1 Introduction
Mathematical modeling of many applied problems of science and engineering leads to numerical solving inverse problems for equations with partial derivatives Alifanov (2011); Lavrent’ev et al. (1986). In the theoretical study of such problems, the main attention is given to issues of wellposedness of problems, the uniqueness of the solution and its stability.
For parabolic equations, inverse coefficient problems attract particular interest. In these problems, identification of coefficients of equations and/or their righthand side is conducted using some additional information about the solution. It is possible to identify dependence of coefficients on time or on spatial variables Isakov (2017); Prilepko et al. (2000). Problems of identifying the righthand side of the equation belong to the class of linear inverse problems. Other inverse coefficient problems are nonlinear that complicate significantly their study.
Among inverse problems of coefficient identification for parabolic equations we can highlight problems of determining the dependence of the lowest coefficient (reaction coefficient) on spatial variables. As a rule, additional conditions are formulated as the solution value at the final time moment and so, in this case, we speak of the final redefinition. In a more general case, a redefinition condition is formulated as some integral timeaverage relation (integral redefinition). The existence and uniqueness of the solution to such an inverse problem and the wellposedness of this problem are considered in a number of works. The pioneer works Isakov (1991); Prilepko and Solov’ev (1987) are devoted to problems with the final redefinition in Hölder classes and are based on the Schauder principle. Later works Kamynin (2013); Kamynin and Kostin (2010); Prilepko and Kostin (1992, 1993) deal with problems with integral redefinition and so, they are studied in Sobolev classes.
In works Kostin (2015); Prilepko and Kostin (1992) (see also (Isakov, 2017, Theorem 9.1.4)), the existence of the solution to the inverse problem of finding the lowest coefficient of a parabolic equation is proved constructively. Namely, an iterative process is used with solving the standard initialboundary parabolic problem at each iteration. It seems natural to implement this approach in a corresponding computational algorithm.
The standard approach to numerical solving inverse coefficient problems for partial differential equations is associated with the minimization of the residual functional using regularization procedures Samarskii and Vabishchevich (2007); Vogel (2002). Computational algorithms are based on the employment of gradient iterative methods, where we solve initialboundary value problems both for the initial parabolic equation and the equation that is conjugate to it. For problems of identifying the lowest coefficient of parabolic equations, which depends only on spatial variables, the optimization method in a combination with finite element approximations in space is used in the work Chen and Liu (2006). Among later studies in this direction, we mention Cao and Lesnic (2018); Yang et al. (2008).
In the work Liu and Chang (2011), an iterative process for the identification of the reaction coefficient in the diffusionreaction equation is proposed without any exact mathematical justification. For model one and twodimensional boundary value problems, using finitedifference approximations in space, the efficiency of this computational algorithm has been demonstrated. This approach has been also applied to some other inverse problems for parabolic equations, in particular, for identifying the highest coefficient Liu (2008).
In the present paper, we construct a computational algorithm for identifying the lowest coefficient with the final redefinition, which is based on an iterative adjustment of the reaction coefficient similarly to Isakov (2017); Kostin (2015). The main attention is paid to obtaining new conditions for the monotonicity of the iterative process for finding the lower coefficient of the parabolic equation, when the coefficient approaches from above. This study continues the work Vabishchevich (2017), where we consider iterative methods for the approximate solving the linear inverse problem of identifying the righthand side for the parabolic equation.
The paper is organized as follows. Statements of direct and inverse problems for the secondorder parabolic equation are given in Section 2. The identification of the reaction coefficient that is independent of time is considered for the twodimensional diffusionreaction equation. An additional information on the solution of the equation is given at the final time moment. An iterative adjustment algorithm for the desired coefficient is investigated in Section 3. The proof of its monotonicity is based on the fulfillment of the maximum principle not only for the solution, but also for derivative of the solution with respect to time. In Section 4, we construct a computational algorithm for approximate solving the identification problem for the lowest coefficient of the parabolic equation, and a discrete problem is formulated using finiteelement approximations in space and twolevel timestepping schemes. Results of computational experiments for a model boundaryvalue problem are represented in Section 5. The findings of the work are summarized in Section 6.
2 Problem formulation
The inverse problem of identifying the lowest coefficient of a parabolic equation is considered. We confine ourselves to the twodimensional case. Generalization to the 3D case is trivial. Let and be a bounded polygon. The direct problem is formulated as follows. We search , such that it is the solution of the homogeneous parabolic equation of second order:
(1) 
with coefficient . The boundary conditions are also specified:
(2) 
where and is the normal to . The initial conditions are
(3) 
The formulation (1)–(3) presents the direct problem, where the coefficients of the equation as well as the boundary conditions are specified.
