Comparative Analysis of Methods for Regularizing an Initial Boundary Value Problem for the Helmholtz Equation

1 Institute of Computational Mathematics and Mathematical Geophysics, Akademika Lavrentjeva, No. 6, Novosibirsk 630090, Russia 2 Novosibirsk State University, Pirogova Street 2, Novosibirsk 630090, Russia 3 Sobolev Institute of Mathematics, 4 Academy Koptyug Avenue, Novosibirsk 630090, Russia 4National Open Research Laboratory of Information and Space Technologies, Kazakh National Technical University after K.I. Satpaev, Seifullin Street 122/22, Almaty 050013, Kazakhstan 5 Kazakh National Pedagogical University Abai, 13 Dostyk Avenue, Almaty 050010, Kazakhstan

The continuation problem is ill-posed problem; its solution is unique, but it does not depend continuously on the Cauchy data [1][2][3][4][5][6][7][8][9][10].Note that the problem was studied by many authors.For example, Tuan and Quan [11] considered the case 0 <  < 1 and proposed a regularization technique which allows one to obtain a stable solution in a two-dimensional domain.Regińska and Regiński [12] showed that if  satisfies a certain condition, then the Cauchy problem for the Helmholtz equation has a stable solution in a three-dimensional domain.Isakov and Kindermann [13] used the singular value decomposition to prove that in a simple domain the considered problem becomes more stable with increasing .The same result was obtained numerically for the general case.The uniqueness of the solution of the investigated problem was proved, for example, by Arendt and Regińska [14], where the concept of weak normal derivative was introduced in formulating the problem.In [15,16] singular values of the continuation problem were obtained for the two-dimensional Helmholtz equation with complex wave number for simple geometry.
We consider two approaches to the numerical solution of the problem (1).The first consists of formulating problem (1) in an operator form A =  and minimizing the coast functional () = ⟨A − , A − ⟩ by the Landweber iteration [7].In the second approach, problem (1) is reduced to the system of linear algebraic equations which is solved and note that the continuation problem (1) can be reduced to the inverse problem of finding function () from ( 2)-( 5) using the additional information Let us consider some theoretical results [7,17].
Let us describe the iterative algorithm.First we choose the initial approximation  0 and we suppose that we calculated successively   by formula (28).Assuming that we have found   , we show below how to calculate  +1 .
Similarly, the discrete adjoint problem (29) has the form As above, this problem can be reformulated in a matrix form where is an unknown vector and   is the data vector (boundary and additional conditions of the adjoint problem).

Results of the Numerical Experiment.
Let  = 1,   =   = 20.We choose the parameter  = 0.9.In order to test the algorithm, we assume that the exact solution has the form () = 1 − cos(2) and calculate the corresponding additional information .Then, let  0 () = 0.1 be the initial approximation; we try to restore the original exact solution using the Landweber iteration with  = 0.01.If the data are given with an error , we choose the following one as a stopping criterion: (  ) ≤  2 [21].The computational experiment was carried out for different noise levels.Tables 1, 2, and 3 show the calculation results obtained using PC Intel(R)Core(TM) i7 processor with a frequency 3.9 GHz.
The approximate solution in the case of  = 0.05 is shown in Figure 1.
We observe that in the case of no noise the functional () decreases monotonically, while in the other cases the decrease stops after 100 iterations.This phenomenon can be explained by the error that arises in solving the direct and adjoint problems.Note that in the case of noise, the stopping criterion does not guarantee the minimal error in the solution of the inverse problem.However, the criterion ensures that the error is of the same order as that of the minimal one, since further calculation leads to an increase in the error.
The matrix  is ill-conditioned [23] (see decreasing of its singular values in Figure 4).The condition number and matrix norma of the discrete direct problem are presented in  5. We see that the direct problem is wellposed.
In view of the ill-conditioning of the matrix , that is, the ill-posedness of the original problem, we will use regularization methods.
As above, we put   =   = 20,  = 0.9, and () = 1 − cos(2) and calculate with  = 0,  = 0.01, and  = 0.05.The approximate solution in the case of  = 0.05 is shown in Figure 2. whereas  contains some a priori information concerning the inverse problem solution.We take as a priori information the existence of the second derivative of the solution [25].We choose  to minimize ‖   ‖ − ‖  −1 ‖.

3. 1 .
Formulation in the Operator Form and Description of the Algorithm.Let us reduce the inverse problem (2)-(6) to the operator equation.Let us consider the operator A such that

Figure 1 :
Figure 1:  indicates exact solution;  indicates solution obtained by the Landweber iteration.

Table 1 :
Calculation results in the case of no noise  = 0.

Table 2 :
Calculation results in the case of noise within 1%  = 0.01.

Table 3 :
Calculation results in the case of noise within 5%  = 0.05.

Table 4 :
The characteristics of the matrices  and   .

Table 4 and
Figure