<tf 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 [4–9] 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 ad-dimensional problem into a d+1-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 t=0 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 space-time 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.


## 2. 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:(1)Δux,y=fx,y,x,y∈Ω=0,ℓx×0,ℓy,ux,0=g2x,x∈0,ℓx,u0,y=h1y,y∈0,ℓy,uℓx,y=h2y,y∈0,ℓy,ux,ℓint=g1x,x∈0,ℓx.The internal data for this problem is described byux,ℓint=g1xwith ℓint<ℓy and x∈0,ℓx. No boundary condition is taken on the fourth side y=ℓy;0<x<ℓx. Figure 1 describes the domain considered with the boundary and internal conditions.Figure 1
Domain for the first stationary heat problem.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:(2)Δux,y=fx,y,x,y∈Ω=0,ℓx×0,ℓy,∂u∂xx,0=g1x,x∈0,ℓx,ux,0=g2x,x∈0,ℓx,u0,y=h1y,y∈0,ℓy,uℓx,y=h2y,y∈0,ℓy.Figure2 describes the domain considered with the boundary conditions. For these two problems, g1x, g2x, h1x, and h2x are known functions.Figure 2
Domains for the second stationary heat problem.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:(3)∂u∂tx,t−a∂2u∂x2x,t=0,∀x,t∈0,1×x,tf,u0,t=f0t,∀t∈0,tf,u1,t=f1t,∀t∈0,tf,ux,tf=g1x,∀x∈0,1,where f0, f1, and g1 are functions that describe the boundary and initial conditions, respectively, and tf is a given positive value of the final time and a is a positive number. The boundary temperatures f0 and f1 and the final temperature g1 are known while the initial temperature ux,0 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.Figure 3
Domain and boundary conditions for the BHCP.


## 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 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.


### 3.1. 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. Thed-dimensional space evolving problem is then transformed into a d+1-dimensional problem and solved in a space-time variable [18]. To apply the technique in the case of a given (BHCP) problem (3) in the space-time domain Ωtf=Ω×0,tf with a boundary defined by ∂Ω×0,tf, Ω×t=0, and Ω×t=tf, we require to formulate the boundary conditions of the new formulated system of equations:
(4)∂u∂tx,t+Lux,t=fx,t,for the equation in the space-time domain Ω×0,tf, and
(5)Bux,t=gx,t,ux,t=utfx,on ∂Ω×0,tf and Ω×t=tf, respectively. As the problem is still ill-posed for the space-time domain since it needs a boundary condition on Ω×t=0, Equation (4) can be considered a boundary condition on Ω×t=0.(6)∂u∂tx,t+Lux,t=fx,tonΩ×t=0.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 pointx,t (see [18] and the next section for more details).


