# Convergence analysis of a Crank-Nicolson Galerkin method for an inverse source problem for parabolic systems with boundary observations

This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank-Nicolson Galerkin method is applied to the least squares functional with quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach to zero with appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and corresponding convergence rates are proved. Finally, two numerical experiments are presented to illustrate the efficiency of the theoretical findings.

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• ### Inverse source in two-parameter anomalous diffusion, numerical algorithms and simulations over graded time-meshes

We consider an inverse source two-parameter sub-diffusion model subject ...
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• ### Multiple coefficient identification in electrical impedance tomography with energy functional method

In this paper we investigate the problem of simultaneously identifying t...
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We apply the nonconforming discretisation of Wu and Xu (2019) to the tri...
01/11/2021 ∙ by Andreas Bock, et al. ∙ 0

• ### Identifying source term in the subdiffusion equation with L^2-TV regularization

In this paper, we consider the inverse source problem for the time-fract...
05/07/2021 ∙ by Bin Fan, et al. ∙ 0

• ### A Unified Divergence Analysis on Projection Method for Symm's Integral Equation of the First Kind

A large amount of literatures analyze (SIE) which concerns the construct...
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## 1 Introduction

The problem of identifying a source in a heat transfer or diffusion process has got attention of many researchers during last years. This problem leads to determining a term in the right hand side of parabolic equations from some observations of the solution which is well known to be ill-posed. For surveys on the subject, we refer the reader to the books [6, 18, 24, 25, 31], the recent papers [19, 32] and the references therein.

Although there have been many papers devoted to the source identification problems with observations in the whole domain or at the final moment, those with boundary observations are quite few. Furthermore, the sought term depends either on the spatial variable as in

[5, 7, 8, 9, 10, 11, 14, 16, 17, 18, 41, 46, 47], or only on the time variable as in [20]. In this paper, we consider the problem of determining the right hand side depending on both spatial and time variables by a variational method. We also treat the case when the sought term depends either on the spatial or time variable. Indeed, let be an open bounded connected set of with boundary and be a given constant. We investigate the problem of identifying the source term in the Robin boundary value problem for the parabolic equation

 ∂u∂t(x,t)+Lu(x,t)=f(x,t)% ~{}in~{}ΩT:=Ω×(0,T], (1.1) ∂u(x,t)∂→n+σ(x,t)u(x,t)=g(x,t)~{}on~{}S:=∂Ω×(0,T], u(x,0)=q(x)~{}in~{}Ω

from a partial boundary measurement of the solution on the surface satisfying

 ∥Z−zδ∥L2(Σ)≤δ, (1.2)

where , is a relatively open subset of and the positive constant stands for the measurement error.

In (1.1) is a time-dependent, second order self-adjoint elliptic operator of the form

 Lu(x,t):=−d∑i,j=1∂∂xi(aij(x,t)∂u(x,t)∂xj)+b(x,t)u(x,t),

where is a symmetric diffusion matrix satisfying the uniformly elliptic condition

 A(x,t)ξ⋅ξ=d∑i,j=1aij(x,t)ξiξj≥a––d∑i,j=1|ξi|2~{}in~{}¯¯¯¯¯¯¯ΩT (1.3)

for all with some constant and

is a non-negative function. The vector

is the unit outward normal on and

 ∂u(x,t)∂→n:=A(x,t)∇u(x,t)⋅→n

with . In addition, the functions , and with in are assumed to be given. The source term is sought in the space .

The contents of this paper are as follows: For any fixed let denote the unique weak solution of the system (1.1), see Section 2 for the definition of related functional spaces. Adopting the output least squares method combined with the Tikhonov regularization, we consider the (unique) minimizer of the minimization problem

 minf∈L2(ΩT)Jρ,δ(f)withJρ,δ(f):=∥u(f)−zδ∥2L2(Σ)+ρ∥f−f∗∥2L2(ΩT)

as a reconstruction, where is the regularization parameter and

is an a priori estimate of the true source which is identified. It is shown that the cost functional

is Fréchet differentiable and for each the -gradient of at is given by

 ∇Jρ,δ(f)=2p(f)+2ρ(f−f∗),

i.e. there holds the relation

 J′ρ,δ(f)ξ=(2p(f)+2ρ(f−f∗),ξ)L2(ΩT)

for all , where is the adjoint state of that is discussed the detail in the next section.

