MONDAL Subhankar and NAIR M.Thamban

Department of Mathematics,IIT Madras,Chennai 600036,India.

Abstract. We consider the inverse problem of identifying a general source term,which is a function of both time variable and the spatial variable,in a parabolic PDE from the knowledge of boundary measurements of the solution on some portion of the lateral boundary. We transform this inverse problem into a problem of solving a compact linear operator equation. For the regularization of the operator equation with noisy data,we employ the standard Tikhonov regularization,and its finite dimensional realization is done using a discretization procedure involving the space L2(0,τ;L2(Ω)). For illustrating the specification of an a priori source condition,we have explicitly obtained the range space of the adjoint of the operator involved in the operator equation.

Key Words: Ill-posed;source identification;Tikhonov regularization;weak solution.

1 Introduction

Letd≥1 and Ω be a bounded domain in Rdwith Lipschitz boundary.For a fixedτ>0 we denote the cylindrical domain Ω×[0,τ] by Ωτand its lateral surface∂Ω×[0,τ] by∂Ωτ.Let Σ be a relatively open subset of∂Ω.We denote the boundary surface Σ×[0,τ]by Στ.For

we consider the parabolic PDE

whereQ∈(L∞(Ω))d×dis a symmetric matrix with entries fromL∞(Ω) satisfying the uniform ellipticity condition,that is,there exist a constantκ0>0 such that

where|ξ|2=ξ12+...+ξd2forξ=(ξ1,...,ξd)∈Rdand→nis the outward unit normal to∂Ω.

Throughout the paper, for a Banach spaceX,φ∈L2(0,τ;X) meansφis anX-valued function on [0,τ] such thatt■→‖φ(t)‖Xbelongs toL2[0,τ]. Also, throughout we use the standard notations of the function spacesL2(Ω) and the Sobolev spacesH1(Ω)(see[1–3]).

For results related to existence and uniqueness of the classical solution corresponding to the forward problem associated with (1.1), namely, that of findingusatisfying (1.1)from the knowledge off,g,has considered above, one may refer to [4–6]. In certain cases, a classical solution may not exist for the forward problem, but we may have aweak solution. In[5, Theorem 2.4], the authors have given an existence result for a weak solution of(1.1). We first state the existence result precisely, whose proof follows along similar lines as in[5,Theorem 2.4].

Theorem 1.1. ([5,Theorem 2.4])Let f∈L2(0,τ;L2(Ω)),g∈L2(0,τ;L2(∂Ω))and h∈L2(Ω).

Also, let Q∈(L∞(Ω))d×d be symmetric satisfying the uniform ellipticity condition(1.2). Then there exists a unique u∈L2(0,τ;H1(Ω))with ut∈L2(0,τ;(H1(Ω))′)satisfying

for all ϕ∈H1(Ω)and for a.a.(almost all)t∈[0,τ]with u(·,0)=h a.e. inΩ. Further,there exists a constant C1>0,independent of f,such that

In(1.3),the notation〈·,·〉stands for the duality action betweenH1(Ω)and(H1(Ω))′,where(H1(Ω))′stands for the dual ofH1(Ω).Also,utdenotes the distributional derivative ofuwith respect tot,that is,utis the unique element inL2(0,τ;(H1(Ω))′)such that

Following[5],an elementu∈L2(0,τ;H1(Ω))withut∈L2(0,τ;(H1(Ω))′)is called a weak solution of the PDE(1.1),if it satisfies(1.3).

For a givenf∈L2(0,τ;L2(Ω)),g∈L2(0,τ;L2(∂Ω)) andh∈L2(Ω), letube the unique weak solution of(1.1). Let

Note that,hereu(x,t) on Στhas to be understood in the sense of trace(see[1,2,7]). We are interested in the inverse problem of determining thesource function ffrom the knowledge ofz, i.e., from the knowledge ofuon the portion Στof the boundary of Ωτ. More precisely,our inverse problem at hand is the following:

(IP): For the given boundary dataz∈L2(0,τ;L2(Στ)), determine the source functionfinL2(0,τ;L2(Ω)) such that the corresponding unique weak solutionuof(1.1)satisfiesu|Στ=z.

In general,the solution of the inverse problem,if it exists need not be unique. To see this,we consider a simple example below.

Example 1.1. Let Ω=(0,1)andh=0 on(0,1). Fort∈[0,1],let

LetQ=1 on Ω,andz(1,t)=sinπtfort∈[0,1]. Then for

we have

but the source functions corresponding tou1andu2are respectively,

Thus,the source functionf,if exists,from our considered boundary observation is not unique,in general. But,for certain specific cases offand for different type of boundary measurements,the uniqueness results are well-established;see[8–10]. In[8],the author has considered the case whenfdepends only on the spatial variable, whereas in [9]the authors have considered the case, wherefcan be written as infinite sum of certain type of functions. Also, in [10], the authors have considered the case, wherefcan be written asf(x,t):=σ(t)φ(x) whereσ(t)=e−λt,λ>0 and considered an inverse problem of identifying the functionφ. We carry out our analysis for a general source functionfassuming only its existence,and obtain regularized approximations for thatfwhich has minimum norm.