### 3.2. Localized RBF Method for the Well-Posed Problem

Let us consider the following boundary value problem:(7)Lux=fx,x∈Ω′,(8)Bux=gx,x∈∂Ω′,where Ω′=Ωtfif dealing with the time-dependent problem (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, letxjj=1N∈Ω′∪∂Ω′ be center points; for any point xs∈Ω′∪∂Ω′, a localized influence domain Ωs is created (see Figure 4). It contains ns nodal points xksk=1ns, including xs. Following the method of particular solutions (MPS) [20, 21], the solution uxs can be approximated in Ωs by a linear combination of ns radial basis functions in the following form:
(9)uxs≃ûxs=∑k=1nsαkΦxs−xks,where αkk=1ns are undetermined coefficients and · is the Euclidean norm. Using Equation (9) and collocating at all xksk=1ns⊂Ωs, we get the following system:
(10)ûs=Φsαs,where Φs=Φxis−xjs1≤i,j≤ns, ûs=ûx1s,⋯,ûxnssT, and αs=α1,α2,⋯,αnsT.From Equation (10), αs can be written as follows:
(11)αs=Φs−1ûs.Figure 4
Schematic showing the five-node, nine-node, and thirteen-node local domainsΩs for 2D (a) and the nine-node local domain for 3D (b).
(a)(b)Forxs∈Ωs, we apply the differential operator L to Equation (9) to obtain the following equation:
(12)Lûxs=∑k=1nsαkLΦxs−xks=LΦαs=Λsûs=Λû,where û=ux1,ux2,⋯,uxN and Λs=ΘsΦs−1. Note that by adding zeros at the proper locations based on the mapping of ûs to û, Λ1×N is the global expansion of Λ1×nss.Similarly, forxs∈∂Ω′, an influence domain Ωs containing xs will be created. Then, we have
(13)Bûxs=∑k=1nsαkBΦxs−xks=BΦsαs=Ξsûs=Ξû,where Ξs=BΦsΦs−1 and Ξ is the expansion of Ξs by adding zeros.By substituting Equation (12) into Equation (7) for xs∈Ω′ and Equation (13) into Equation (8) for xs∈∂Ω′, we obtain the following equations:
(14)fxs=Lûxs=Λxsû,gxs=Bûxs=Ξxsû.By collocating all the interpolation pointsxjj=1N using Equation (14), we get the following sparse linear system:
(15)MU=F,where
(16)M=Λx1,Λx2,⋯,ΛxNi,ΞxNi+1,⋯,ΞxNT,U=ûx1,ûx2,⋯,ûxNi,ûxNi+1,⋯,ûxNT,F=fx1,fx2,⋯,fxNi,gxNi+1,⋯,gxNT.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.


### 3.3. 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
(17)Δux,y=fx,y,x,y∈Ω=0,ℓx×0,ℓy,ux,0=g2x,x∈0,ℓx,u0,y=h1y,y∈0,ℓy,uℓx,y=h2y,y∈0,ℓy.We can then remark that on the boundary part defined by]0,ℓx[×ℓy, no condition is defined. To solve the problem, more data or boundary condition is needed. For problem (1), potential values at M interior points uxj,yj, j=1,⋯,M, are given. They are defined by ux,ℓint=g1x in system (1). And, for problem (2), another prescribed boundary condition on the part of the boundary 0,ℓx×0 is given and defined by
(18)∂u∂xx,0=g1x.Generally, these PDEs have the following form:(19)Lux=fx,x∈Ω,B1ux=h1x,x∈Γ1,B2ux=h2x,x∈Γ2.The domainΩ is a subset of ℝd with a boundary ∂Ω=Γ1∪Γ3, Γ1∩Γ3=∅. No boundary condition is given on Γ3. Γ2 can be a part of ∂Ω or even Γ2⊂Ω̇ (interior data). Problem (19) is ill-posed when Γ2⊂Ω or Γ2⊂∂Ω and B1≠B2. The operators B1 and B2 can be a Dirichlet, Neumann, or mixed boundary condition. B2 can be the identity operator when h2 is an interior data.To solve numerically this problem, we chooseN distinct nodes xjj=1N∈Ω∪∂Ω as centers with N=Nd+Nb. Nd and Nb are the numbers of centers in Ω and on the boundary ∂Ω, respectively. For the collocation nodes, we take xjj=1Nd∈Ω, xjj=Nd+1N1∈Γ1, and N2=N−N1 nodes xkk∈Γ2. 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 cardΓ2=cardΓ3 (the number of centers in Γ3 is equal to the number of collocation nodes in Γ2). 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.


## 3.1. 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. Thed-dimensional space evolving problem is then transformed into a d+1-dimensional problem and solved in a space-time variable [18]. To apply the technique in the case of a given (BHCP) problem (3) in the space-time domain Ωtf=Ω×0,tf with a boundary defined by ∂Ω×0,tf, Ω×t=0, and Ω×t=tf, we require to formulate the boundary conditions of the new formulated system of equations:
(4)∂u∂tx,t+Lux,t=fx,t,for the equation in the space-time domain Ω×0,tf, and
(5)Bux,t=gx,t,ux,t=utfx,on ∂Ω×0,tf and Ω×t=tf, respectively. As the problem is still ill-posed for the space-time domain since it needs a boundary condition on Ω×t=0, Equation (4) can be considered a boundary condition on Ω×t=0.(6)∂u∂tx,t+Lux,t=fx,tonΩ×t=0.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 pointx,t (see [18] and the next section for more details).


