Conditional Well-Posedness for an Inverse Source Problem in the Diffusion Equation Using the Variational Adjoint Method

This article deals with an inverse problem of determining a linear source term in the multidimensional diffusion equation using the variational adjoint method. A variational identity connecting the known data with the unknown is established based on an adjoint problem, and a conditional uniqueness for the inverse source problem is proved by the approximate controllability to the adjoint problem under the condition that the unknowns can keep orders locally. Furthermore, a bilinear form is set forth also based on the variational identity and then a norm for the unknowns is well-defined by which a conditional Lipschitz stability is established.


Introduction
Air pollution and fog-haze problems have attracted much attention during the last decade in North China.There are some important aspects in the research of air pollution, such as the component and property of the air-pollutant, the accumulating process, and the migrating rule and the diffusion model.It is an effective method to analyze the transport and diffusion behaviors utilizing suitable advectiondiffusion equations (cf.[1][2][3], for instance).However, for reallife diffusion problems, some physical parameters are always unknown, or can not be measured directly, or can be obtained but expanding much cost, such as the diffusion coefficient, the initial distribution of the pollutant, and the physical/chemical source term.Therefore, it becomes feasible and necessary to put forward inverse problems using some additional data to determine those parameters with less cost in mathematics.An inverse problem arising in a diffusion equation belongs to inverse problems of parabolic type of partial differential equations.
By the literatures we have, there are quite a few of studies on inverse problems for parabolic equations since 1970s.As for general researches and summary, we refer to the monograph [4], and, for the research methods, we refer to [5][6][7][8][9][10][11][12][13][14][15] for the fixed point method based on the solution's expression of the forward problem, refer to [16][17][18][19] for the orthogonality method and energy estimates method, refer to [20][21][22][23] for utilizing the maximum principle, and refer to [24][25][26][27][28] for Carleman-type estimates method, and so forth.It is noted that stability analysis is still a trouble for inverse problems, especially for the construction of Lipschitz stability.By using classical estimates for parabolic problems in Hölder spaces, Hölder stability can be obtained based on the maximum principles and Sobolev embedding theorems; see also [4,21,22] and so forth.However, such method always needs more conditions for the initial boundary value functions, and it always involves complicated integral estimates.On the other hand, the variational identity method, also known as monotonicity method or the adjoint method, see, for example, [29][30][31][32][33][34][35], has been applied to parameter identification problems in the parabolic equations, by which uniqueness results can be proved using approximate controllability for the adjoint problems based on integral identities.The author discussed some inverse source problems for parabolic equations in 1D case (see [36][37][38] for instance) and gave conditional stability estimates for determining the source term or source coefficient using the variational identity method.Recently, an inverse problem of determining the first-order coefficient in an advection-dispersion equation in 2D/3D case using the final observations was discussed also using the variational adjoint method [39], but the research domain is limited to be regular and the unknown should keep a form of variables separable, and the solution to the adjoint problem should have an explicit expression in order to prove the Lipschitz stability of the inverse problem.
In this paper we continue to deal with the inverse source problem for the multidimensional diffusion equation in a bounded open domain in R  ( ≥ 1).Here the inverse source problem is to determine the space-dependent source coefficient using the flux at partial boundary.A variational identity connecting the known data with the unknown source coefficient is established based on a suitable adjoint problem, and the uniqueness of the inverse problem is proved by the approximate controllability to the adjoint problem under the condition that the unknowns can keep orders locally.Furthermore, a bilinear form is set forth by the variational identity and then a norm for the unknowns is well-defined by which a conditional Lipschitz stability is constructed.It is noted that the maximum-minimum principle for parabolic equations is employed to keep sign of the solution to the adjoint problem instead of utilizing the explicit expression of the solution to the adjoint problem in the proof of the stability.Such kind of research approach can give more wide applications of the variational adjoint method to study well-posedness of inverse problems arising in partial differential equations.
Consider the multidimensional diffusion equation: where  = (, ) denotes the concentration of the pollutant at the space point  and the time , Δ is the Laplace operator,  > 0 is the diffusion coefficient, and (, ) is a linear source term.Generally speaking, the source term has a variables separable form: (, ) = ()(), where () > 0 is a timedependent attenuation factor and () is a space-dependent source magnitude.The initial boundary value conditions are given as follows: If the model parameters and the initial boundary value functions in the forward problem (1)-( 2) are all known and satisfy suitable consistency conditions, then it is a well-posed deterministic problem by the theory of parabolic equations, and there exists a unique solution on Ω  .Here we omit some related assertions on the forward problem (1)-( 2) and focus on the inverse source problem given in the following.

