This paper discusses the inverse problem of determining an unknown source in a second order differential equation from measured final data. This problem is ill-posed; that is, the solution (if it exists) does not depend continuously on the data. In order to solve the considered problem, an iterative method is proposed. Using this method a regularized solution is constructed and an a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.
1. Introduction
Let H be a separable Hilbert space with the inner product (·,·) and the norm ·. Consider the problem of finding the source term f∈H in the following system: (1)u′′t+2Au′t+A2ut=f,0<t<T,u0=0,u′0=0,with the additional data(2)uT=g,where A:D(A)⊂H→H is a positive self-adjoint linear operator with a compact resolvent; we denote by σ(A) the spectrum of the operator A.
The problem (1) is an abstract version of the system(3)uttx,t-2Δutx,t+Δ2ux,t=fx,0<t<T,x∈Ω,ux,t=Δux,t=0,0≤t≤T,x∈∂Ω,ux,0=utx,0=0,x∈Ω,which arises in the mathematical study of structural damped vibrations of string or a beam [1–3]. Also this problem can be considered as a biparabolic problem in the abstract setting. For physical motivation we cite the biparabolic model proposed in [4] for more adequate mathematical description of heat and diffusion processes than the classical heat equation. For other models we refer the reader to [5–7].
For most classical partial differential equations, the reconstruction of source functions from the final data or a partial boundary data is an inverse problem with many applications in several branches of sciences and engineering, such as geophysical prospecting and pollutant detection [8–12].
The main difficulty of inverse source identification problems is that they are ill-posed, that is, even if a solution exists, it does not depend continuously on the data; in other words, small error in the data measurement can induce enormous error to the solution. Thus, special regularization methods that restore the stability with respect to measurements errors are needed. In the present work, we focus on an iterative method proposed by Kozlov and Maz’ya [13, 14] for solving the problem; it is based on solving a sequence of well-posed boundary value problems such that the sequence of solutions converges to the solution for the original problem. It has been successfully used for solving various classes of ill-posed elliptic, parabolic, and hyperbolic problems [5, 15–21].
We note that although the interest in inverse problem has rapidly increased during this decade, the literature devoted to the class of problems (1) is quite scarce.
The paper is organized as follows. Section 2 gives some tools which are useful for this study; in Section 3 we introduce some basic results and we show the ill-posedness of the inverse problem; Section 4 gives a regularization solution and error estimation between the approximate solution and the exact one; the numerical implementation is described in Section 5 to illustrate the accuracy and efficiency of this method.
2. Preliminaries
Let (φn)n≥1⊂H be an orthonormal eigenbasis corresponding to the eigenvalues (λn)n≥1 such that (4)Aφn=λnφn,n∈N∗,0<λ1≤λ2⋯≤⋯,limn→∞λn=+∞.ξ=∑n=1∞Enξ,Enξ=ξ,φnφn,∀ξ∈H.We denote by {T(t)=e-tA}t≥0 the analytic semigroup generated by -A on H, (5)Ttξ=∑n=1∞e-λntEnξ,∀ξ∈H.For α>0, the space Hα is given by (6)Hα=ξ∈H:∑n=1∞1+λn2αEnξ2<∞,with the norm (7)ξHα=∑n=1∞1+λn2αEnξ21/2,ξ∈Hα.We achieve this section by a result concerning nonexpansive operators.
Definition 1.
A linear bounded operator L:H→H is called nonexpansive if L≤1.
Let L be an nonexpansive operator; to solve the equation(8)I-Lφ=ψ,we state a convergence theorem for a successive approximation method.
Theorem 2 (see [22], p. 66).
Let L be a nonexpansive, self-adjoint positive operator on H. Let ψ∈H be such that (8) has a solution. If 1 is not eigenvalue of L, then the successive approximations (9)φn+1=Lφn+ψ,n=0,1,2,…converge to a solution to (8) for any initial data φ0∈H. Moreover, Lnφ→0 for every φ∈H, as n→∞.
3. Basic Results3.1. The Direct Problem
Let Z=D(A)×H with the norm UZ2=Aξ12+ξ22, U=ξ1ξ2∈Z.