## 3.2. Localized RBF Method for the Well-Posed Problem

Let us consider the following boundary value problem:(7)Lux=fx,x∈Ω′,(8)Bux=gx,x∈∂Ω′,where Ω′=Ωtfif dealing with the time-dependent problem (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, letxjj=1N∈Ω′∪∂Ω′ be center points; for any point xs∈Ω′∪∂Ω′, a localized influence domain Ωs is created (see Figure 4). It contains ns nodal points xksk=1ns, including xs. Following the method of particular solutions (MPS) [20, 21], the solution uxs can be approximated in Ωs by a linear combination of ns radial basis functions in the following form:
(9)uxs≃ûxs=∑k=1nsαkΦxs−xks,where αkk=1ns are undetermined coefficients and · is the Euclidean norm. Using Equation (9) and collocating at all xksk=1ns⊂Ωs, we get the following system:
(10)ûs=Φsαs,where Φs=Φxis−xjs1≤i,j≤ns, ûs=ûx1s,⋯,ûxnssT, and αs=α1,α2,⋯,αnsT.From Equation (10), αs can be written as follows:
(11)αs=Φs−1ûs.Figure 4
Schematic showing the five-node, nine-node, and thirteen-node local domainsΩs for 2D (a) and the nine-node local domain for 3D (b).
(a)(b)Forxs∈Ωs, we apply the differential operator L to Equation (9) to obtain the following equation:
(12)Lûxs=∑k=1nsαkLΦxs−xks=LΦαs=Λsûs=Λû,where û=ux1,ux2,⋯,uxN and Λs=ΘsΦs−1. Note that by adding zeros at the proper locations based on the mapping of ûs to û, Λ1×N is the global expansion of Λ1×nss.Similarly, forxs∈∂Ω′, an influence domain Ωs containing xs will be created. Then, we have
(13)Bûxs=∑k=1nsαkBΦxs−xks=BΦsαs=Ξsûs=Ξû,where Ξs=BΦsΦs−1 and Ξ is the expansion of Ξs by adding zeros.By substituting Equation (12) into Equation (7) for xs∈Ω′ and Equation (13) into Equation (8) for xs∈∂Ω′, we obtain the following equations:
(14)fxs=Lûxs=Λxsû,gxs=Bûxs=Ξxsû.By collocating all the interpolation pointsxjj=1N using Equation (14), we get the following sparse linear system:
(15)MU=F,where
(16)M=Λx1,Λx2,⋯,ΛxNi,ΞxNi+1,⋯,ΞxNT,U=ûx1,ûx2,⋯,ûxNi,ûxNi+1,⋯,ûxNT,F=fx1,fx2,⋯,fxNi,gxNi+1,⋯,gxNT.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.


## 3.3. 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
(17)Δux,y=fx,y,x,y∈Ω=0,ℓx×0,ℓy,ux,0=g2x,x∈0,ℓx,u0,y=h1y,y∈0,ℓy,uℓx,y=h2y,y∈0,ℓy.We can then remark that on the boundary part defined by]0,ℓx[×ℓy, no condition is defined. To solve the problem, more data or boundary condition is needed. For problem (1), potential values at M interior points uxj,yj, j=1,⋯,M, are given. They are defined by ux,ℓint=g1x in system (1). And, for problem (2), another prescribed boundary condition on the part of the boundary 0,ℓx×0 is given and defined by
(18)∂u∂xx,0=g1x.Generally, these PDEs have the following form:(19)Lux=fx,x∈Ω,B1ux=h1x,x∈Γ1,B2ux=h2x,x∈Γ2.The domainΩ is a subset of ℝd with a boundary ∂Ω=Γ1∪Γ3, Γ1∩Γ3=∅. No boundary condition is given on Γ3. Γ2 can be a part of ∂Ω or even Γ2⊂Ω̇ (interior data). Problem (19) is ill-posed when Γ2⊂Ω or Γ2⊂∂Ω and B1≠B2. The operators B1 and B2 can be a Dirichlet, Neumann, or mixed boundary condition. B2 can be the identity operator when h2 is an interior data.To solve numerically this problem, we chooseN distinct nodes xjj=1N∈Ω∪∂Ω as centers with N=Nd+Nb. Nd and Nb are the numbers of centers in Ω and on the boundary ∂Ω, respectively. For the collocation nodes, we take xjj=1Nd∈Ω, xjj=Nd+1N1∈Γ1, and N2=N−N1 nodes xkk∈Γ2. 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 cardΓ2=cardΓ3 (the number of centers in Γ3 is equal to the number of collocation nodes in Γ2). 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.


