A. Lapin and S. Lapin
Lobachevskii Journal of Mathematics Vol. 14, 2004, 69–84
©Alexander Lapin and Serguei Lapin
2000 Mathematical Subject Classification. 65M32, 65N21.
Key words and phrases. nonlinear coefficient identification, transport equation, final observation, finite difference scheme, multilevel algorithm.
This work was supported by RFBR, grant 01-01-00070.
ABSTRACT. Considered a problem of identification a nonlinear coefficient in a first order PDE via final observation. The problem is stated as an optimal control problem and solved numerically. Implicit finite difference scheme is used for the approximation of the state equation. A space of control variables is approximated by a sequence of finite-dimensional spaces with increaing dimensions. Finite dimensional problems are solved by a gradient method and numerical results are presented.
In this paper we consider the following nonlinear initial boundary-value problem
which models a convective transport of sorption chemical through a porous medium. Here is the dissolved concentration of a chemical and is a so-called sorption isotherm. Function is the unknown of the problem, so we consider a structure identification problem. For physical reasons we assume to be continuous and non-decreasing and . Under these assumptions there exists a unique solution to the problem (1) which takes its values from segment . In order, to define on , we use a final observation : we try to choose in such a way that and where is a non-negative continuous function. It is obvious that the formulated inverse problem is ill-posed, because it is not solvable for the arbitrary function
We set up the above problem as an optimal control problem . This approach is well-known in parameter identification problems (e.g. , ) and it is often called the output least squares method (cf. , ). In this paper we will concentrate on the numerical solution to the problem rather than its theoretical study.
In order to solve the optimal control problem we approximate the set of admissible coefficients by a finite dimensional set. We also approximate problem (1) by a finite difference scheme. The existence of an unique solution to the finite-dimensional optimal control problem is shown. We use a gradient-type method for its solution, where the gradient information is calculated via the solution of an adjoint state problem. Due to the highly ill-posedness of the problem we chose an approach which is characterized by increasing the dimension of the set for admissible coefficient (cf.). The results of the numerical experiments are presented.
We consider problem (1) with non-linear ”coefficient” which belongs to the following subset of the Lipshitz-continuous on functions:
Let us define the cost functional for the control optimization problem by
The problem under consideration can be posed as the following optimal control problem:
find such that it minimizes when satisfies the initial boundary-value problem (1).
Now in order to solve and stabilize the above stated optimal control problem we approximate the set by a finite-dimensional set . is constructed by discretization of the coefficient space. Namely, let where the functions compose a basis of a finite-dimensional space , containing , while belongs to a set of admissible parameters: After an approximation of set we derive the problem of minimization to the functional
with satisfying (1) and
For the differentiable function the functional is also differentiable and
Here is the solution of the state problem (1) and is the solution of the corresponding adjoint problem:
We can rewrite (6) in pointwise form:
In order to solve numerically the optimal control problem we discretize the state problem (1) and construct corresponding discrete adjoint state problem. Let us divide the segment by the uniform mesh with stepsize , , and define finite-dimensional spaces
Now we consider set which is the basis of . This set consists of the piecewise polynomial functions which are defined the following way:
Use of the basis (7) instead of usual Courant basis allows us to obtain the simplest form of the set via the nodal parameters of the functions :
Now, let be an uniform mesh on with mesh step-size and . We also introduce being uniform mesh on with step-size ,
We define the finite differences in time as:
and in space as:
State equation (1) is approximated by the following implicit scheme:
where Applying the quadrature formula to (4), we derive the following finite-dimensional cost functional:
Futher we consider the optimal control problem (OCP):
Lemma 1. Discretization scheme (8) has a unique solution for any and for all .
Proof. For a fixed time level problem (8) can be written as
Because , we solve recurrently the equations with monotone increasing and continuous functions to find for all , whence the unique solvability follows. Further, owing to the non-negativeness of the initial and boundary conditions we obviously have . Now, let and for . If we suppose that , then from (8) it follows , thus, and we get a contradiction. □
In addition, we prove that is a Lipschitz-continuous function of . To do this we rewrite equation (8) for a fixed time level in an algebraic form. Let be the vector of nodal values to for , while be the vector with coordinates for and Let further be the two-diagonal matrix with diagonal elements and off-diagonal ones . Then (8) for a fixed time level has the form
Lemma 2. Let and be the solutions of (11), corresponding to Then there exists a constant such that
where and are maximum and -norm of vectors, respectively.
Proof. Let be unit matrix and . From equation (11) we obtain
and estimate the difference
Let be the diagonal matrix with entries, defined the following way: if then the -th diagonal entry of is ; otherwise it is equal to . With this definition one has
Owing to the definition of the functions
Direct calculations show that
Because and similarly to the previous estimates we have
whence inequality (12) follows with . □
Theorem 1. Problem (10) has a solution for any
Proof. Owing to the previous Lemma 2 and definition of , the cost function is continuous, and obviously it is coercive: , whence the result. □
A function is not differentiable in . However, it is Lipcshitz-continuous and has left and right derivatives . Below we use the piecewise constant function (by taking either left or right derivative) to construct adojnt state and to receive ”gradient” information. To construct the adjoint state problem we define the following Lagrange function
where the mesh function vanishes when or Stationary points of Lagrange function are defined from the following system
where and trial mesh functions and vanish when or .
Equation (20) gives us the adjoint state problem which has the following pointwise form
From the equation (21) we derive the formula for calculation the gradient
Now to minimize the functional we use the gradient method:
where an initial guess , and iterative parameter is defined via the line search technique. To improve the convergence of the method we use “multilevel” implementation; namely, we first solve the problem with the dimension of the space and then increase . This approach gives us the possibility to achieve better results than if using fixed dimension .
In this section, we will describe numerical experiments that we have performed for the solution of the above defined optimal control problem.
We discretize domain by uniform mesh with step size and time step . Then, we define discrete control space with the dimension . For the numerical experiments we have used the following objective functions :
We have performed calculations for variable , where on each step after increasing we used the previous solution as an initial data.
Figures 1 - 6 show evaluated coefficient and comparison between objective function and computed values of for three considered functions and for different mesh and time step sizes.
Remark 1. We also performed calculations ”directly” for without multilevel approach. The calculated results were similar to the previous ones, but the number of iterations and especially the time of calculation were much greater than for the multilevel method.
 Knabner P., Igler B. Structural Identification of Nonlinear Coefficient Functions in Transport Processes through porous media, in Lectures on Applied Mathematics, Springer Verlag, 2000, pp.157-178
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