For a given f∈H, consider the direct problem(10)w′′t+2Aw′t+A2wt=f,0<t<T,w0=0,w′0=0.Making the change of variable w′=v, we can write the second order equation in (10) as a first order system in the space Z as follows:(11)z′t=Azt+F,0<t<T,z0=0,where z=wv, F=0f, and A=0I-A2-2A.
The linear operator A is unbounded with the domain D(A)=D(A2)×D(A) and it is the infinitesimal generator of strongly continuous semigroup {S(t)=etA}t≥0. Moreover {S(t)}t≥0 is analytic (see [1]) and it admits the following explicit form: (12)StU=∑n=1∞etBnPnU,U=ξ1ξ2∈Z,where Bn=01-λn2-2λn and {Pn}n≥1 is a complete family of orthogonal projections in Z given by Pn=diag(En,En).
Using matrix algebra, we obtain (13)etBn=e-λnt+λnte-λntte-λnt-λn2te-λnt-λnte-λnt+e-λnt.From the semigroup theory (see [23]), the problem (11) admits a unique solution z∈C([0,T),Z) given by (14)z=∫0tSt-sFds.Hence, (15)z=∫0t∑n=1∞et-sBnPnFds=∫0t∑n=1∞σn1t,sσn2t,sσn3t,sσn4t,s·0f,φnφnds,such that (16)σn1t,s=e-λnt-s+λnt-se-λnt-s,σn2t,s=t-se-λnt-s,σn3t,s=-λn2t-se-λnt-s,σn4t,s=-λnt-se-λnt-s+e-λnt-s.As a consequence, we obtain the following theorem.
Theorem 3.
The problem (10) admits a unique solution w∈C([0,T),D(A))∩C1([0,T),H) given by(17)wt=Ktf=A-2I-I+tAe-tAf=∑n=1∞1-1+tλne-tλnλn2f,φnφn.
3.2. Ill-Posedness of the Inverse Problem
Now, we wish to solve the inverse problem, that is, find the source term f in the system (1). Making use of the supplementary condition (2) and defining the operator K(T):f→g, we have (18)g=uT=KTf=∑n=1∞σnEnf,where σn=(1-(1+Tλn)e-Tλn)/λn2.
It is easy to see that K(T) is a self-adjoint compact linear operator. On the other hand, (19)g=∑n=1∞Eng=∑n=1∞σnEnf,so(20)σnEnf=Eng,which implies (21)Enf=1σnEng,and therefore (22)f=KT-1g=∑n=1∞1σnEng.Note that 1/σn→∞ as n→∞, so the inverse problem is ill-posed; that is, the solution does not depend continuously on the given data. Hence this problem cannot be solved by using classical numerical methods.
Remark 4.
As many boundary inverse value problems for partial differential equations which are ill-posed, the study of the problem (1) is reduced to the study of the equation K(T)f=g, where K(T) is a compact self-adjoint operator in the Hilbert space H. This equation can be rewritten in the following way: (23)f=I-γKTf+γg=Lf+γg,where γ is a positive number satisfying γ<1/K(T).
In the next section, we will show that the operator L is nonexpansive and 1 is not eigenvalue of L, so it follows from Theorem 2 that (fn)n∈N∗ converges and (I-γK(T))nf→0, for every f∈H, as n→∞.
4. Iterative Procedure and Convergence Results
The alternating iterative method is based on reducing the ill-posed problem (1) to a sequence of well-posed boundary value problems and consists of the following steps.
First, we start by letting f0∈H be arbitrary; the initial approximation u0 is the solution to the direct problem (24)u0′′+2Au0′+A2u0=f0,0<t<T,u00=0,u0′0=0.Then, if the pair (fk,uk) has been constructed, let(25)fk+1=fk-γukT-g,where γ is such that (26)0<γ<1KT,and K(T)=supn∈N∗(1-(1+Tλn)e-λnT)/λn2.
Finally, we get uk+1 by solving the problem (27)uk+1′′+2Auk+1′+A2uk+1=fk+1,0<t<T,uk+10=0,uk+1′0=0.Let us iterate backwards in (25) to obtain(28)fk+1=fk-γKTfk+γg=I-γKTfk+γg=I-γKTk+1f0+γ∑j=0kI-γKTjg.Now, we introduce some properties and tools which are useful for our main theorems.
