Wenyan WANGMasahiro YAMAMOTOBo HAN

1 Introduction

Inverse source identification problems are important in many branches of engineering sciences.For example,an accurate estimation of pollutant source is crucial to environmental safeguard in cities with high populations.Taking more and more important roles in the migration of groundwater,identification and control of pollution source and environmental protection(see[1]),the inverse source problems attracted much attention and have been studied by many authors(see[2–13]).

We consider the following inverse source problem of determining a pair offunctionsw(x,t)andp(x)satisfying

and the overdetermination condition

where Ω is a bounded domain in Rnwith smooth boundary∂Ω,g(x,t)andh(x)are known and sufficiently smooth functions,r(x)is measurement data,andOn the right-hand side of(1.1),p(x)g(x,t)is interpreted as a heat source.Also,in the modeling of air pollution phenomena,p(x)g(x,t)is considered as the source pollutant.

The existence and uniqueness of the solutions to this inverse problem are discussed in[11–13].Various methods(see[14–20])are developed for this inverse source problem and related inverse parabolic problems.In this paper,we use a reproducing kernel method to obtain the analytical solutions.

In recent years,there are broader interests in the use of reproducing kernels for the solutions to diverse inverse problems(see[21–22]).Those papers indicate that the reproducing kernel method(RKM)(see[23])has many outstanding advantages.The most important advantages of RKM are as follows:

(i)The approximate solutions and their derivatives can be proved to converge uniformly to the exact solutions and their derivatives.

(ii)The structure of numerical programming is simple and the calculations are very fast.

In this paper,we represent an exact solution to problem(1.1)–(1.4)in a reproducing kernel space,and improve the existing methods as follows:First,we obtain reproducing kernel spaces by re-defining the inner products appropriately,and our reproducing kernel is simpler than[24].Numerical calculations indicate that the method using our reproducing kernel can improve the precision and decrease the runtime,compared to the case where we use the reproducing kernel in[24];second,this approach reduces problem(1.1)–(1.4)to a system of linear equations,and avoids the Gram-Schmidt orthogonalization process which is needed in[25].Numerical calculations indicate that our method decreases the runtime,compared to the method using Gram-Schmidt orthogonalization process in[25].

Before applying our method to problem(1.1)–(1.4),we first transform(1.1)to an equation which is easy to solve by using the RKM.For simplicity,we take Ω=[0,1]×[0,1]⊂R2andT=1.

From(1.1)and(1.4),we obtain

Substituting(1.5)into(1.1)yields

Again we set

Then we can further transform the original problem to

Here we set

In our method,the datumr(x,y)with∆r(x,y)is involved in(1.7).Thus we have to assume thatr(x,y)∈C2(Ω).This can be proved by the classical result for an initial-boundary value problem if we assume suitable smoothness assumptions ong(x,y,t),h(x,y)and compatibility conditions.In this paper,we omit the detailed proofs.In Section 4,we add noises inL∞(Ω)tor(x,y)and test the robustness of our method with Tikhonov regularization for reconstructingp(x,y).

We apply our numerical method to problem(1.7)–(1.10).Our method is composed of:

(i)Solving(1.7)–(1.10)which is an initial-boundary value problem for a non-classical heat equation.

(ii)Findingp(x,y)by(1.5)–(1.6).

This paper is organized as follows:In Section 2,we construct reproducing kernel spaces according to(1.7)–(1.10).In Section 3,we give the exact and approximate solutions in the reproducing kernel space and show the convergence.In Section 4,numerical tests are done and we can conclude that our method is robust against errors.Finally a conclusion is given in Section 5.

2 Several Reproducing Kernel Spaces

In this section,referring to[23],we construct the reproducing kernel spaceW(4,4,3)(D)according to(1.7)–(1.10),which gives a simpler method than[24].Henceforth“·′”denotes the derivative in the variable under consideration.

First we define the reproducing kernel spacesWk[0,1],k=4,3,2.

We setW4[0,1]={u|u,u′,u′′andu(3)are absolutely continuous real-valued functions in[0,1],u(4)∈L2[0,1],u(0)=0,u(1)=0},and we define the inner product inW4[0,1]by

Next we setW3[0,1]={u|u,u′andu′′are absolutely continuous real-valued functions in[0,1],u(3)∈L2[0,1],u(0)=0},and we define the inner product inW3[0,1]by

We setW2[0,1]={u|uandu′are absolutely continuous real-valued functions in[0,1],u′′∈L2[0,1]},and we define the inner product inW2[0,1]by

We define the norms byfork=4,3,2.We can prove thatW4[0,1],W3[0,1],W2[0,1]are reproducing kernel spaces and the reproducing kernelsR1,R2,R3 are defined respectively by(2.1)–(2.3)in[26].