## 4. Numerical Results and Discussions

In this work, we consider two kinds of ill-posed heat problem. The first one, described in Figures1 and 2, is a two-dimensional stationary heat equation defined by
(20)Δux,y=0,x,y∈Ω⊂ℝ2,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(21)∂u∂tx,t−a∂2u∂x2x,t=fx,t,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,ns denotes the number of neighboring points in an influence domain for the used localized collocation method. The numbers Nx, Ny, and Nt 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 φr=1+ε2r2 or the inverse multiquadric function φr=1/1+ε2r2. 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 theLer1 relative error defined as follows:
(22)MAE=max1≤j≤Nûxĵ−uxĵ,RMSE=1N∑j=1Nûxĵ−uxĵ2,Ler1=∑j=1Nûxĵ−uxĵ∑j=1Nuxĵ,where uxĵ and ûxĵ are the exact and approximate solutions at the node xĵ, respectively. We consider xĵ=xj,yj for the 2D space examples and xĵ=xj,tj 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 formΔu=0 in the two-dimensional domain.


### 4.1. Problems with Missing Boundary Condition and Internal Data

First, we start by the following ill-posed problem of potential flow investigated by Onishi [22] on a square domain 0,ℓ2. Based on its exact solution given by ux,y=x2−y2, Dirichlet conditions are prescribed on three sides of the boundary, ux,0=x2, u0,y=−y2, and uℓ,y=ℓ2−y2. No boundary condition is given on the fourth side {y=ℓ;0<x<l}. In the interior, the potential values are known for y=ℓ/2; ux,ℓ/2=x2−ℓ2/4.We solve the problem without using any boundary condition on0,ℓ×y=ℓ 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 takeℓ=1 and ns=9. Tables 1 and 2 show the numerical results obtained for some ϵ, Nx, and Ny 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=Nx×Ny=11×11. It can be remarked that accurate results are obtained in all different cases.Table 1
Errors for example4.1 for some values of Nx, Ny, and ϵ using MQ.
ϵNxNyMAERMSELer10.0211119.88E−051.50E−051.73E−050.0421213.37E−044.34E−054.45E−050.0931314.74E−045.76E−057.25E−05Table 2
Errors for example4.1 for some values of Nx, Ny, and ϵ using IMQ.
ϵNxNyMAERMSELer10.0811111.79E−043.55E−054.52E−050.0221216.19E−047.73E−051.02E−040.0431312.83E−042.97E−054.40E−050.0941415.29E−049.32E−051.42E−04Figure 5
MAE for example4.1 using MQ with ϵ=0.02 (a) and IMQ with ϵ=0.08 (b) with N=11×11.
(a)(b)