The Inverse Source Problem.
Suppose that there occurs some polluting phenomenon in a bounded region, and the pollutant is mainly produced by some continuous source distributed in the region.If the source term can not be measured directly, an inverse problem is encountered.Let the timedependent factor () in the source term (, ) = ()() be known; we are to determine the space-dependent source magnitude function () using the additional flux data measured at the partial boundary of Γ 1 ⊂ Γ.That is, we have the additional condition: where ] denotes the normal outward vector at the boundary Γ 1 .Thus an inverse source problem is formulated by the problem ( 1)-( 2) together with the overposed information (3). Denote and let   = { ∈  2 (Ω) : ‖‖ 2 ≤ } be an admissible set for the unknowns, where  > 0 is a positive constant.For any given  ∈   , there is a unique solution to the forward problem, denoted by ()(, ).Together with the additional information, we define a mapping  :  2 (Ω) →  2 (Γ  1 ): Then the inverse problem ( 1)-( 3) can be transformed to the problem of solving the operator equation ( 4).From the viewpoint of numerics, solving the inverse problem is reduced to solve the minimization problem with regularization strategy min where  > 0 is the regularization parameter.This paper is devoted to the well-posedness of the inverse source problem; numerical inversions will be presented in another occasion.Therefore, the following lemma on the approximate controllability to the heat equation is useful.
Lemma 1 (see [40]).Let Ω ⊂ R  be a bounded open subset with piecewise-smooth boundary Ω having finite  − 1dimensional volume,  be the set of all admissible boundary functions  : Ω × (0, ) → R, and A be any nonempty set of Ω × (0, ).For  ∈ , let (⋅) = (⋅; ) : (0, ) →  2 (Ω) be the solution to the following initial boundary value problem: Then for any given  ∈  2 (Ω) and  > 0, there exists a function  * , continuous in A, such that The assertion of this lemma is also valid to general parabolic equations, and it reveals that the final value of a homogeneous linear parabolic model can be approximately controlled by its boundary value.

Uniqueness of the Inverse Source Problem
3.1.The Identity.We give a variational identity connecting the known data with the unknown source function based on a suitable adjoint problem.
Proof.Denote  =  1 −  2 ; we have and the additional information By smooth test function  = (, ), multiplying the two sides of the first equation in (10), and integrating on Ω  , there holds Noting the homogeneous boundary conditions in (10), we have by integration by parts for the left-hand side of ( 12) Let  = (, ) be the solution to the adjoint problem (9); then it follows that (8) is valid by (13) together with the condition (11).Since the adjoint problem ( 9) is uniquely determined by its boundary input (, ), we denote its solution as  = ()(, ).The proof is over.

Conditional Uniqueness.
Uniqueness is an important aspect in the research of inverse problems.With the help of the variational identity (8) and Lemma 1, we will prove a conditional uniqueness for the inverse source problem given above.The required condition is that the unknown function should keep its sign in a small region of the considered domain.
Theorem 3.Under the conditions of Theorem 2, also assume that () ∈ ([0, ]) and ( 1 , then there is no positive measurable subdomain of Ω where  1 −  2 is of one sign. Proof.By the assumption of ℎ 1 = ℎ 2 on Γ  1 , we have by identity ( 8) Suppose that there is a subdomain Ω 0 ⊂ Ω with positive measure such that  1 −  2 keeps one sign.Let ( 1 −  2 )| Ω 0 ≥ 0 for convenience.We will deduce a contradiction with ( 14) in the following.Denote Take a partition of the time interval [0, ]: and denote Δ =   −  −1 ,  = 1, 2, . . ., ; we have where  1 (Δ) 2 denotes the truncated error and  1 > 0 is a positive constant.Noting the condition ()| = = 0, here the term of  =  is omitted.By the assumption, there exists a nonnegative integrable function () on Ω given by where () is a smooth positive function on Ω 0 .Thus we get