For discretization we employ the Crank-Nicolson Galerkin method, where the finite dimensional space of piecewise linear, continuous finite elements is used to discretize the state with respect to the spatial variable. Further, to discretize the state with respect to the time variable, we divide the time interval into equal subintervals and introduce a time step together with time levels

 tn:=nτ~{}with~{}n∈I0:={0,1,…,M}.

As a result, the state is then approximated by the finite sequence in which for each the element

 Unh,τ(f)∈V1h:={φh∈C(¯¯¯¯Ω) | φh|T∈P1(T)~{% }for all~{}T∈Th}.

With these notions at hand, we examine the discrete regularized problem corresponding to i.e. the following strictly convex minimization problem

 minf∈L2(ΩT)Jρ,δ,h,τ(f)withJρ,δ,h,τ(f):=M∑n=1∫tntn−1∥Unh,τ(f)−zδ∥2L2(Γ)dt+ρ∥f−f∗∥2L2(ΩT)

which admits a unique solution obeying the relation (Section 3)

 fρ,δ,h,τ|Ω×(tn−1,tn]=f∗−1ρPn−1h,τ(fρ,δ,h,τ) (1.4)

for any , where is the approximation of the adjoint state . Using the variational discretization concept introduced in [21], the minimizer automatically belongs to the finite dimensional space

 V1,0h,τ:={Φ∈L2(0,T;V1h) | Φ|Ω×(tn−1,tn]:=φnh∈V1h~{}for all~{}n∈I}

provided an a priori estimate and hence a discretization of the admissible set can be avoided. Furthermore, due to the equation (1.4) the -gradient of the discrete cost functional at is given by

 ∇Jρ,δ,h,τ(fρ,δ,h,τ)=2GJ(fρ,δ,h,τ)+2ρ(fρ,δ,h,τ−f∗)

with

 GJ(fρ,δ,h,τ)|Ω×[tn−1,tn):=Pn−1h,τ(fρ,δ,h,τ)andn∈I.

As and with an appropriate a priori regularization parameter choice , we show in Section 4 that the whole sequence converges in the -norm to the unique -minimum-norm solution of the identification problem defined by

 f†=argminf∈{f∈L2(ΩT) | u(f)|Σ=Z}∥f−f∗∥L2(ΩT).

The corresponding state sequence then converges in the -norm to the exact state of the problem (1.1).

Section 5 is devoted to convergence rates for the discretized problem, where we first show that if and there exists a function such that , where is the unique weak solution of the parabolic system

 −∂F∂t(x,t)+LF(x,t)=0~{}% in~{}ΩT, ∂F(x,t)∂→n+σ(x,t)F(x,t)=wχΣ~{}on~{}S, F(x,T)=0~{}in~{}Ω

with

being the characteristic function of

, then , i.e. it is the unique -minimum-norm solution of the above . Furthermore, if the data appearing in the system (1.1) are regular enough the convergence rate

 ∥fρ,δ,h,τ−f†∥2L2(ΩT)≤C(h3ρ−1+τ2h−1ρ−1+δ+ρ+δ2ρ−1)

is established, where is the unique minimizer of .

For the numerical solution of the discrete regularized problem we in Section 6 utilize a conjugate gradient algorithm. Numerical studies are presented for two cases where the sought source is smooth and discontinuous as well, that illustrates the efficiency of our theoretical findings.

In some practical situations the source term has the form

 F1(x,t)f(x,t)+F2(x,t). (1.5)

Motivating by this reason we in Section 7 present briefly some related results for the problem of identifying the part in the source term expressed by (1.5), where the functions and are known.

