We consider an illposed initial boundary value problem for the Helmholtz equation. This problem is reduced to the inverse continuation problem for the Helmholtz equation. We prove the wellposedness of the direct problem and obtain a stability estimate of its solution. We solve numerically the inverse problem using the Tikhonov regularization, Godunov approach, and the Landweber iteration. Comparative analysis of these methods is presented.
Let us consider the initial boundary value problem (continuation problem) for the Helmholtz equation in the domain
The continuation problem is illposed problem; its solution is unique, but it does not depend continuously on the Cauchy data [
We consider two approaches to the numerical solution of the problem (
Let us consider the direct (wellposed) problem of finding the function
A function
If
Let us introduce the auxiliary problem
Due to (
Thus, we have proved the wellposedness of the direct problem, which allows us to apply wellelaborated computational methods. Also, a stability estimate has been obtained in
Let us reduce the inverse problem (
Let us describe the iterative algorithm. First we choose the initial approximation
Solve the direct problem (
Calculate
Check the stopping criterion
Solve the adjoint problem
Calculate the gradient
Calculate the next approximation
The direct and adjoint problems are solved using the direct finitedifference method. For discretizing the direct problem, we construct a grid in
Thus, we obtain the system of algebraic equations
Similarly, the discrete adjoint problem (
Let
The computational experiment was carried out for different noise levels. Tables
Calculation results in the case of no noise
Number  The number of iterations 


Runtime 

1  10  0.8243  0.094  4 sec 
2  100  0.0632 

40 sec 
3  1000  0.0414 

6 min 40 sec 
4  5000  0.0311 

33 min 30 sec 
5  7318  0.0304 

49 min 
Calculation results in the case of noise within 1%
Number  The number of iterations 


Runtime 

1  10  0.83  0.09  4 sec 
2  100  0.077 

40 sec 
3  1000  0.047 

6 min 33 sec 
4  1508  0.051 

10 min 
Calculation results in the case of noise within 5%
Number  The number of iterations 


Runtime 

1  10  0.86 

4 sec 
2  100  0.165 

40 sec 
3  1000  0.210 

6 min 32 sec 
4  2000  0.297 

13 min 4 sec 
5  2418  0.326 

15 min 48 sec 
The approximate solution in the case of
We observe that in the case of no noise the functional
In this section we consider a discrete analog of problem (
We reduce the continuation problem (
Assuming
The matrix
The characteristics of the matrices
Matrix 





321098.0 


321098.0 
In view of the illconditioning of the matrix
The Tikhonov regularization consists of replacing the system
As above, we put
S. K. Godunov proposed considering the extended system
We choose
For
Singular values of matrix
Singular values of matrix
Tables
Comparative analysis of methods in the case of no noise.
Number  Name methods 

Runtime 

1  Landweber iteration  0.030  49 min 
2  Tikhonov regularization  0.021  23 sec 
3  Godunov regularization  0.019  46 sec 
Comparative analysis of methods in the case of noise within 1%.
Number  Name methods 

Runtime 

1  Landweber iteration  0.051  10 min 
2  Tikhonov regularization  0.190  15 sec 
3  Godunov regularization  0.055  17 sec 
Comparative analysis of methods in the case of noise within 5%.
Number  Name methods 

Runtime 

1  Landweber iteration  0.5349  18 min 24 sec 
2  Tikhonov regularization  0.4088  7 sec 
3  Godunov regularization  0.2989  12 sec 
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work was partially supported by the Ministry of Education and Science of the Russian Federation, joint Project SB RAS and NAS of Ukraine, 2013, no. 12, and RFBR Grant 140100208.