Moreover we assume thatare respectively the orthonormal bases of the reproducing kernel spacesW4[0,1]andW3[0,1].Then we defineW(4,4,3)(D)by

The inner product ofW(4,4,3)(D)is defined dy

whereandThe norm is denoted by

According to[23],we can obtain the following theorem.

Theorem 2.1W(4,4,3)(D)is a reproducing kernel space and its reproducing kernel is given by

where R1(·,·)and R2(·,·)are respectively the reproducing kernel of W4[0,1]and W3[0,1].

Similarly toW(4,4,3)(D),we can define the reproducing kernel spaceW(2,2,2)(D).It is easy to show that its reproducing kernel is

whereR3(·,·)is the reproducing kernel ofW2[0,1].

3 The Solution of(1.7)–(1.10)

In this section,the exact solution of problem(1.7)–(1.10)is given in the reproducing kernel spaceW(4,4,3)(D).

We choose a countable dense subset(xi,yi,ti)∈D,i=1,2,3,···.Put

whereis the reproducing kernel ofW(2,2,2)(D)andL∗is the formal adjoint operator ofL:for eachHere,noting thatbyby the Sobolev embedding,we see thatL:W(4,4,3)(D)→W(2,2,2)(D)is a bounded linear operator.LetK(x,ξ,y,ζ,t,η)be the reproducing kernel ofW(4,4,3)(D).

We show the following Theorems 3.1–3.2.

Theorem 3.1Assume the uniqueness for the inverse problem(1.1)–(1.4).Then the systemis complete in W(4,4,3)(D),and

We construct an orthonormal systemofW(4,4,3)(D)by the Gram-Schmidt orthogonalization process to

Theorem 3.2A unique solution to(1.7)–(1.10)is expressed by

in W(4,4,3)(D).

The proofs of the theorems are given in appendix and similar to those in[26].

Next we will find approximate solutionsunin the form of

which is then-term truncated Fourier series of the exact solutionuin(1.7)–(1.10).

Theorem 3.3If u is the solution to(1.7)–(1.10)and un=Pnu,where Pnis the orthogonalprojection from W(4,4,3)toSpanthen Lun(xi,yi,ti)=f(xi,yi,ti),i=1,2,···,n.

Since

the proof of Theorem 3.3 is seen.

On the other hand,we have

Herewhich is verified as follows.

From(3.2),we have

LetThen we obtain

Thus the proofis completed.

Then,from Theorem 3.3,we have

Thus,from(3.4),we can obtainCi,i=1,2,···,n.Then we take them into(3.3)and obtain the approximate solutionun(x,y,t)to(1.7)–(1.10).Finally we may obtain the approximation of(w(x,y,t),p(x,y))to the original inverse problem from(1.5)–(1.6).

Using(3.3)and(3.4)to solve(1.7)–(1.10),we can avoid the Gram-Schmidt orthogonalization process in[25]ofThus we can improve the precision and considerably decrease the runtime,compared to the method using the Gram-Schmidt orthogonalization process in[25].Our method is efficiently applied for solving some model problems,and is of high precision.

We conclude this section with the convergence of the approximate solutionsun.The exact solution and then-term approximation solution to(1.7)–(1.10)are respectively denoted byuandun.Similarly to[27],we can obtain the following theorem.

Theorem 3.4Assume that u∈W(4,4,3)(D).Then

(i)Moreover the sequenceis monotonicallydecreasing in n.

(ii)

for i,j=0,1,2,k=0,1,i+j+k=0,1,2.

4 Numerical Examples

In this section,the numerical examples are studied to demonstrate that our method is effective and the accuracy of approximate solution is high.All computations are performed by Mathematica 5.0.

The domainDis divided intom1×m2×m3meshes with the step sizein thetdirection,and the step sizesandin thexandydirections,respectively,in whichm1,m2,m3∈N.

Example 4.1Consider problem(1.1)–(1.4)with the following conditions:

The exact solution is

w(x,y,t)=exp(t)sin(x)sin(1−x)sin(y)sin(1−y)

and

With gridm1×m2×m3=5×5×5,the absolute errors ofw(x,y,t)andp(x,y)are respectively presented in Tables 1–2.Also the root mean square(RMS,for short)errors ofw(x,y,t)andp(x,y),and CPU time are given in Table 3.

In the reproducing kernel method computations,the Gram-Schmidt orthogonal step is not included in the CPU time.We note that only the proof needs the Gram-Schmidt orthogonal step,but the numerical computations do not depend on such a step(see(3.3)–(3.4)).