To conclude this introduction we wish to mention that to the best of our knowledge, although there have been many papers devoted to source identification problems for parabolic equations, we however have not yet found investigations on the discretization analysis for those with boundary observations — which is more realistic from the practical point of view, a fact that motivated the research presented in the paper. Concerning the identification problem in elliptic equations utilizing boundary measurements, we here would like to comment briefly some previously published works. In [42, 44] the authors used finite element methods to numerically recover the fluxes on the inaccessible boundary from measurement data of the state on the accessible boundary , while the problem of identifying the Robin coefficient on is also investigated in [43]. Recently, authors of [22, 23] have adopted the variational approach of Kohn and Vogelius to the source term and scalar diffusion coefficient identification, respectively, using observations available on the whole boundary. Finite element analysis for the reaction coefficient identification problem from partial observations is carried out in [35], while a survey of the problem of simultaneously identifying the source term and coefficients from distributed observations can be found in [34].

Throughout the paper we use the standard notion of Sobolev spaces , , , etc. from, for example, [1, 39].

## 2 Problem setting and preliminaries

To formulate the identification problem, we first give some notations [40]. Let be a Banach space, we denote by

 C([0,T],X):={w:[0,T]→X | w~{}% is continuous on~{}[0,T],~{}i.e.~{}limτ→t∥w(τ)−w(t)∥X=0~{}for all~{}t∈[0,T]}

which is also a Banach space with respect to the norm

 ∥w∥C([0,T],X):=max0≤t≤T∥w(t)∥X.

We define for the Banach space

 Lp(0,T;X):={w:[0,T]→X | ∥w∥Lp(0,T;X)<∞},

where

 ∥w∥Lp(0,T;X):=(∫T0∥w(t)∥pXdt)1/p~{}if~{}1≤p<∞~{}and~{}∥w∥L∞(0,T;X):=ess~{}sup0≤t≤T∥w(t)∥X.

Let be the dual space of , we use the notation

 W(0,T):={w∈L2(0,T;H1(Ω)) ∣∣ ∂w∂t∈L2(0,T;H1(Ω)∗)}.

It is a Banach space equipped with the norm

 ∥w∥W(0,T):=(∥w∥2L2(0,T;H1(Ω))+∥∥∥∂w∂t∥∥∥2L2(0,T;H1(Ω)∗))1/2. (2.1)

We note that, since with respect to the norm (2.1) is a closed subspace of the reflexive space , it is itself reflexive. We now quote the following useful result.

###### Lemma 2.1 ([15, 48]).

(i) The embedding is continuous, meanwhile the one is compact.

(ii) Let . The mapping is absolutely continuous and

 ddt∥v(t)∥2L2(Ω)=2⟨dv(t)dt,v(t)⟩(H1(Ω)∗,H1(Ω)) (2.2)

for a.e. .

(iii) For all and the equation

 ∫βα⟨∂v∂t(t),w(t)⟩(H1(Ω)∗,H1(Ω))dt +∫βα⟨∂w∂t(t),v(t)⟩(H1(Ω)∗,H1(Ω))dt =∫Ωv(β)w(β)dx−∫Ωv(α)w(α)dx (2.3)

holds.

### 2.1 Direct and inverse problems

For considering the problem (1.1), we set

 a(t;v,w) :=∫ΩA(x,t)∇v(x)⋅∇w(x)dx+∫Ωb(x,t)v(x)w(x)dx+∫∂Ωσ(x,t)v(x)w(x)dx (2.4) :=a(v,w),

where and . Then, for each the Robin boundary value problem (1.1) defines a unique weak solution in the sense that with for a.e. and the following variational equation is satisfied (cf. [40, 45])

 ⟨∂u∂t,φ⟩(H1(Ω)∗,H1(Ω))+a(u,φ)=(f,φ)L2(Ω)+(g,φ)L2(∂Ω)∀φ∈H1(Ω), a.e. t∈(0,T]. (2.5)

Furthermore, the estimate

 ∥u∥W(0,T)≤CR(∥f∥L2(ΩT)+∥g∥L2(S)+∥q∥L2(Ω)) (2.6)

holds, where is a positive constant independent of and . To emphasize the dependence, we sometimes write or if there is not confusion.