The inverse problems of source identification from boundary measurements have vast literature and they have real world applications. For inverse problems related to source identification, one may look into[8,9,11–15] and also the recent work in[16]. In fact, in [14], the authors have mentioned explicitly various inverse problems on source identifications from boundary measurements.

Usually the inverse source identification problems from boundary measurements or final time observations are ill-posed in nature(see[14]),that is,either the inverse problem do not have a unique solution,or even if the solution exist,it does not depend continuously on the data. In[16],the authors have considered the inverse problem of identifying source function from a boundary measurement for a parabolic PDE with Robin boundary conditions. For obtaining stable approximate solutions, they have used the output least square method combined withCrank-Nicolson Galerkin methodto obtain numerical approximations for the source function.

In this paper,we convert the inverse problem(IP)into a linear operator equation first,

where the associated operator is compact and is of infinite rank so that it is an ill-posed operator equation. We use Tikhonov regularization (see [17,18]) in the infinite dimensional setting for obtaining stable approximate solutions corresponding to the noisy data.In order to obtain approximations in a finite dimensional setting, we employ Galerkinprojection method to the regularized operator equation, by using different projections corresponding to space variable and time variable. Making use of the fact that the operator involved in the infinite dimensional setting is compact,we derive order optimal error estimates by choosing the regularization parameter and the level of approximation appropriately. Thus,our method of obtaining regularized approximations is much simpler compared to the one considered in[16].

Also,we would like to mention that in[9],similar operator theoretic formulation has been adopted for the problem of identifying the functionϕoccurring in the source functionf(x,t)=√tϕ(x),from the knowledge of a lateral boundary measurement ˜zdefined on Σ×[0,τ], where Σ is a relatively open subset of∂Ω and the corresponding governing PDE is same as(1.1)withg=0 andh=0.But in this paper,we are considering the identification of a general source function which depends on both space and time variable,andg,hare not necessarily zero. Further, in [9], no finite dimensional analysis is done whereas in this paper we have given a finite dimensional analysis for obtaining stable approximations to the sought source function.

The rest of the paper is organized as follows: In Section 2, we introduce some notations that we shall use throughout and obtain some results related to continuity and compactness of the linear operator involved in the formulation of the inverse problem.Section 3 deals with the regularization analysis corresponding to the noisy data. In Section 4, we have considered the finite dimensional realization of our proposed method,which is one of the main objectives of this work. Finally,in Section 5,titled as appendix,we have explicitly obtained the range space of the adjoint of the linear operator.

2 Operator equation formulation

Letγ:H1(Ω)→L2(∂Ω)be the trace map(see[1–3]). It is known thatγis a continuous linear operator and the range ofγisH1/2(∂Ω). We define Γ:W →Y by

We first observe some properties of the map Γ.

Theorem 2.1.The mapΓ:W →Ydefined as in(2.1)is a bounded linear operator.

Proof.Linearity of Γ follows from the linearity ofγ.LetC2>0 be such that‖γϕ‖L2(∂Ω)≤C2‖ϕ‖H1(Ω)for allϕ∈H1(Ω). Then,for allφ∈W,we have

This shows that Γ is a bounded linear operator with‖Γ‖≤C2.

For proving one more property of Γ, we first put on record some of the embedding results.For the proofs,one may refer to[19].

Sinceγ=γ0onHs(Ω),we have Γ=˜Γ0onL2(0,τ;Hs(Ω)),where(˜Γ0φ)(t)=(Γ0φ)(t)|Σ.

Theorem 2.2.LetΓbe as defined in(2.1). ThenΓis a compact linear operator.

LetQ(x)=(qij(x))∈(L∞(Ω))d×dbe symmetric,i.e.,qij=qjia.e. on Ω for all 1≤i,j≤dandQsatisfies(1.2). Now consider the PDE

Then by Theorem 1.1,we know that for each Φ∈X, there exists a unique weak solutionvΦ∈W of(2.2)satisfying

whereC1is as in Theorem 1.1. We now define a mapS:X →W by

wherevΦ∈W. By the nature of PDE(2.2),it is clear thatSis a linear operator. Also,by the estimate in(2.3),it follows thatSis a continuous linear operator.

We now define the mapT:X →Y by

Theorem 2.3.Let T be as defined in(2.5). Then T is a compact linear operator and‖T‖≤C3,

where C3=C1C2with C1,C2are constants as in Theorem1.1and Theorem2.1,respectively.

Proof.The linearity and continuity ofTfollows from the fact that Γ andSare both linear and continuous. SinceTis a composition of a compact operator Γ(see Theorem 2.2)and a bounded linear operatorS, the compactness ofTfollows. Finally, the estimate can be easily obtained by applying the estimates in Theorem 2.1 and(2.3).

Next we show thatTis of infinite rank. In order to show this, we shall make use of the representation ofT∗which is given in Section 4,as an appendix.