Advances in Mathematical Physics
In addition, we a series of integrable functions on Ω by the assumption on (): Noting that the adjoint problem ( 9) is a well-posed backward problem and the initial state is zero at  = , we deduce that its solution ()(, ) at  =  −1 can be approximated to any objective function  −1 () ∈  2 (Ω) by Lemma 1 by suitably choosing the boundary input (, ).Subsequently, there exists a boundary input on each subinterval [ −1 ,   ] such that for any  > 0 we have Based on ( 17) and ( 19), utilizing the basic inequality and noting Thanks to () = (  )  (), there holds the estimate for the center term of the right-hand side in ( 22) by ( 21) and Cauchy-Schwartz inequality Henceforth, we have It follows that there must be || > 0 as long as  and |Δ| are small enough, which is a contradiction with (14).The proof is completed.

Conditional Lipschitz Stability
Firstly we give a basic assertion.
With a complete method as that used in Theorem 2, we can prove the following assertion.
For construction of stability for the inverse source problem, we need extra conditions for the known and unknown data functions.For the unknown source coefficient function  ∈   , we further assume that () ≥ 0 and  ̸ ≡ 0 for  ∈ Ω.In addition we assume that the attenuation factor () is continuous and positive on [0, ] and ‖‖ ∞ ≥  0 > 0; here  0 is a positive constant.Now, based on the variational identity (8), we define a functional on the source function () and the controllable input (, ): where ()(, ) is the solution to the adjoint problem (9) which only depends upon the boundary input  = (, ) for (, ) ∈ Γ  1 .
It is not difficult to show that bilinear functional B(, ) is bounded on  and  in  2 norm.Then we define a norm on  which is called B-norm given by         B fl sup We need to prove that the norm given by ( 27) is welldefined.Obviously, B(, ) is nonnegative and satisfies the absolute homogeneity and triangle inequality due to the linearity and additivity of the integration.We now prove that if ‖‖ B = 0, then there must be  = 0 a.e. ∈ Ω in the case of  ≥ 0 and  ̸ ≡ 0. In fact, assume that ‖‖ B = 0; that is, there holds By the assumptions that  ≥ 0 for (, ) ∈ Γ  we deduce that the solution  = () to the adjoint problem ( 9) takes nonnegative values by the maximum-minimum principle for the parabolic equations (cf.[41], for instance).Together with () ≥ 0 and () ≥  0 > 0 it follows that the integrated function in (28) takes nonnegative values; that is, we get Noting the assumption on the attenuation factor (), there must be  = 0 a.e. ∈ Ω or () = 0 a.e.(, ) ∈ Ω  by Lemma 4. Since the boundary input satisfies the condition  ≥ 0 and  ̸ ≡ 0, there holds () ≥ 0 a.e.(, ) ∈ Ω  also by the maximum-minimum principle.So there must be  = 0 a.e. ∈ Ω.Thus the norm given by ( 27) is well-defined.Theorem 6.Under the conditions of Theorem 5, let  ∈   and take nonnegative values on  ∈ Ω, and let () be continuous and positive on [0, ] and ‖‖ ∞ ≥  0 > 0. Then there exists a positive constant  = (Ω  , , (), ) such that where the norm ‖ ⋅ ‖ B is defined by (27).
Proof.By the variational identity (25) and the norm definition (27), utilizing Cauchy-Schwartz inequality, we have + Noting (31) and the condition ‖‖ ∞ ≥  0 > 0, by setting it follows that the assertion of this theorem is valid.

Conclusions
An inverse problem of determining a space-dependent source coefficient in the multidimensional diffusion equation using the variational adjoint method is investigated.A variational identity connecting the known data with the unknown is established based on an adjoint problem, and uniqueness for the inverse source problem is proved by the approximate controllability to the adjoint problem under suitable conditions.This is the conditional uniqueness.Also, based on the variational identity, a bilinear form is set forth by which a norm for the unknown source is well-defined, and then a Lipschitz stability can be proved if the unknown source is of one sign.Such method can be generalized to study the Lipschitz stability for other kinds of inverse coefficient problems arising in partial differential equations.