Table 1 Absolute errors of w(x,y,t)for Example 4.1

Table 2 Absolute errors of p(x,y)for Example 4.1

Table 3 RMS errors of w(x,y,t)and p(x,y),and CPU time for Example 4.1

The above results show that as the step sizes decrease,the precision is improved.

In addition,in order to test the robustness of our method,uniform random noise creates noisy data byr(x,y)+d×random(x,y),where random(x,y)describes a uniform random function with the range[0,1]×[0,1].The figures 1–4 show the results with gridm1×m2×m3=5×5×5 and the noise factorsd=0.1,0.2,0.3.Our numerical method needs the second derivatives ofr,but our available data with noise are not inH2(Ω)in general andL∞(Ω)-noises may cause instability.Therefore we apply the Tikhonov regularization for the stabilized reconstruction.

Figure 1 Error|w(x,y,0.1)−w125(x,y,0.1)|for Example 4.1:(left)d=0.1,(middle)d=0.2,(right)d=0.3

Figure 2 Error|w(x,y,0.5)−w125(x,y,0.5)|for Example 4.1:(left)d=0.1,(middle)d=0.2,(right)d=0.3

Figure 3 Error|w(x,y,0.9)−w125(x,y,0.9)|for Example 4.1:(left)d=0.1,(middle)d=0.2,(right)d=0.3

From the figures,it can be seen that the results become worse as the noise factordbecomes larger,but even in the case of the largest noise level,i.e.,d=0.3,the absolute error|p−p25|is about 0.06,which can be considered as an acceptable numerical result.Thus,the figures confirm the robustness of our method for reconstructingp(x,y).

Figure 4 Error|p−p25|for Example 4.1:(left)d=0.1,(middle)d=0.2,(right)d=0.3

Example 4.2Consider problem(1.1)–(1.4)with the following conditions:

The exact solution isw(x,y,t)=exp(t)(1−x)ysin(x)sin(1−y)andp(x,y)=2(1−x)cos(1−y)sin(x)+y(2cos(x)+3(1−x)sin(x))sin(1−y).

With gridm1×m2×m3=5×5×5,the absolute errors ofw(x,y,t)andp(x,y)are respectively presented in Tables 4–5.Also RMS errors ofw(x,y,t)andp(x,y),and CPU time are given in Table 6.

Table 4 Absolute errors of w(x,y,t)for Example 4.2

From the above results,we see that as the step sizes decrease,the precision improves.

In addition,the results with gridm1×m2×m3=5×5×5 and the noise factorsd=0.1,0.2,0.3 are given in Figures 5–8.

From the figures,it can be seen that results become worse as noise factordbecomes larger,but even in the case whered=0.3,the absolute error|p−p25|is about 0.02.Thus,the figures confirm the robustness of our method of reconstructingp(x,y).

Figure 5 Error|w(x,y,0.1)−w125(x,y,0.1)|for Example 4.2:(left)d=0.1,(middle)d=0.2,(right)d=0.3

Figure 6 Error|w(x,y,0.5)−w125(x,y,0.5)|for Example 4.2:(left)d=0.1,(middle)d=0.2,(right)d=0.3

Figure 7 Error|w(x,y,0.9)−w125(x,y,0.9)|for Example 4.2:(left)d=0.1,(middle)d=0.2,(right)d=0.3

Figure 8 Error|p−p25|for Example 4.2:(left)d=0.1,(middle)d=0.2,(right)d=0.3

Table 5 Absolute errors of p(x,y)for Example 4.2

Table 6 RMS errors of w(x,y,t)and p(x,y),and CPU time for Example 4.2

5 Conclusions

In this article,our method has been successfully applied to a two-dimensional parabolic inverse source problem with the final overdetermination.Our method is based on the reproducing kernel Hilbert space,and improves some existing methods.The numerical results confirm that the accuracy of our method and the error of approximate solution are monotonically decreasing in the sense ofMoreover,our method is applicable to more general inverse source problem for parabolic equations,and we will discuss in a forthcoming paper.

6 The Proof of Theorems 3.1–3.2

6.1 The proof of Theorem 3.1

We have

Clearly,ψi∈W(4,4,3)(D).

For each fixedu∈W(4,4,3)(D),letwhich means that

Note thatis dense inD,and(Lu)(x,y,t)=0.It follows thatu≡0 by the uniqueness assumption for the inverse problem.Thus the proof of Theorem 3.1 is completed.

6.2 The proof of Theorem 3.2

Applying Theorem 3.1,we easily see thatis a complete orthonormal system ofW(4,4,3)(D).

Note thatfor eachv∈W(2,2,2)(D),and we have

The proof of the theorem is completed.

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