Lemma 5.
The norm of the operator K(t) is given by (29)Kt=supn∈N∗1-1+tλne-λntλn2=1-1+tλ1e-λ1tλ12.
Proof.
We aim to find the supremum of the function (1-(1+tλn)e-λnt)/λn2, n∈N∗, and for this purpose, fix t, let μ=λt, and define the function (30)G1μ=1-1+μe-μμ2,for μ≥μ1=λ1t.We compute (31)G1′μ=μ2+2μ+2e-μ-2μ3.Put (32)hμ=μ2+2μ+2e-μ-2.Hence,(33)G1′μ=hμμ3.To study the monotony of G1, it suffices to determine the sign of h. We have (34)h′μ=-μ2e-μ<0,∀μ>0,and then h is decreasing; moreover h(μ)⊂]-2,0[, ∀μ>0. Hence G1′(μ)<0, ∀μ≥μ1, which implies that G1 is decreasing and (35)supμ≥μ1G1μ=G1μ1.Therefore, (36)supn≥11-1+λnte-λntλn2=1-1+λ1te-λ1tλ12.
Proposition 6.
For the linear operator L=I-γK(T), one has the following properties:
Lis positive and self-adjoint,
L is nonexpansive,
1 is not an eigenvalue of L.
Proof.
Form properties of operator A and the definition of L it follows that L is self-adjoint and nonexpansive positive operator and from the inequality (37)0<1-γ1-1+Tλe-λTλ2<1,for λ∈σA,it follows that the point spectrum of L,σp(L)⊂]0,1[. Then 1 is not eigenvalue of the operator L.
Lemma 7.
If λ>0, one has the estimates(38)11+λ2≤max3T2,11-1+Tλe-λTλ2,(39)0<1-1+tλe-λtλ2<T2,∀t∈0,T.
Proof.
To establish (38), let us first prove that(40)13+μ2≤1-1+μe-μμ2,∀μ>0,which is equivalent to prove that (41)G2μ=3-3+μ21+μe-μ≥0,∀μ>0.We have (42)G2′μ=μμ-12e-μ≥0,∀μ>0.Then, G2 is nondecreasing and it follows that G2(μ)⊂]0,3[. SoG2(μ)≥0, ∀μ>0.
Choosing μ=Tλ in (40), we obtain (43)13+Tλ2≤1-Tλ+1e-TλTλ2.So,(44)T2max3,T21+λ2≤1-1+Tλe-Tλλ2.From (44), we deduce (38).
Now, we prove the estimate (39). It is easy to verify that (45)G3μ=1-1+μe-μ-μ2<0,∀μ>0.Then, if we choose μ=tλ, we get(46)1-1+tλe-tλ<t2λ2,∀λ>0,∀t∈0,T.Hence, from (46), (39) follows.
Theorem 8.
Let u be a solution to the inverse problem (1). Let f0∈H be an arbitrary initial data element for the iterative procedure proposed above and let uk be the kth approximate solution. Then
The method converges; that is,(47)supt∈0,Tukt-ut⟶0,as k⟶∞.
Moreover, if, for some α=1+θ, θ>0, f0-f∈Hα, that is, f0-fHα≤E, then the rate of convergence of the method is given by (48)supt∈0,Tukt-ut≤T2CEk-α/2,where C is a positive constant independent of k.
Proof.
(i) From (28), we get (49)fk=I-γKTkf0+I-I-γKTkKT-1g,and then(50)fk=I-γKTkf0-f+f,which implies that (51)ukt-ut=Ktfk-f=KtI-γKTkf0-f.Hence,(52)ukt-ut≤KtI-γKTkf0-f.From Lemma 5 and (39) we have(53)supt∈0,TKt=supt∈0,T1-1+tλ1e-tλ1λ12<T2.Combining (52) and (53) and passing to the supremum with respect to t∈[0,T], we obtain (54)supt∈0,Tukt-ut≤T2I-γKTkf0-f⟶0,as k⟶∞.
