Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation. (English) Zbl 1185.65172
The authors consider the two regularization methods for ill-posed 2D problem named in the title and prove \(L_2\) convergence to the solution assuming apriorily boundedness in \(L_2\) of the solution or of one of its derivatives (in the marching direction of the Cauchy problem). For these results to hold, they are able to specify the regularization parameters (parameter multiplying the fourth derivative, resp.number of terms in the Fourier series) in dependence on the data (solution bound and measuring error).
In their numerical experiments they compare the two methods and a third one, the method of lines, and conclude that the truncation method performs better.
In their numerical experiments they compare the two methods and a third one, the method of lines, and conclude that the truncation method performs better.
Reviewer: Gisbert Stoyan (Budapest)
MSC:
| 65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35R25 | Ill-posed problems for PDEs |
Keywords:
Cauchy problem; Helmholtz equation; quasi-reversibility method; truncation method; convergence; regularization methods; ill-posed problem; numerical experimentsReferences:
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