Therefore, we can define the source-to-state operator

 u:L2(ΩT)→W(0,T)

which maps each to the unique weak solution of the problem (1.1). The inverse problem is stated as follows:

Given the boundary data of the exact solution , find an element such that .

### 2.2 Variational method

In practice only the observation of the exact with an error level

 ∥Z−zδ∥L2(Σ)≤δ,δ>0

is available. Hence, our problem is to reconstruct an element in (1.1) from noisy observation of . For this purpose we use the standard least squares method with Tikhonov regularization, i.e. we consider a minimizer of the minimization problem

 minf∈L2(ΩT)Jρ,δ(f)withJρ,δ(f):=∥u(f)−zδ∥2L2(Σ)+ρ∥f−f∗∥2L2(ΩT)

as a reconstruction.

###### Remark 2.2.

In case or the expression in (2.4) generates a scalar product on the space equivalent to the usual one, i.e. there exist positive constants such that (cf. [29, 33])

 C1∥φ∥2H1(Ω)≤a(t;φ,φ)≤C2∥φ∥2H1(Ω) (2.7)

for all and .

Now we assume that . A change of the variable , the system (1.1) has the form

 ∂v(x,t)∂→n=e−tg(x,t)~{}on~{% }S, v(x,0)=q(x)~{}in~{}Ω.

Therefore, in the sequel we consider the case or only. All results in present paper are still valid for the case .

Now we summarize some useful properties of the source-to-state operator .

###### Lemma 2.3.

The source-to-state operator is Fréchet differentiable. For any fixed the differential in the direction is the unique weak solution in to the problem

 ∂ˆu∂t(x,t)+Lˆu(x,t)=ξ(x,t)~{}in~{}ΩT, (2.8) ∂ˆu(x,t)∂→n+σ(x,t)ˆu(x,t)=0~{}on~{}S, ˆu(x,0)=0~{}in~{}Ω.

Furthermore, there holds the estimate

 ∥ˆu∥W(0,T)≤CR∥ξ∥L2(ΩT).
###### Proof.

We have for a.e. and along with (2.5) also get for all and a.e. that

 (2.9) ⟨∂ˆu(ξ)∂t,φ⟩(H1(Ω)∗,H1(Ω))+a(ˆu(ξ),φ)=(ξ,φ)L2(Ω)

and for a.e. . Let satisfy for a.e. . Further, combining (2.9) with (2.5), we arrive at

 ⟨∂θ∂t,φ⟩(H1(Ω)∗,H1(Ω))+a(θ,φ)=0

for all . Taking and using (2.2), we have

 ⟨∂θ∂t,θ⟩(H1(Ω)∗,H1(Ω))=12ddt∥θ(t)∥2L2(Ω)=∥θ(t)∥L2(Ω)ddt∥θ(t)∥L2(Ω)

which together with (2.7) yield . Thus, by the Gronwall’s inequality, we get for a.e. in , which finishes the proof. ∎

Together with the problems (1.1) and (2.8), we consider the problem

 ∂~u∂t(x,t)+L~u(x,t)=0~{}in~{}ΩT, (2.10) ∂~u(x,t)∂→n+σ(x,t)~u(x,t)=g(x,t)~{}on~{}S, ~u(x,0)=q(x)~{}in~{}Ω.

Then we see that

 u(f)=ˆu(f)+~u~{}for all~{}f∈L2(ΩT), (2.11)

where comes from (2.8) by using in the right hand side instead of , which depends linearly on . Further, for each we consider the adjoint problem

 −∂p∂t(x,t)+Lp(x,t)=0~{}% in~{}ΩT, (2.12) ∂p(x,t)∂→n+σ(x,t)p(x,t)=(u(x,t;f)−zδ(x,t))χΣ~{}on~{}S, p(x,T)=0~{}in~{}Ω,

where is the characteristic function of , i.e. if and equals to zero otherwise. A function is said to be a weak solution to this problem, if for a.e.  and

 −⟨∂p∂t,φ⟩(H1(Ω)∗,H1(Ω))+a(p,φ)=∫Γ(u(x,t;f)−zδ(x,t))φ(x)dx∀φ∈H1(Ω), % a.e. t∈(0,T]. (2.13)

Since , the boundary value belongs to and by changing the time direction we see that (2.12) attains a unique weak solution .