(ii) By part (i), we have (55)ukt-ut2≤T4∑n=1∞1-γ1-1+λnTe-λnTλn22kf0-f,φn2,and hence (56)ukt-ut2≤T4∑n=1∞1-γ1-1+λnTe-λnTλn22k1+λn2-α1+λn2αf0-f,φn2.Using the inequality (38), we obtain (57)ukt-ut2≤T4max3T2,1α∑n=1∞1-γβn2kβnα1+λn2αf0-f,φn2,where βn=((1-(1+λnT)e-λnT)/λn2).
So, it follows that(58)ukt-ut2≤T4max3T2,1αsup0≤βn≤T21-γβn2kβnαf0-fHα2.Put (59)ϕβ=1-γβ2kβα,0≤β≤T2.We compute (60)ϕ′β=1-γβ2k-1βα-1-γ2k+αβ+α.Setting ϕ′(β)=0, it follows that β∗=α/(2k+α)γ is the critical point of ϕ. It is easy to see that the maximum of ϕ is attained at β∗. So (61)sup0≤β≤T2ϕβ≤ϕβ∗=1-γβ∗2kβ∗α≤β∗α=α2k+αγα,and hence(62)sup0≤β≤T2ϕβ≤α2γαk-α.Combining (58) and (62), we obtain (63)supt∈0,Tukt-ut2≤T4α2γmax3T2,1α1kαE2.
Since in practice the measured data g is never known exactly but only up to an error of, say, δ>0, it is our aim to solve the equation K(T)f=g from the knowledge of a perturbed right-hand side gδ satisfying(64)g-gδ<δ,where δ>0 denotes a noise level. In the following theorem, we consider the case of inexact data.
Theorem 9.
Let α=1+θ, (θ>0), f0 be an arbitrary initial data element for the iterative procedure proposed above such that (f0-f)∈Hα, let uk be the kth approximations solution for the exact data g, and let ukδ be the kth approximations solution corresponding to the perturbed data gδ such that (64) holds. Then one has the following estimate:(65)supt∈0,Tukt-ut≤T2δγk+CE1kα/2.
Proof.
Let(66)fk=I-γKTkf0+γ∑j=0k-1I-γKTjg,ukt=Ktfk,fkδ=I-γKTkf0+γ∑j=0k-1I-γKTjgδ,ukδt=Ktfkδ.Using the triangle inequality, we obtain (67)ukδ-u≤ukδ-uk+uk-u.From Theorem 8, we have(68)supt∈0,Tukt-ut≤T2CE1kα/2.On the other hand, (69)ukδt-ukt=Ktfkδ-fk≤T2γ∑j=0k-1I-γKTjgδ-g≤T2δγ∑j=0k-1I-γKTj≤T2δγ∑j=0k-1I-γKTj.Since (70)I-γKT≤1,it follows that(71)supt∈0,Tukδt-ukt≤T2δγk.Combining (68) and (71) and passing to the supremum with respect to t∈[0,T], we obtain the estimate (65).
Remark 10.
If we choose the number of the iterations k(δ) so that k(δ)→0 as δ→0, we obtain (72)supt∈0,Tukδt-ut⟶0,as k⟶+∞.
5. Numerical Implementation
In this section, an example is devised for verifying the effectiveness of the proposed method. Consider the problem of finding a pair of functions (u(x,t),f(x)), in the system(73)∂2∂t2ux,t-2∂2∂x2∂∂tux,t+∂4∂x4ux,t=fx,t,x∈0,1×0,1,u0,t=u1,t=0,t∈0,1,ux,0=utx,0=0,x∈0,1,ux,1=gx,x∈0,1.Denote(74)A=-∂2∂x2,with DA=H010,1∩H20,1⊂H=L20,1.λn=n2π2,φn=2sinnπx,n=1,2,…are eigenvalues and orthonormal eigenfunctions, which form a basis for H.