### 4.2. Problems with Overspecified Boundary Condition

This second ill-posed problem of steady-state heat conduction described below is solved in a square domainΩ=0,12 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 ux,y=cosxcoshy+sinxsinhy. 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 ux,0=cosx, u0,y=coshy, and u1,y=cos1coshy+sin1sinhy. 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 ns=9 and chose different values of ϵ and various numbers of Nx and Ny. 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.Table 3
Errors for example4.2 for different values of Nx, Ny, and ϵ using MQ.
ϵNxNyMAERMSELer10.1110102.98E−037.07E−043.36E−040.2620201.68E−032.29E−047.49E−050.2430305.43E−048.46E−053.47E−050.5540402.12E−033.50E−041.22E−040.3950501.05E−031.24E−043.80E−050.4460602.67E−042.99E−058.94E−06Table 4
Errors for example4.2 for different values of Nx, Ny, and ϵ using IMQ.
ϵNxNyMAERMSELer10.1010103.64E−038.81E−043.92E−040.2020205.46E−041.07E−045.12E−050.3630302.23E−032.63E−047.70E−050.4340403.19E−034.59E−041.42E−040.3650506.46E−049.28E−053.20E−050.3060602.40E−033.32E−041.13E−04Figure 6
MAE for example4.2 using MQ with ϵ=0.44 for N=60×60 (a) and IMQ with ϵ=0.36 for N=50×50 (b).
(a)(b)


### 4.3. 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 ill-conditioning. 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 ux,t=sinπxe−π2t. Following by setting the data f0 and f1 to be zero, it can be remarked that for any large tf, the information given by gx=sinπxe−π2tf are very weak since g approach zero and become smaller than the desired initial condition u0x=sinπx. 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 0,1×0,tf. For this numerical simulation, the obtained results are presented for the MQ-RBF and IMQ-RBF functions and ns=13. The number of nodes on the x-axis and t-axis are chosen Nx=Nt=40. In Tables 5 and 6, we show the errors obtained at different time tf and for different values of shape parameter ϵ.Table 5
Errors for BHCP example4.3 for different values of tf using MQ.
tfϵMAERMSELer10.100.936.53E−041.10E−041.69E−040.200.357.90E−041.77E−044.69E−040.501.2555.83E−047.16E−051.90E−040.751.1644.47E−049.68E−054.65E−041.000.8231.59E−032.73E−041.62E−03Table 6
Errors for BHCP example4.3 for different values of tfusing IMQ.
tfϵMAERMSELer10.101.1082.25E−045.98E−058.85E−050.201.0732.92E−047.39E−051.52E−040.501.0835.61E−041.06E−043.82E−040.750.9268.62E−042.09E−041.18E−031.000.9132.58E−033.20E−041.46E−03Figure7 further demonstrates the accuracy of the MAE error at t=0 for tf=1 and in the entire space-time domain. We remark that even for the big values of tf 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 tf, the same accurate results are obtained using less than 40 nodes on the t-axis.Figure 7
MAE for example4.3 using MQ with ϵ=0.823, N=40×40, and tf=1.
(a)(b)


### 4.4. 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=maxgs/100. s is the percentage of additive noise included in the input data g. Figure 8 shows the numerical obtained approximate solution of initial temperature ux,0 for tf=0.1 and different values of noise s=0%, 1%, 2%, 3%. From these numerical results, we can mention that as the percentage of additive noise s decreases, the numerical solution approximates better the exact initial solution. The same remark has been declared in [24].Figure 8
The numerical solution for the initial temperatureux,0 obtained for tf=0.1 with various levels of noise added into the input data for example 4.3.


## 4.1. Problems with Missing Boundary Condition and Internal Data

First, we start by the following ill-posed problem of potential flow investigated by Onishi [22] on a square domain 0,ℓ2. Based on its exact solution given by ux,y=x2−y2, Dirichlet conditions are prescribed on three sides of the boundary, ux,0=x2, u0,y=−y2, and uℓ,y=ℓ2−y2. No boundary condition is given on the fourth side {y=ℓ;0<x<l}. In the interior, the potential values are known for y=ℓ/2; ux,ℓ/2=x2−ℓ2/4.We solve the problem without using any boundary condition on0,ℓ×y=ℓ 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 takeℓ=1 and ns=9. Tables 1 and 2 show the numerical results obtained for some ϵ, Nx, and Ny 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=Nx×Ny=11×11. It can be remarked that accurate results are obtained in all different cases.Table 1
Errors for example4.1 for some values of Nx, Ny, and ϵ using MQ.
ϵNxNyMAERMSELer10.0211119.88E−051.50E−051.73E−050.0421213.37E−044.34E−054.45E−050.0931314.74E−045.76E−057.25E−05Table 2
Errors for example4.1 for some values of Nx, Ny, and ϵ using IMQ.
ϵNxNyMAERMSELer10.0811111.79E−043.55E−054.52E−050.0221216.19E−047.73E−051.02E−040.0431312.83E−042.97E−054.40E−050.0941415.29E−049.32E−051.42E−04Figure 5
MAE for example4.1 using MQ with ϵ=0.02 (a) and IMQ with ϵ=0.08 (b) with N=11×11.
(a)(b)