Let us consider the inverse problem, where in equation (1), the lowest coefficient that depends on spatial variables only is unknown. An additional condition is often formulated as
(4) 
In this case, we have the case of the final redefinition.
Conditions for the unique solvability of the inverse coefficient problem (1)–(4) and its correctness in various functional classes are established, for example, in the works cited above (see Isakov (1991); Prilepko and Solov’ev (1987)). We focus on using the iterative process to identify the coefficient , which has been employed, in particular, in Isakov (2017); Kostin (2015); Prilepko and Kostin (1992). Let us formulate wider conditions for the monotonicity of the iterative process of defining a new initial approximation, when the desired coefficient approaches from above. In our consideration, we assume that the solution of the problem, the coefficients of the equation, and the boundary conditions are sufficiently smooth, i.e., we have all necessary derivatives with respect to the space variables and time.
On the set of functions satisfying the homogeneous boundary conditions (2), let us define the elliptic operator by the relation
In this case, equations (1), (2) can be written in the compact form:
(5) 
Without loss of generality, we consider the inverse problem (3)–(5) for the definition of the pair under a priori restrictions on the reaction coefficient:
(6) 
If with a constant , it is possible to employ the standard transition to the problem for the function .
3 Iterative process
The inverse problem consists in evaluating the pair of functions from the conditions (3)–(5) under the constraints (6)–(8). The iterative process of identifying the coefficient is implemented as follows. It starts from specifying some initial approximation . With the known , , where is the iteration number, the direct problem is solved:
(9) 
(10) 
A new approximation for the desired coefficient is evaluated from the equation at the final time moment using the redefinition (4):
(11) 
In the works Isakov (2017); Kostin (2015); Prilepko and Kostin (1992), the initial approximation is given in the form
(12) 
In this case, the monotone approach to the required coefficient (, approaching from below) holds, if this monotonicity condition holds for :
(13) 
The condition (13) is strong enough, but it can be removed. To do this, consider the algorithm for monotone approaching the reaction coefficient from above.
For the initialboundary value problem (3), (5), in assumption (7), we have
(14) 
The nonnegativity of the solution follows from the maximum principle and the nonnegativity of the righthand side (). The nonnegativity of the time derivative is established similarly when considering problem for . Differentiation of equation (5) by time gives
For , from equation (5) and the first condition in (7), we get
From the maximum principle for this problem, it follows that .
In view of (14), from equation (5), for , we obtain
(15) 
Thus, the inverse problem (3)–(5) is considered with twoside restrictions (5) and (15) for the lowest coefficient .
Let us consider the iterative process (9)–(11) with the initial approximation
(16) 
To find , we solve the problem
For , similarly to (14), we have
In view of this, from (11), we obtain
By (16), we arrive at
(17) 
Let us formulate a problem for the solution difference between two adjacent iterations:
(18) 
(19) 
From (11), we get
(20) 
Similarly to (14), we prove that
(21) 
for . Considering the problem (18), (19) for , in view of (17), on the basis of the maximum principle, we obtain
(22) 
An analogous property of the monotonicity of the approximate solution also holds for other :
(23) 
The proof is by induction on . For , it is satisfied (see (22)). Let us show that from the fulfillment (23) for some this holds also for . If (23) holds, taking into account the second inequality (21), after differentiating (18) with respect to , we obtain
Under these conditions, directly from (20), it follows that
and from (18), (19), for , we get
For , we obtain
(27) 
The second inequality follows immediately from (15), (16). The first inequality is established for the solution of the problem (24), (25) with using the maximum principle. Further, similarly to (23), on the basis of induction, the property of monotonicity is established for other :
(28) 
The result of our consideration is the following statement on the monotonicity of the iteration process (9)–(11).
Theorem 1
4 Computational implementation
It seems reasonable to recall some general points of numerical solving the inverse coefficient problem (1)–(4) on the basis of the iterative adjustment of the desired reaction coefficient. The monotonicity of the iterative process is established in Theorem 1 using the maximum principle for the solution and its time derivative (14). In constructing discretizations in space and time, we need to preserve this basic property of the differential problem, i.e., an approximate solution of the problem should satisfy the maximum principle.
Special attention should be given to monotone approximations in space (approximations of the diffusionreaction operators) and discretizations in time. The maximum principle is formulated in the most simple way (see, e.g., Samarskii (2001)) for difference schemes on rectangular grids. For steadystate problems, its implementation is associated with a diagonal dominance for the corresponding matrix and nonpositivity of offdiagonal elements. Some possibilities for constructing monotone approximations on general irregular grids (using the finite volume method) and the maximum principle for convectionreaction problems with anisotropic diffusion coefficients are discussed in the work Droniou (2014).