The solution of the above problem is given by (75)ux,t=∑n=1∞1-1+nπ2te-nπ2tnπ4fnφn,where fn=(f,φn)=2∫01f(s)sin(nπs)ds, n=1,2,…
Now, to solve the inverse problem, making use of the supplementary condition and defining the operator K:f→g, we have(76)gx=ux,1=Kfx=2∑n=1∞1-1+nπ2e-nπ2nπ4∫01fssinnπsdssinnπx.
Example 11.
In the following, we first selected the exact solution f(x) and obtained the exact data function g(x) through solving the forward problem. Then we added a normally distributed perturbation to each data function and obtained vectors gδ(x). Finally we obtained the regularization solutions through solving the inverse problem with noisy data gδ(x) satisfying(77)g-gδL20,12≤δ.It is easy to see that if f(x)=sinπx, then (78)ux,t=1-1+π2te-π2tπ4sinπxis the exact solution of the problem (73). Consequently, g(x)=((1-(1+π2)e-π2)/π4)sin(πx).
Now, we propose to approximate the first and second space derivatives by using central difference and we consider an equidistant grid points to a spatial step size x0=0<x1<⋯<xN+1=1, (h=1/(N+1)), where N is a positive integer. We get the following semidiscrete problem:(79)u′′xi,t+2Ahu′xi,t+Ah2uxi,t=fxi,xi=ih,i=1,…,N,0<t<1,u0,t=u1,t=0,0<t<1,uxi,0=u′xi,0=0,xi=ih,i=1,…,N,uxi,1=gxi,xi=ih,i=1,…,N,where Ah is the discretisation matrix stemming from the operator A=-d2/dx2, and(80)Ah=1h2Tridiag-1,2,-1is a symmetric, positive definite matrix, with eigenvalues (81)μj=4N+12sin2jπ2N+1,j=1,…,N,and orthonormal eigenvalues (82)vj=sinmjπN+11≤m≤N,j=1,…,N.We assume that it is fine enough so that the discretization errors are small compared to the uncertainty δ of the data; this means that Ah is a good approximation of the differential operator A whose unboundedness is reflected in a large norm of Ah (see [24]).
Adding a random distributed perturbation to each data function, we obtain (83)gδ=g+εrandnsizeg,where ε indicates the noise level of the measurements data and the function randn(·) generates arrays of random numbers whose elements are normally distributed with mean 0, variance σ2=1, and standard deviation σ=1. randn(size(g)) returns an array of random entries that is of the same size as g. The noise level δ can be measured in the sense of root mean square error (RMSE) according to (84)δ=gδ-gl2=1N+1∑i=0Ngxi-gδxi21/2.The relative error is given as follows: (85)REf=fapproximate-fexactl2fexactl2.The discrete iterative approximation of (66) is given by (86)fkδxi=I-γKhkf0xi+γ∑j=0k-1I-γKhjgδxi,i=1,…,N,where Kh=Ah-2(IN-(IN+Ah)e-Ah) and γ<1/Kh=(μ12/(1-(1+μ1)e-μ1)).
Figures 1–4 display that as the amount of noise ε decreases, the regularized solutions approximate better the exact solution.
The comparison between the exact solution fe and its computed approximations fa for N=60k=4 and noisy level ε=10-3.
The comparison between the exact solution fe and its computed approximations fa for N=60k=4 and noisy level ε=10-4.
The comparison between the exact solution fe and its computed approximations fa forN=60k=5 and noisy level ε=10-3.
The comparison between the exact solution fe and its computed approximations fa for N=60k=5 and noisy level ε=10-4.
Table 1 shows that for k=4 or k=5 the relative error decreases with the decease of epsilon which is consistent with our regularization.
Relative error RE(f).
N
k
ε
RE(f)
60
4
10−3
0.2039
60
4
10−4
0.0945
60
5
10−3
0.3032
60
5
10−4
0.0305
6. Conclusion
In this paper, we have extended the iterative method to identify the unknown source term in a second order differential equation, convergence results were established, and error estimates have been obtained under an a priori bound of the exact solution. Some numerical tests have been given to verify the validity of the method.
Conflict of Interests
The authors declare that they have no conflict of interests.
Authors’ Contribution
All authors read and approved the paper.
Acknowledgments
The authors would like to thank the anonymous referees for their suggestions.
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