## 4.2. Problems with Overspecified Boundary Condition

This second ill-posed problem of steady-state heat conduction described below is solved in a square domainΩ=0,12 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 ux,y=cosxcoshy+sinxsinhy. 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 ux,0=cosx, u0,y=coshy, and u1,y=cos1coshy+sin1sinhy. 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 ns=9 and chose different values of ϵ and various numbers of Nx and Ny. 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.Table 3
Errors for example4.2 for different values of Nx, Ny, and ϵ using MQ.
ϵNxNyMAERMSELer10.1110102.98E−037.07E−043.36E−040.2620201.68E−032.29E−047.49E−050.2430305.43E−048.46E−053.47E−050.5540402.12E−033.50E−041.22E−040.3950501.05E−031.24E−043.80E−050.4460602.67E−042.99E−058.94E−06Table 4
Errors for example4.2 for different values of Nx, Ny, and ϵ using IMQ.
ϵNxNyMAERMSELer10.1010103.64E−038.81E−043.92E−040.2020205.46E−041.07E−045.12E−050.3630302.23E−032.63E−047.70E−050.4340403.19E−034.59E−041.42E−040.3650506.46E−049.28E−053.20E−050.3060602.40E−033.32E−041.13E−04Figure 6
MAE for example4.2 using MQ with ϵ=0.44 for N=60×60 (a) and IMQ with ϵ=0.36 for N=50×50 (b).
(a)(b)


## 4.3. 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 ill-conditioning. 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 ux,t=sinπxe−π2t. Following by setting the data f0 and f1 to be zero, it can be remarked that for any large tf, the information given by gx=sinπxe−π2tf are very weak since g approach zero and become smaller than the desired initial condition u0x=sinπx. 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 0,1×0,tf. For this numerical simulation, the obtained results are presented for the MQ-RBF and IMQ-RBF functions and ns=13. The number of nodes on the x-axis and t-axis are chosen Nx=Nt=40. In Tables 5 and 6, we show the errors obtained at different time tf and for different values of shape parameter ϵ.Table 5
Errors for BHCP example4.3 for different values of tf using MQ.
tfϵMAERMSELer10.100.936.53E−041.10E−041.69E−040.200.357.90E−041.77E−044.69E−040.501.2555.83E−047.16E−051.90E−040.751.1644.47E−049.68E−054.65E−041.000.8231.59E−032.73E−041.62E−03Table 6
Errors for BHCP example4.3 for different values of tfusing IMQ.
tfϵMAERMSELer10.101.1082.25E−045.98E−058.85E−050.201.0732.92E−047.39E−051.52E−040.501.0835.61E−041.06E−043.82E−040.750.9268.62E−042.09E−041.18E−031.000.9132.58E−033.20E−041.46E−03Figure7 further demonstrates the accuracy of the MAE error at t=0 for tf=1 and in the entire space-time domain. We remark that even for the big values of tf 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 tf, the same accurate results are obtained using less than 40 nodes on the t-axis.Figure 7
MAE for example4.3 using MQ with ϵ=0.823, N=40×40, and tf=1.
(a)(b)


## 4.4. 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=maxgs/100. s is the percentage of additive noise included in the input data g. Figure 8 shows the numerical obtained approximate solution of initial temperature ux,0 for tf=0.1 and different values of noise s=0%, 1%, 2%, 3%. From these numerical results, we can mention that as the percentage of additive noise s decreases, the numerical solution approximates better the exact initial solution. The same remark has been declared in [24].Figure 8
The numerical solution for the initial temperatureux,0 obtained for tf=0.1 with various levels of noise added into the input data for example 4.3.


## 5. 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.



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