Discretization in time leads to additional restrictions on monotonicity. For example, a typical situation is the case, where the monotonicity of the approximate solution in twolevel schemes is ensured by using a small enough step in time. Unconditionally monotone time approximations for parabolic problems occure (see, for example, Hundsdorfer and Verwer (2003); Samarskii (2001)) when using fully implicit twolevel schemes (backward Euler).
Here, we focus on the application of the finite element method. Monotone approximations in space for linear finite elements can be constructed with restrictions on a computational grid (Delaunaytype mesh, see, e.g., Huang (2011); Letniowski (1992)). Some additional restrictions arise (see, for instance, Brandts et al. (2008); Ciarlet (1973)) from the reaction coefficient. They can be removed using the standard approach based on a correction of the approximations of the coefficient at the time derivative and reaction coefficient employing lumping procedures (see, e.g., Chatzipantelidis et al. (2015); Thomée (2006)).
To solve numerically the problem (1)–(4), we employ finite element approximations in space Brenner and Scott (2008); Thomée (2006). In the Hilbert space , we define the scalar product and norm in the standard way:
We define the bilinear form
Define a subspace of finite elements . Let be triangulation points for the domain . When using Lagrange finite elements of the first order (piecewise linear approximation), we can define pyramid function , where
For , we have
where .
Let us define a uniform grid in time
and denote . Define an approximate solution of the inverse problem (1)–(4) as
For the fully implicit scheme, the solution is evaluated from
(30) 
(31) 
The computational algorithm for solving the problem (30), (31) is based on the iterative method (9)–(11), (16) for identifying the lowest coefficient. The calculation starts from specifying the initial approximation for the desired coefficient:
(32) 
For known , we solve the direct problem for evaluating :
(33) 
(34) 
After this, the reaction coefficient is adjusted:
(35) 
5 Numerical experiments
To illustrate the capabilities of the iterative technique for solving inverse problems of identifying the lowest coefficient of parabolic equations, we present the results of numerical experiments for a test problem. Let us consider model problem (1)–(3), where
The problem is solved in the unit square
The data at the final time moment (see (4)) are obtained from the solution of the direct problem with a given coefficient .
In our case, the coefficient is the piecewise constant (see Fig. 1): inside a circle of radius 0.3 with the center (0.6,0.4), we put ; inside the square with side 0.2 and the center (0.3,0.8), we have ; and otherwise, we put .
The solution of the direct problem with this coefficient at the final time moment (the function ) is used as input data for the inverse problem. In our analysis, we focus on iterative solving the identification problem after finite element discretizations in space. Because of this, we do not discuss the dependence of the accuracy of the numerical solution on approximations in space, it seems appropriate to do in a separate study. The effect of computational errors is studied via calculations on different time grids, when the input data is derived from the solution of the direct problem on more fine time grids and with higherorder approximations in time.
To solve the direct problem, we employ the time step . The division into 50 intervals in each direction is used to construct the uniform spatial grid, the Lagrangian finite elements of first degree are applied. The solution at the finite time moment is shown in Fig. 2.
The inverse problem is solved using the fully implicit scheme (see (33)–(35)). The error of the approximate solution of the identification problem on a separate iteration is evaluated as follows:
The main issue is to evaluate an actual convergence rate for the iterative processes under the consideration. We need to recognize clearly how quickly the accuracy of the approximate solution is stabilized with increasing the iteration number. The obtained error itself depends on a time step, namely, the smaller time step, the higher accuracy of the approximate solution. Influence of the time step of the iterative process (33)–(35) with the initial approximation (32) on accuracy is shown in Fig. 3. We observe a high convergence of the iterative process and the improvement of the accuracy of the approximate solution by reducing the time step. Similar results for the iterative process with the initial approximation (12) are presented in Fig. 4.
The convergence of the approximate solution for the reaction coefficient is shown in Fig. 5 and Fig. 6. For the iterative process with initial approximation (32), we observe a monotone convergence from above (see Fig. 5). The iterative process with the initial approximation (12) is nonmonotone. In particular, for , on a part of the domain , the function is negative (see Fig. 6).
6 Conclusion

A nonlinear inverse problem of identifying the lowest coefficient that depends only on spatial variables is studied for a secondorder parabolic equation. The solution of the parabolic equation at the final time moment is given, i.e., the final redefinition is considered.

An iterative process of identifying an unknown coefficient is conducted by solving the standard initialboundary value problem at each iteration. The main result is in establishing the monotonicity of the iterative process, where the desired lower coefficient approaches from above.

The computational algorithm is based on standard approximations in space by linear finite elements, whereas timestepping is implemented using the fully implicit twolevel schemes.

Possibilities of the proposed algorithms were demonstrated by numerical solving a test twodimensional problem.
Acknowledgements
The work was supported by the megagrant of the Russian Federation Government (# 14.Y26.31.0013) and RFBR (project 170100689).
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