Theorem 2.4.Let T be as defined in(2.5). Then R(T),the range of T,is infinite dimensional.

Letz∈Y. Recall that our inverse problem is to determine anf∈X such that the corresponding unique weak solutionuof(1.1)satisfiesu|Στ=z.As pointed out in Section 1, there may be more than one solution for this inverse problem. We assume that our inverse problem has a solutionfz. Letuzbe the corresponding unique weak solution of(1.1)for the source functionfz.

Letf0∈X be anya prioriknown function andu0∈W be the unique weak solution of(1.1)forf=f0.Then it can be seen thatuz−u0is the unique weak solution of the PDE

Letz0=u0|Στ.Then it follows thatfz−f0is a solution of the operator equation

Thus,our inverse problem has been transformed into the problem of solving the operator equation(2.7),which has a solution,namelyfz−f0.Also,the solution of the operator equation(2.7)need not be unique. We would like to identify the uniquef†,where

In practical application, the exact datazmay not be known. Instead,we may have a noisy measured data. But by Theorem 2.3 and Theorem 2.4,it follows that(2.7)is an illposed operator equation. Therefore solving(2.7)with perturbed right hand side may not give stable solutions,that is,small perturbation in the data may produce large deviation in the solution. So,as mentioned in the introductory section,we shall use the theory of Tikhonov regularization to obtain stable solutions,while dealing with noisy data.

3 Regularization with noisy data

Letz∈Y be the exact data as considered in our inverse problem(IP).Forδ>0,letzδ∈Y be the measured noisy data satisfying

We now consider the perturbed operator equation

AsTis a linear compact operator of infinite rank, solving the operator equation is illposed.To obtain stable approximations with the help of the noisy datazδ,we shall make use of the standard theory of Tikhonov regularization. For eachα>0,letfαandfδαbe the unique elements in X such that

The following result is known in the literature([17,18]).

Remark 3.2. In order to obtain an estimate for the quantity‖f†−fα‖,we need to assume somea prioricondition onf†. It is well known in the theory of Tikhonov regularization that iff†∈R(T∗),the range ofT∗,then‖f†−fα‖≤c√αfor some constantc>0,so that,in this case,we also have the rate

for the choiceα～δ. More generally, ifϕ:(0,∞)→[0,∞) is a monotonically increasing function such that

for somec0>0 and iff†∈R(ϕ(T∗T)),then it is known that(see[18,pp. 195])

4 Finite dimensional analysis

For eachn∈N,letPn:L2(Ω)→L2(Ω)be an orthogonal projection of ranknand for eachm∈N,let Πm:L2(0,τ)→L2(0,τ)be an orthogonal projection of rankmsuch that

Proof.Since{g1,...,gm}⊂L2(0,τ)is an orthonormal basis ofR(Πm),we have

Let Φij(t)be as in(4.2). Then,we see that,Φij∈X for alli=1,...,n,j=1,...,m,and

whereδpq=1 forp=qandδpq=0 forp/=q. Therefore,{Φij:i=1,...,n,j=1,...,m} is an orthonormal set in X.

Since {Φij:i=1,...,n,j=1,...,m} is an orthonormal set, thereforeQmnis an orthogonal projection of ranknm.

Therefore,we have the matrix equation

where→c=[c11,...,cnm]tis the unique solution of(4.8).

This completes the proof.

Since Πm→Ipointwise inL2(0,τ),for eachn∈N,we have

Also,since{ϕ1,...,ϕn}is an orthonormal basis ofR(Pn),for eacht∈(0,τ),we have

SincePn→Ipointwise inL2(Ω) and ‖PnΦ(t)−Φ(t)‖≤2‖Φ(t)‖, by dominated convergence theorem we have

Therefore

for every Φ∈X. This completes the proof.

From the above theorem we obtain the following corollary,by simply using the definition of double limits as obtained in the theorem.

In view of Theorem 4.4 and Theorem 4.2 we have the following.

The above theorem along with Remark 3.2 leads to the following theorem.

Theorem 4.6.For δ>0,letˆn:=nδ, ˆm:=mδ be as in Theorem4.5. Then we have the following.

(i)If f†∈R(T∗)and α:=cδ for some c>0,then

and in that caseT∗(ˆφ)=f,where ˆφ=Q∇f·→n|Στ.

Appendix

In this section, we shall find explicitly the range space ofT∗. Recall that for Φ ∈X andφ∈Y,we have

whereγis the trace map andSis the linear map defined in Section 2,by assigning each Φ∈X,the unique weak solutionvΦ:=SΦ of(2.2).

Forφ∈Y,we consider the PDE

whereQ∈(L∞(Ω))d×dis as in(2.2). Then reversing the time direction and following the lines of the proof of Theorem 1.1 and following Part 3 of the proof of Theorem 3 in[3, pg.357],(A.1)has a unique weak solutionwφ∈W,that is,

Acknowledgments

The authors gratefully acknowledge the useful comments of the referees on the first version of this paper.