###### Lemma 2.4.

Let us denote by

 J0(f):=∥u(f)−zδ∥2L2(Σ)

with . Then the Fréchet derivative of is given by

 ∇J0(f)=2p(f). (2.14)
###### Proof.

In view of Lemma 2.3, for each the action of the Fréchet derivative in the direction is given by

 12J′0(f)ξ =(u(f)−zδ,u′(f)ξ)L2(Σ)=(u(f)−zδ,ˆu(ξ))L2(Σ) =∫T0⟨−∂p(f)∂t,ˆu(ξ)⟩(H1(Ω)∗,H1(Ω))dt+∫T0a(p(f),ˆu(ξ))dt =∫T0⟨∂ˆu(ξ)∂t,p(f)⟩(H1(Ω)∗,H1(Ω))dt+∫Ωp(x,0;f)ˆu(x,0;ξ)dx −∫Ωp(x,T;f)ˆu(x,T;ξ)dx+∫T0a(ˆu(ξ),p(f))dt =∫T0⟨∂ˆu(ξ)∂t,p(f)⟩(H1(Ω)∗,H1(Ω))dt+∫T0a(ˆu(ξ),p(f))dt =∫ΩTξp(f)dxdt,

here we used (2.1). ∎

Before going farther we state the following result.

###### Lemma 2.5.

Assume that the sequence weakly converges in to an element . Then the sequence weakly converges in (and strongly in ) to .

###### Proof.

Since the sequence is weakly convergent, it is a bounded sequence in the -norm. Due to (2.6), the sequence is bounded in the reflexive space . Hence, there exists a subsequence of it denoted the same symbol such that weakly converges to an element in . For all and we have that

 ∫T0⟨∂u(fk)∂t,ϕ⟩(H1(Ω)∗,H1(Ω))dt+∫T0a(u(fk),ϕ)dt=∫T0(fk,ϕ)L2(Ω)dt+∫T0(g,ϕ)L2(∂Ω)dt. (2.15)

Sending to , we thus obtain that

 ⟨∂u∂t,φ⟩(H1(Ω)∗,H1(Ω))+a(u,φ)=(f,φ)L2(Ω)+(g,φ)L2(∂Ω) (2.16)

for all and a.e. . We show that for a.e. . In fact, let be arbitrary with for a.e. . By (2.1), we have from (2.15) for all that

 +∫T0a(u(fk),Φ)dt =∫T0(fk,Φ)L2(Ω)dt+∫T0(g,Φ)L2(∂Ω)dt+∫Ωu(x,0;fk)Φ(x,0)dx.

Noting for a.e. , sending to in the last equation, we get

 −∫T0(∂Φ∂t,u)L2(Ω)dt +∫T0a(u,Φ)dt =∫T0(f,Φ)L2(Ω)dt+∫T0(g,Φ)L2(∂Ω)dt+∫Ωq(x)Φ(x,0)dx. (2.17)

Likewise, using (2.1) and (2.16), we deduce

 −∫T0(∂Φ∂t,u)L2(Ω)dt +∫T0a(u,Φ)dt =∫T0(f,Φ)L2(Ω)dt+∫T0(g,Φ)L2(∂Ω)dt+∫Ωu(x,0)Φ(x,0)dx. (2.18)

We thus obtain from (2.2)–(2.2) that , where is arbitrary. This results that for a.e. and so . Since is unique, we get that the whole sequence weakly converges in to .

Further, since the trace operator is compact (cf. [28, pp. 31]), so is also compact (cf. [48, pp. 18]). We therefore conclude that strongly converges to in , that finishes the proof. ∎

Now, we are in a position to prove the main result of this section.

###### Theorem 2.6.

The minimization problem attains a unique minimizer which satisfies the equation

 fρ,δ=f∗−1ρp(fρ,δ